Friday, November 9  (Lecture #20) 

This is eligible for L'H since the top and bottom separately > as x>. So let's try it, differentiating the top and bottom separately. The result is lim_{x>}2x/e^{x} and everyone now realizes that this should itself be analyzed using L'H. This quotient is certainly eligible, since again the top and bottom >. Differentiation of the top and bottom separately gives this as result: lim_{x>}2/e^{x}. And now the top is a constant, so the bottom, which remained unchanged through differentiation and certainly still goes to , tells us that this limit and therefore the original limit is 0.
A less obvious use of L'H
We looked at two functions. I considered their values when x=2. Below
are numerical approximations of their values, and then possible
physical (?) interpretations of these numbers. I severely
underestimated the number for the top row in terms of physical
understanding. I actually don't think I can understand that number
physically! (It is interesting that such numbers, maybe not quite so
large, do occur in cryptographic considerations, so they are not
completely irrelevant to reality.)
Formulas for the functions  Value when x=2  If 1 unit is an inch, then ... 
x^{10,000}  1.995·10^{3010}  More than 3·10^{2960} times the theoretical diameter of the universe! 

e^{.001x}  1.002003001...  A bit more than an inch 
The numbers don't tell the whole story. I wanted to compare the rates of growth of these two functions. The real reason is that there are chemical processes, more typically biochemical processes, that have exponential and polynomial growth rates, and there are real world considerations about which one is faster. If x represents the time scale, then one might consider that x could be in nanoseconds, so large values of x could be more relevant than one might initially suspect.
One way to compare these functions is to consider their ratio and see
what happens when x gets large. So we looked at
lim_{x>}x^{10,000}/e^{.001x}. Of
course people said that the top> and the bottom>
separately. So this is eligible for L'H. The result, after
separate differentiation of the top and bottom is
lim_{x>}[10,000x^{9,999}]/[.001e^{.001x}]. Is
this any improvement? A completely naive person might not see much
improvement, but again this is eligible for L'H since the top
and bottom both>. So the separate differentiations, done
carefully, give
lim_{x>}[10,000·9,999x^{9,998}]/[(.001)^{2}e^{.001x}]. Now I hoped that people would see a pattern.
(L'H)^{10,000}
We can compute
lim_{x>}x^{10,000}/e^{.001x} by
applying L'H ten thousand times in succession. I hope that you can see
the result, almost. After 10,000 separate differentiations of the top and bottom we need to evaluate
lim_{x>}(10,000!)/[(.001)^{10,000}e^{.001x}]
Here by 10,000! I mean the product of the integers 10,000 and 9,999
and 9,998 and 9,997 and ... all the way down to 1. This is called a
factorial and you will see many factorials in Math 152. Actually, for the purposes of this computation I don't care much about specific constants, and really I think of the limit as
lim_{x>}CONSTANT_{1}/[CONSTANT_{2}e^{.001x}]
because then it is clear that the exponential, which is really growing
and growing and growing will make the limit equal to 0.
The final result
So, in fact, eventually e^{.001x} grows faster than
x^{10,000}.
Angels and humans
We could think of functions with exponential growth, so e^{Ax}
with A>0, as angels, and functions with power of x growh,
x^{A} with A>0, as humans. (If A happens to be an integer
we get a monomial, and sums [families?] would be polynomials.) So if
you think about it, eventually any angel is stronger than even the
most powerful human.
Another not so obvious use of L'H
Here was another pair of functions we considered.
Formulas for the functions  Value when x=5  If 1 unit is an inch, then ... 
(ln(x))^{300}  1.005·10^{62}  This is
only about 1.7·10^{44} times the theoretical diameter of the universe! 

x^{1/3}  1.7099...  Almost two inches! 
We use L'H to compare the rates of growth. So look at
lim_{x>}[(ln(x))^{300}]/x^{1/3}. You need to really
believe in (?) logs and powers of x, but, if you do, you should see
that the top and bottom both > separately. So apply L'H by
computing derivatives, and this limit results:
lim_{x>}[300(ln(x))^{299}(1/x)]/[(1/3)x^{2/3}]. This
computation seems more intricate to me than the earlier one. The Chain
Rule is needed. The result is a compound fraction, and here it is
worthwhile to simplify the compound fraction. The (1/x) makes an
additional power of x appear "downstairs" and we get (after we
recognize that 1+(2/3)=1/3 !)
lim_{x>}[300(ln(x))^{299}]/[(1/3)x^{1/3}]. Again L'H
applies (yes, we check the eligibility!) and the result is
lim_{x>}[300·299(ln(x))^{298}](1/x)/[(1/3)^{2}x^{2/3}].
Look! 1/x is on top, which changes the power on the bottom from
x^{2/3} to x^{1/3}. The structure of the computation
should be observed.
(L'H)^{300}
We can compute lim_{x>}[(ln(x))^{300}]/x^{1/3} by using L'H three
hundred times in a row, each time carefully (!) checking (!!) the
resulting quotient for the eligibility (!!!) of the top and bottom. Ha. The result will be
lim_{x>}(300!)/[(1/3)^{300}x^{1/3}]
and this easily has limit equal to 0. Maybe it is amusing to notice we
can get the exact values of certain constants, but for the final
result of this computation, the exact values of the constants don't
matter.
The final result
So, in fact, eventually x^{1/3} grows faster than (ln(x))^{300}.
And now demons ...
Moral (?) lessons concerning this hierarchy 
Antiderivatives
A function F is an antiderivative of f if F´=f. I
think the name itself is almost the definition. This turns out to be a
very useful concept. It has accumulated other names. For example, F is
also called an indefinite integral of f. The word "primitive"
is also used, as in, "F is a primitive of f".
Example I think we should begin with a very simple example. Look at f(x)=x3, quite simple. Then F(x)=(1/2)x^{2}3x is an antiderivative of f. Why it this true? Well, just compute the derivative of F. The 2 in the exponent moves down, and cancels the 2 in the 1/2. And 3x differentiates to 3. So the claim is verified. Actually (and this is useful to note!), you can always verify a claim that F is an antiderivative of f by just differentiating. Some qualitative aspects of even this "simple" example are worth noticing. The line is negative to the left of x=3, so the parabola (which happens to have roots at 0 and 6) must be decreasing to the left of x=3. The line is positive to the right of x=3, and there the parabola is increasing. Certainly, when x=3, the parabola must have a minimum, because to the left its derivative is negative and to the right its derivative is positive. The transition in first derivative sign from  to + indicates a local minimum (First Derivative Test).  
More example and discussion Well, as several (more than several, actually) tried to tell me, there are other antidervatives. For example, (1/2)x^{2}3x+2 is also an antiderivative of f(x)=x3. The graph now to the right I hope shows y=(1/2)x^{2}3x+2. It is the original parabola shoved up. I hope that you notice it has the same qualitative ({inde}creasing) properties in the same intervals as the original parabola. And we could push the parabola down (I think y=(1/2)x^{2}3x1.75 is displayed). Etc. What does this "Etc." mean here?
+C 
Here is a first very brief list of antiderivatives. This information is obtained just be reversing some derivative formulas. Please notice that if you don't believe a given formula, you can always differentiate and check. Also, the "+C" is generally omitted from tables of this type because it takes room and "one" should know it is there.
Function  An antiderivative 

x^{n}, n not equal to 1  (1/[n+1])x 
1/x  ln(x) 
sin(x)  cos(x) 
cos(x)  sin(x) 
e^{x}  e^{x} 
There will be many more entries! 
Notation
There is special notation for antiderivatives. For reasons which are
not clear now but which will be clear (I hope!) in about two
weeks, and which come from hundreds of years ago (really, this
notation is an abbreviation of a Latin word) we would write the
phrase, "F(x) is an antiderivative of f(x)" as f(x) dx=F(x)+C. So, for example,
3e^{x}+x^{5}dx=3e^{x}+(1/6)x^{6}+C.
Example and vocabulary
A function can certainly have many antiderivatives (hey, if it has
one, it must have infinitely many!). So how can we separate the
various antiderivatives? The idea and the vocabulary come from simple
physical considerations, where the derivative is velocity and the
function is position (and x represents time). If the velocity is
prescribed, then we should be able to tell where the object is if the
starting position is given.
So this is the idea of an initial value problem. A differential equation is given for an "unknown" function, and then some value of the unknown function is given at some value of its argument (the initial condition). Let's look at a simple example.
Initial Value Problem
Differential equation
y´=2x^{7}+8x.
Initial
condition y(1)=3.
Since y´=2x^{7}+8x I know (consult the table above) that
y must be (2/7)x^{8}+4x^{2}+C. I put in the
"+C" so that all possibilities for y are covered. Now plug in x=1 and
set this equal to the given value of y. We get: (2/7)+4+C=3, so
C=34(2/7)=(9/7). The particular solution, the one specific solution
for this initial value problem is therefore
y=(2/7)x^{8}+4x^{2}(9/7).
The idea here is that the differential equation tells how a quantity changes or evolves (?) with respect to its independent variable, and that giving a starting value together with a description of the rate of change should be enough to deduce the value of the quantity at any other time. Of course, whether or not this is practical or gives any useful information is something we need to investigate. Here is a very direct version of this idea: if you tell me that my weight last January 1^{st} was 180 lbs, and then you keep track of, say, my monthly gains and losses, you certainly should be able to tell my weight at the end of the year.
Superman
Suppose that Superman leaps straight up at 40 ft/sec from the top of a
20 ft tall hill. What is his maximum height, and when does he get
highest? In this case, the presumed gravity of earth pulls down
and we will say that g is 32 ft/sec^{2}.
It was my feeling that most students know how to do a problem like this using formulas, and I wanted to plod through it principally as a way of checking our vocabulary.
Translation
Let's measure height from ground level in feet, with + meaning above
ground level. And we will measure time in seconds from when Superman
jumped. What do we know? There are facts about three functions,
and confusion is easy.
Solution
Comments on this problem and its solution
This is not a difficult problem, and I am sure that lots of formulas memorized for physics can be used to get the answers. I want to be sure that you understand the language and the process, so that more interesting problems can be analyzed successfully.
The graph to the right has time in seconds as the horizontal axis and has height in feet as the vertical axis. The units differ. I plotted the height functon for the time interval [0,3]. My Superman has somehow landed below ground level on his flight. Oh well. In fact, I think my Superman would hit the darn hill he leaped from (unless someone else has moved it during his flight) at time 2.5. I've never found this sort of problem too believable.
Car acceleration So I need to buy a car. One car I am thinking about goes from 0 to 60 mph in 9.5 seconds. How many g's (g is acceleration of gravity) is this? How can we set this up as a differential equation problem? Again, right now, I am more interested in the process than the answer. Well, 60 mph is 88 feet per second. Assume we begin at t=0 seconds, with both position and velocity equal to 0: s(0)=0 and v(0)=0. We know that v(9.5)=88. Also assume acceleration is constant, so a(t)=K. Now v(t)=Kt+C, and (using v(0)=0) we get v(t)=Kt. Since v(9.5)=88, I see that K(9.5)=88, and K=88/9.5, approximately 9.263. Now g (in these units) is 32, so this car accelerates (for this task) at 9.263/32, approximately .289 g. And then I wondered what distance the car takes to get to 60 MPH, surely a relevant safety question. For this we need to know s(t). But s(t)=(1/2)Kt^{2}+C, and again C0 (since s(0)=0). So s(9.5)=(1/2)(9.263)(9.5)^{2}, about 417.993 ft, or a bit less than a tenth of a mile (hey, more than a football field). This car's list price is $16,500. Now another car surely which should be bought by a faculty member will go from 0 to 60 in 3.4 seconds. Let's go through the numbers: K(3.4)=88, so this K is 25.882, and g is 25.882, about .81 g. Huh: about three times the acceleration. And the distance needed to get to 60 mph is s(3.4)=(1/2)(25.882)(3.4)^{2}, about 150 feet, half the length of a football field. This list price of this reasonable vehicle is $123,600. Indeed. Which should I buy? 
QotD
Suppose we know:
d^{2}y/dx^{2}=5x^{2}+cos(x).
Note: d^{2}y/dx^{2} means y´´
and we want a solution satisfying these initial conditions:
y´(0)=1 and y(0)=3.
Some warnings There are three functions around. One mistake might be to confuse the initial conditions for y and y´. Another mistake might be to interchange x and y values for the initial conditions. The initial conditions mean that when x=0 then the value of y is 3 and the value of y´ is 1.
Solution
Tuesday, November 6  (Lecture #19) 

A line segment joins a point on the positive xaxis with a point on the positive yaxis. The line segment goes through the point (3,2). Find the shortest such line segment.
Let me discuss this a bit. We sketched the geometric situation relatively quickly, but then students seem to have found making the transition from the geometry to an algebraic description not so easy. There's a diagram to the right. It shows a "typical" line segment which joins a point on the positive xaxis with a point on the positive yaxis and which also goes through the point (3,2). 
As I wrote in a discussion of a previous geometric "story", I would
play around with the problem. When the point on the xaxis is very
close to (3,0), then the line would tild "way up", very steeply. The
length of the line segment would be large. Similarly, if we were to
pull the point on the xaxis far away from (3,0), the slope would be
close to 0, and the length of the line segment would be large.

This would seem to suggest that the length varies, and that "somewhere in the middle" there is a minimum, and the minimum will occur at a critical point.
How about the transition from picture to an algebraic description? As I wrote above, we had some difficulty. The length of the line segment (what we need to maximize) is the distance from (x,0) to (0,y): this is sqrt(x^{2}+y^{2}) (the objective function). But what's missing are relationships between the variables and any restrictions on the variables, the constraint(s). If a diagram of the situation is labeled, then some restrictions almost yell at the reader. Look to the right. Certainly x>3 and y>2. And there are bunch (well, three) similar triangles which give some further relationships between the variables. For example, the big triangle and the lower right small triangle tell me that y/x=2/(x3), so that y=(2x)/(x3). And therefore the function we need to minimize is f(x)=sqrt(x^{2}+[(2x)/(x3)]^{2}). What is the domain of this function? Certainly, x>3. There is no other restriction. Now we have turned this into a calculus problem.
A calculus problem
What is the minimum of
f(x)=sqrt(x^{2}+[(2x)/(x3)]^{2}) for
3<x<?
• Endpoint evaluation There are no
endpoints! But I would still begin with some sort of analysis or
understanding of what happens out towards the edges. So maybe I should
call this
Edge analysis What happens as
x>? Well,
lim_{x>}f(x)=lim_{x>}sqrt(x^{2}+[(2x)/(x3)]^{2}).
When x gets very large, x_{2} certainly does also. But
(2x)/(x3)>2. But both pieces are squared and added: I think the
result gets very large. So lim_{x>y}f(x)=.
Now what happens as x>3^{+}? Well,
x^{2}>3. But (2x)/(x3) is like 6/(something small and
positive), which is large and positive. So I bet that the sum of the
squares of these also gets very large. So
lim_{x>3+}f(x)=.
• Critical point analysis If
f(x)=sqrt(x^{2}+[(2x)/(x3)]^{2}) then
f´(x)=(1/2)STUFF^{1/2}(2x+2[(2x)/(x3)]^{1}[(2(x3)2x)/(x3)^{2}]).
We want to choose x so that this whole "mess" is 0. But notice that
f&&180;(x) is really a fraction, and STUFF^{1/2} is "on the bottom" (because of
the minus sign in the original exponent). We only need to look at the
TOP of the fraction to see when it is 0. So let's solve:
2x+4x[(2(x3)2x)/(x3)^{3}]=0. We could multiply by
(x3)^{3}. Here is the result:
2x(x3)^{3}+4x(2x62x)=0.
2x[(x3)^{3}+2(6)]=0.
This is 0 when x=0, which is outside of our domain. Or it is 0 when
(x3)^{3}+2(6)=0, which happens when x=3+12^{1/3}.
I computed the approximate numerical value of f(3+12^{1/3}). It is 7.02348, and I don't get much insight from the specific value.
Resembles...?
The last two stories had pictures which resembled each other quite a
bit. We minimized a length, and we maximized an area. The two problems
are actually related to each other. They are examples of what are
called dual problems. In economics, such problems occur when we
want to maximize profit. This might be the same as minimizing cost. In
physics, the ideas are minimizing work and maximizing potential
energy.
My darling, struggling in the ocean!
The last story for class.
I am standing on a straight beach and my darling is swimming in the ocean, a quarter mile from shore. The closest point to my darling is two miles down the beach. The sharks attack, and I must get to my darling as soon as possible. I can run a mile in 10 minutes and swim a mile in 40 minutes. How can I get to my darling in the least time?
As I mentioned in class, this is actually not such a toy problem. Similar problems arise in optics frequently: minimizing travel time when the speed in different materials varies. I've attempted to sketch the situation, as seen from "above". How can I get from my initial position to my darling?  
Pure strategy #1 Swim all the way! Swim directly. The distance is sqrt((1/4)^{2}+2^{2}), about 2.01555 miles. At 40 minutes per mile, this takes about 80.6226 minutes. 

Pure strategy #2 Run as much as possible. I run 2 miles down the beach, and then swim. So the 2 miles running take me 20 minutes, and the 1/4 mile swimming takes 10 minutes. The total time is 30 minutes. 

A "mixed" strategy? Run for a while, then swim. It is not clear but maybe some blend of the two is faster. So I could run part of the way, and then swim directly to my darling. 

Suppose the "breakpoint" between these two activities is x miles from the point on the beach which is closest to my darling. Then I'd run 2x miles, which would take 10(2x) minutes. I would need to swim the length of the hypotenuse of a triangle with legs x and 1/4 long: that's sqrt(x^{2}+(1/4)^{2}) miles, and that would take 40sqrt(x^{2}+(1/4)^{2}) minutes. The total time would be f(x)=10(2x)+40sqrt(x^{2}+(1/4)^{2}). The domain of interest is [0,2]. 
Amazing!!! To the right is a fairly careful graph of f(x). Notice that there is a critical point in the interval [0,2]. The critical point is close to 0. I used my "friend" Maple to find the critical point. Partly this is because I am lazy, but it is more because I am tired. This computation could be done by hand, because the most "difficult" part of it is just solving a quadratic question. The first instruction defines f as the algebraic mess we have above. Maple echos the definition so that I can check it is what I want (I make lots of typing errors!). The second instruction differentiates this formula. The third instruction, using the word solve sets the previous expression (that is what % means) equal to 0. The answer is exact and comes from the quadratic formula. Then I substituted this into the expression for f. Since I didn't "understand" the expression, I used evalf finds a 10 digit approximation to the answer. > f:=10*(2x)+40*sqrt(x^2+(1/4)^2); 2 1/2 f := 20  10 x + 10 (16 x + 1) > diff(f,x); 160 x 10 +  2 1/2 (16 x + 1) > solve(%); 1/2 15  60 > subs(x=%,f); 1/2 1/2 1/2 15 2 16 15 20   +  6 3 > evalf(%); 29.68245837At the all swimming endpoint, the time needed was about 80.6226 minutes. At the most running endpoint, the time needed was 30 minutes. The ideal strategy (at least for minizing time!) gets me to my darling in less time than that: 29.6824 ... minutes. I will certainly happily admit that this is not a great difference in time. But you should see that there is a difference, and in other problems the difference might be significant. The speed of light in vacuum (air is about the same) is 299,792,458 meters per second (I copied this!). This speed is frequently called c. "Denser media, such as water and glass, can slow light much more, to fractions such as 3/4 and 2/3 of c." This difference is responsible for light "bending" or refracting, because light travels (!) to minimize time. 
L'Hôp
L'Hôpital's (also L'Hopital's
or L'Hospital's) Rule is a method
of evaluating certain very special limits. It is worth showing to you
because it is another neat application of "local linearization", and
some really important limits become easy to evaluate. However, it must
be used with some care. One quote I found on the Internet declared,
"Giving l'Hopital's Rule to a calculus student is like handing a
chainsaw to a three year old."
So now
An example
lim_{x>0}[sin(9x)/(e^{6x}1)]
Of course I began my consideration of the limit, which is made up of
familiar functions, by just plugging in x=0 in the hope that the
functions, which are individually continuous, will be continuous near
and at 0. Since this is an invented example, we get 0/0: no hope.
We can do the following. Remember that f(0+h)=f(0)+f´(0)h+Err_{f}(h)·h where the error term, Err_{f}(h), goes to 0 as h>0. Also g(0+h)=g(0)+g´(0)h+Err_{g}(h)·h where that error term, Err_{g}(h), goes to 0 also as h>0.
Consider the quotient:
f(x) f(0)+f´(0)h+Err_{f}(h)·h  =  g(x) g(0)+g´(0)h+Err_{g}(h)·hIf both f(0) and g(0) are 0 then:
f(x) f´(0)h+Err_{f}(h)·h f´(0)+Err_{f}(h)  =  =  g(x) g´(0)h+Err_{g}(h)·h g´(0)+Err_{g}(h)(dividing top and bottom by h). Then let h>0. The result seems to be f´(0)/g´(0).
L'Hôpital's Rule (version 1)
Suppose that f and g have continuous derivatives, and that
f(a)=g(a)=0 (the eligibility criterion). If
lim_{x>a}f´(x)/g´(x) exists, then
lim_{x>a}f(x)/g(x) exists and is the same value.
Return to lim_{x>0}[sin(9x)/(e^{6x}1)] which we've already checked satisfies the eligibility criteria that the top and bottom are 0 when x=0. So we consider lim_{x>0}[cos(9x)9/(e^{6x}6] and the value of this limit is easy by plugging in: 9/6. So L'H allows us to "compute" an otherwise rather difficult limit in an easy way.
Some textbook examples
I did some examples from the text. I've forgotten which ones, sorry.
More realistic
Here is a more realistic use of L'H. It may look a little bit weird,
but please just read on for a while. Suppose you want to compute the
function
sqrt(1+x) for small x. The functions that are easiest to compute are
polynomials. So ideally, I would like to find polynomials which are
really close to sqrt(1+x) when x is small. So this means finding A and
B and C and ... so that A+Bx+Cx^{2}+Dx^{2}+... is
close to sqrt(1+x).
QotD
What is
lim_{x>}sin(5x^{2})/[x^{2}+x^{3}]? If
you apply L'H, be sure to indicate why the limit satisfies the
eligibility criterion (the fraction should be an appropriate
indeterminate form).
Friday, November 2  (Lecture #18) 

Thinking about the graph of y=f(x)
The process is important. I'll try to go slowly through every bit of
information. I may make mistakes, and I may have to fix things
up. I'll only be able to sketch somethng which is qualitatively
correct.
Fact f is a differentiable function whose domain is all real
numbers except for 2 and 1.
Response I'd probably try to indicate somehow to myself that the graph I'll draw should not appear on x=2 and on x=1.  
Fact f´ is positive only in the intervals x>3 and
2<x<0.
Response f is increasing in those intervals where the derivative is positive. An important word in the "Fact" sentence is only. To me this says that f is increasing only in the intervals from 2 to 0 and from 3 upwards. In interval notation, these are (2,0) and (3,). Probably f will be decreasing in other intervals.  
Fact lim_{x>1}f(x)=.
Response The symbols x>1^{} mean x is getting close to 1 from the negative (left) side. And then  means that f(x) is getting large, but large negative, so the graph is going down. What I would expect is something like what is shown here.  
Fact lim_{x>1+}f(x)=+.
Response Here we have x>1^{+} and we should consider x getting close to 1 from the right, positive side. The + means that f(x) will be getting very large positive. The direction of approach and the largeness combine to get a sort of piece of a graph which looks like what is shown.  
Fact
lim_{x>2+}f(x)=;
lim_{x>2}f(x)=.
Response I'll do these next two as a pair. The notation is somewhat intricate here. We've got x>2^{+} and x>2^{}. So x is getting close to 2 from the right (+) and from the left (). The result in both cases is . The f(x) is getting large negative as x gets close to 2.  
Fact lim_{x>+}f(x)=1.
Response This is new. The x>+ indicates the idea that x is traveling far to the right. As it does, f(x), the y coordinate on the graph, is getting close to 1. I don't really know exactly what I should draw. I drew two possible candidate graphs in magenta. Both of the pieces of curves drawn satisfy the limit statement. BUT we can select exactly one as a valid candidate using other information. This piece of y=f(x) is drawn in a region where the derivative is positive, so the graph should be increasing. Therefore we can drop the top candidate.  
Fact lim_{x>}f(x)=0.
Response We've gotten to the final limit statement. Here x> means that x is moving "far" to the left. And the =0 tells me that the graph is getting close to the xaxis. Again I've drawn two possible candidates. Which one will be appropriate? Here we need knowledge once more about {inde}creasing behavior to make a selection. We are outside of the region where we know f(x) is increasing, and since we had that word "only" we should expect in this region that f(x) is decreasing. The lower alternative, the bottom candidate, is the one we should select. Please note that in the behavior on the right, I have drawn only the surviving candidate as supported by the previous picture's discussion.  
Further discussion We've got to complete the graph. Let me move from left to right. I know that the graph must be differentiable and therefore also continuous. In the region to the left of 2, I think the graph should be decreasing. I have indicated a tentative way of joining the two pieces we've already drawn, and you should check that what's suggested is indeed decreasing. Now consider the region between 2 and 1. This is a bit more complicated. We need to be increasing until 0, and then, probably, decreasing. And we should draw a continuous graph which connects what the limit statements gave us. So I have tried to suggest a good candidate. Notice, please, that although I am fairly sure f has a local maximum when x=0, I really have no idea where the local maximum is (I don't even know if the value is positive or negative). I've only been given quantitative information. So my suggestion is only one of many which would be consistent with what's required.  
And more discussion What happens between 1 and 4 in the picture? To the right of 3, that is, for x>3, f should be increasing. So we guess something like the magenta piece shown. And to the left of 3, in the region with 1<x<3, we know that f is not increasing, so probably (!) it should be decreasing. The result could be what is shown in this magenta piece. but I've inserted large question marks because something strange has happened: if we accept both suggestions as drawn the result will certainly not be continuous. The graph should not have a jump in it. How can we fix this? You need to think a bit.  
The last piece A curve which completes the graph validly, satisfying all of the specifications, is shown to the right. From 1 to 3, the graph decreases, dropping below the horizontal asymptote y=1. Then, at x=3, the curve begins increasing, and blends into the required asymptotic behavior as x>+. This graph has two vertical asymptotes (x=2 and x=1) and two horizontal asymptotes (y=0 and y=1). To the right is sketched one correct answer to this problem. Again, the problem is mostly qualitative, and there could be many specific graphs which are correct answers to the problem.  
Limits at +/
I wanted to give examples of some of the algebraic manipulations
resulting in horizontal asysmptotes which you should know about. So
here they are.
1. What is
lim_{x>}[x^{3}+7x17]/[x^{4}+3x+9]? Here I would take the fraction [x^{3}+7x17]/[x^{4}+3x+9] and multiply the top and bottom by 1/x^{4}. The result is a fraction equal to the original. If the algebra is done carefully, this result is [{1/x}+{7/x^{3}}{17/x^{4}]/[1+{3/x^{3}}+{9/x^{4}}]. Now consider carefully and separately the pieces of this fraction. These terms: {1/x} and {7/x^{3}} and {17/x^{4} and {3/x^{3}} and {9/x^{4} are >0 as x>. The only term that "survives" is the 1 on the bottom. I think that the limit is [0+0+0]/[1+0+0], and this is 1.  
I would like to illustrate what's happening with a graph. There are
some difficulties in presenting the information graphically. I'll give
two pictures. One of them is what we "think" the situation looks
like. This graph is to the right. Please look carefully at the scales
on the horizontal and vertical axes. They are not the same. x goes
from 4 to 20 and y goes from 0 to 0.35. There is a disproportion  a
factor of 45 to 1. That's remarkable. Below is the graph shown with
its true proportions. Look at how flat it is!
Generally I will show you pictures similar to what is on the
right. The flat picture below doesn't "show" me very much.
 
2. What is
lim_{x>}[5x^{4}+2x^{2}+33]/[x^{4}+3x+9]? Here divide again top and bottom by x^{4}. The result is [5+{2/x^{2}}+{33/x^{4}}]/[1+{3/x^{3}+{9/x^{4}}]. What happens here is slightly different. A bunch of terms ({2/x^{2}} and {33/x^{4}} and {3/x^{3} and {9/x^{4}}) still go individually to 0 as x>. But there are two nonzero terms, one each in the top and bottom. As x>, the ratio>[5+0+0]/[1+0+0]. The limit should be 1. To the right is a graph, about as good as I can have the machine draw for this example. It is a graph of the ratio for 4 between 4 and 20. The graph gets quit close to 5 rather rapidly. It is not clear to me how helpful this picture is. 
3. What is
lim_{x>}[8x^{5}5x^{3}+88x]/[x^{4}+3x+9]? In this case the degree of the top is one greater than the degree of the bottom. Dividing the top and bottom by x^{4} gives [8x{5/x}+{88/x^{3}}]/[1+{3/x^{3}}+{9/x^{4}}]. The bottom is 1+{3/x^{3}}+{9/x^{4}} and I think as x> the bottom gets close to 1 rather rapidly. The top is [8x{5/x}+{88/x^{3}}. The second and third terms are negligible as x gets large, and the top seems to be nearly 8x. I think that the quotient will behave as 8x/1 for large x, and the limit will be . A graph is shown to the right. Please notice that in this case, x goes from 0 to 10 and y, from 0 to 80. The graph appears to resemble closely a straight line of slope 8, which is what the algebraic manipulation suggests. 
A weird one ...
I also consider the function f(x)=(5x+7)/sqrt(x^{2}+3). Here
let me begin by displaying a graph, since the algebraic
analysis did not proceed very well in class. Below is a graph of
y=f(x) for x between 30 and 30. Also on display (the dashed green lines) are the horizontal lines y=5 and
y=5. Look at the graph, and observe (I hope!) that the limit as x
gets large positive of f(x) seems to be 5, and the limit as x gets
large negative of f(x) seems to be 5. The situation is more
complicated than the previous examples.
What's happening? Look at sqrt(x^{2}+5) and "massage" it algebraically. Well, sqrt(x^{2}+5)=sqrt(x^{2}·(1+{5/x^{2}})).
Square root facts 

Example The value of sqrt(4) is 2. The equation x^{2}=4 has two roots. We could indicate these two roots by writing +/sqrt(4), but sqrt(4) without any "decoration" always means 2. 
Example sqrt(400)=sqrt(4·100)=sqrt(4)·sqrt(100)=2·10=20. 
Example sqrt(25)=5 and sqrt(16)=4 and sqrt(9)=3. Although 16+9=25, notice that 3+4=7 which is not equal to 5. That is: sqrt(16+9) and sqrt(16)+sqrt(9) are very different. 
Therefore sqrt(x^{2}·(1+{5/x^{2}})) is the same as
sqrt(x^{2})·sqrt(1+{5/x^{2}}).
Please notice that if x>0, then sqrt(x^{2}) is the same as
x. So:
If x>0 then
f(x)=(5x+7)/sqrt(x^{2}+3)=(5x+7)/[x·sqrt(1+{5/x^{2}})]
.
Divide the top and bottom by x and the result is
(5+{7/x})/[sqrt(1+{5/x^{2}})] and I hope that now you can see
the limit as x> is 5.
If x<0 then sqrt(x^{2}) is the same as x. You can
check this: try, say, x=5 and see what happens. So look at f(x):
f(x)=(5x+7)/sqrt(x^{2}+3)=(5x+7)/[x·sqrt(1+{5/x^{2}})].
Dividing top and bottom by x gets
(5+{7/x})/[–sqrt(1+{5/x^{2}})]. As
x>, this becomes 5.
Damped oscillation
The function f(x)=sin(x)/x is actually something I'm more comfortable
discussing with engineering students, since things that it "models"
are easily observed. As x>, the bottom, x, grows, and the
top, sin(x), is caught between 1 and 1. This function should approach
0. In fact, an appropriate version of the Sandwich Theorem can
be used. By this I mean, please, just realize that:
1/x<=sin(x)/x<=1/x.
Since both 1/x and 1/x >0 as x>, f(x), which is
caught in between, also approaches 0.
A graph of y=f(x) is shown to the right. The "envelope curves", y=+/1/x, are dashed green curves. You will study oscillations whose amplitude go to 0 (think of a spring vibrating in a viscous fluid).
A story about numbers
We should translate this problem statement into algebra, and then we will use methods of calculus to solve the problem. This is a "toy" problem, but the process remains valid even in much more complication situations. The translation is important. We may not always be able to solve a problem, but calculus can usually move the analysis of the problem closer to a solution.The sum of two nonnegative numbers is 20. How should the numbers be selected so that the product of the square of one with the other is largest?
So the two nonnegative numbers will be
called x and y. We know that x>=0 and y>=0. We also know that
their sum ... is 20 so x+y=20.
We want to know How should the numbers be
selected to make something largest. What's the quantity we should try to
maximize? It is the product of the square of
one with the other and this is, I think, x^{2}y.
Now the problem can be written in the following way:
Suppose x+y=20 and x>=0 and y>=0. Find the maximum value of
x^{2}y.
In economics and some other disciplines, the first sentence is called
the constraint. It relates the variables of the problem and
restricts which values should be considered. It usually is related to
the appropriate domain of the problem. The second sentence is the
objective function. It describes what should be "extremized" 
in this case, what should be maximized or made largest. Here the
objective involves two variables, x and y. But the constraint tells us
that y=20x. And therefore the problem description changes.
Suppose f(x)=x^{2}(20x). The domain for this problem is
[0,20]. How can we get the largest value of f?
Notice that the constraint, which shouldn't be forgotten, has
resulted in the domain statement when we write this version of the problem.
Now we can use calculus. The maximum of f on [0,20] occurs either at
endpoints or critical points.
• Endpoint evaluation
f(0)=0^{2}(200)=0; f(20)=(20)^{2}(2020)=0. I bet the
max is inside, at a c.p.
• Critical point analysis If
f(x)=x^{2}(20x), then f´(x)=2x(20x)x^{2}. This
is 0 when =2x(20x)x^{2}=0 or x(402xx)=0 or x(403x)=0. So
we get x=0 (already checked) and x=40/3. Since
f(40/3)=(40/3)^{2}(2040/3) is positive (20 is 60/3),
this is where the max value of f occurs.
The answer to the question is 40/3 and 2040/3 (we were asked How should the numbers be selected).
Another story
A line segment joins the points (3,0) and (0,2) in the first quadrant. A rectangle in the first quadrant has sides on the x and yaxes, and a vertex on the line segment. What is the largest area that such a rectangle can have?
Probably the first thing I would do when meeting a problem of this type is to draw a picture. I think my sketch would look something like what's to the right. Certainly, I would need the line segment joins the points (3,0) and (0,2) in the first quadrant which is drawn. Then I likely would try to draw a typical rectangle. Although I tried to write the specification of the rectangle in a direct way, my desire to be brief may have made this difficult to understand. The rectangle has sides on the x and yaxes as shown. It also has a vertex on the line segment. Legitimate questions include, "What's a vertex?" and "What line segment?" I hope, even if you've not heard the word before or this use of the word, that you would know "vertex" means a corner of the rectangle. And the first sentence of the problem defines "the line segment".
Almost surely in my mind I would play around with the problem as
pictured, and try to see some of the eligible rectangles. The pictures
I would look at would include the two below. If the vertex were very
near (0,2), as in the left picture, I see that one side of the
rectangle would be almost 2 units long, while the other is very
small. The area of that rectangle would be small. On the other hand
(right picture), if the vertex were very near (3,0), one side of the
rectangle would be almost 3 units long, while the other is very
small. Again the area would be small. This encourages me to think
that, yes, there is a maximum and almost surely the max area will
occur for some rectangle with vertex "inside" the segment (at a
critical point of the algebraic model we will build).
Two "extreme" rectangles  



Building the algebraic model
We will need the equation for the line segment. In class I tried to
describe how I would "guess" this equation. Since this is top secret
and also silly, I will just write it here: (x/3)+(y/2)=1. Well, o.k.,
my guessing method goes like this: the equation will involve both
variables since the line is neither vertical (x=const) nor horizontal
(y=const). Therefore it could be written
(x/something)+(y/something else)=1. Then plugging
in both (0,2) and (3,0) give me the values of something and
something else. What about the area of the rectangle? If
the coordinates of the point p are (x,y), then the area is xy.
I'm almost "there". The objective function is xy. The constraint
situation certainly involves (x/3)+(y/2)=1 but there is just a bit
more. Notice we are in the first quadrant, so that both x and y are
nonnegative. Now my model follows:
Constraint(x/3)+(y/2)=1 and x>=0 and y>=0.
Objective Maximize xy.
As before I'll use the constraint to reduce the number of variables in
the objective. So (x/3)+(y/2)=1 becomes y=2(1(x/3) and xy becomes
2x(1(x/3)). But don't forget the remainder of the constraint, which
becomes a domain restriction: x is in the interval [0,3].
The resulting calculus problem
Find the max value of f(x)=2x(1(x/3)) when x is in the interval
[0,3]. This was the QotD.
The maximum of f on [0,3] occurs either at
endpoints or critical points.
• Endpoint evaluation
f(0)=2·0(10)=0; f(3)=2·3^{2}(1(3/3))=0. I bet
the max is inside, at a c.p.
• Critical point analysis Since
f(x)=2x(1(x/3)) I know that
f´=2(1(x/3))+2x(1/3)=2(4/3)x. Therefore the only c.p. is where
2(4/3)x=0 so x must be 3/2. f(3/2)=2(3/2)(1(1/2)) is positive, so
this is the maximum value. Yes, it "simplifies" to 3/2. The answer to
the problem is 3/2. Notice that the problem requests the largest area. We should attempt to answer
the question that is asked (so remarking that the dimensions of the
rectangle are ... and ... is not responsive!).
You should ...
Do 8 or 10 of these problems yourself, possibly together with other
students. Certainly what I wrote about is much more than I expect you
to write, but I would hope that, after sufficient experience,
you would think through such problems in much the same way as what's
above. I wanted to help you learn the process.
I will do one or two more of these problems in the next class, but
your practice is essential to your own ability to answer
questions of this type. The "translation" skill, that is, constructing
the correct mathematical model, is very important in applications.
Tuesday, October 30  (Lecture #17) 

What second derivatives can tell you
The sign of the second derivative on an interval also has some neat
geometric information. Here is the logic:
If f´´>0 on an interval then f´ is increasing
on that interval. So moving from left to right means that the slope of
the tangent line increases. I don't know enough about animations to
put my walking demonstration of this on the web (I wish I did, but I
am lazy), but to the right is a static picture of what I am trying to
describe. This curve is concave up.
Of course, if f´´<0 on an interval then f´ is decreasing on that interval. So moving from left to right means that the slope of the tangent line decreases. A typical geometric picture of this situation is to the right. (Since I am lazy, all I did was flip the previous picture  drawing programs are wonderful.) This curve is concave down.
The Second Derivative Zoo
Before we went on, I thought giving several (relatively) simple
examples would be useful. I learn far more from examples than from
definitions, and even from theorems. So here is my "zoo" or collection
of concavity examples.
Examples and discussion  Graphs 

Since f´´(x)=6x, I know that f is concave up where 6x>0: this is (0,). Similarly, f is concave down where 6x<0. This is (,0). At 0, the xaxis (equation:y=0) is a horizontal tangent line. Also at x=0, the concavity changes from down to up. This is called an inflection point.  
Since f´(x)=(1/3)x^{2/3}, we know that f´´(x)=(4/9)x^{5/3}. Again, the domain is nonzero x's. But what is the sign of f´´(x) for those nonzero x's? Here please notice that 5 and 3 are both odd. This is important. A negative number raised to an odd integer power or an odd integer root is negative. But look closely: in front of the formula, before the (4/9) is –: a minus sign. So f´´ is positive when x is negative, and therefore the graph is concave up for negative x. For similar reasons, the graph is concave down when x is positive. And x=0 is a point on the graph where concavity changes, and therefore is an inflection point.  
This means f is decreasing inside (,0) and f is decreasing inside (0,). Notice, though, that f(1)=1 and f(01)=1. Even though 1<1, we have f(1)<f(1): f is not decreasing "across" the intervals. If we compute correctly, f´´(x)=2/x^{3}. 3 is an odd power, so the graph is concave down in (,0) and concave up in (0,). The domain of f does not include 0, so there is no point of inflection! 

Comment 
Inflection point
x is an inflection point of f if
The Gaussian (bellshaped) curve
I wanted to graph a "random" (not!) curve defined by a formula. What
we looked at was f(x)= e^{(x2)}. This turns out to
be a rather important function in analyzing experimental
results. Any "real" repeatable experiment with a measurable
outcome is likely to have the numbers scattered so they resemble
this bellshaped curve. This phenomenon is not obvious at all!!
The exponential function's outputs are never negative and never 0. Therefore no part of the graph will be on or below the xaxis. Since x^{2} is always less than or equal to 0, and these numbers serve as inputs to exp, I bet that the outputs from exp will be less than or equal to 1, and will be equal to 1 only at x=0. Also this graph is even, symmetric with respect to the x=axis, since (x)^{2}=x^{2}. I hope this explains to you why the graph looks the way it does. But let me try analyzing the graph using the first and second derivative.
If f(x)=e^{x2} then f´(x)=(e^{x2})(2x). The exponential function is very nice. It is never 0 and always positive. Therefore the only x for which f´(x)=0 is when 2x=0. So x=0 is the only critical number. Now reasoning using the Intermediate Value Theorem says that f (which is certainly continuous!) can have only one sign for x<0 and one sign for x>0 (or else f(x) would have to have to be 0 again). We can check signs at, say, x=1 (where the derivative is negative) and x=1 (where the derivative is positive). f is increasing in (,0) and f is decreasing in (0,). Naturally 0 represents a local (and indeed, absolute!) maximum.
What information can we get from the second derivative? If we use the product and the chain rule correctly, then f´´(x)=(e^{x2})(4x^{2}2). Logic similar to the preceding asserts that this is 0 exactly when the nonexponential factor is 0. But 4x^{2}2=0 when x={+}1/sqrt(2). Again, we can check signs in between the 0's of f'', and f will be concave up for x<1/sqrt(2) and for x>1/sqrt(2). For x between 1/sqrt(2) and +1/sqrt(2), the graph will be concave down. The points where x={+}1/sqrt(2) are where the concavity of f changes: these are called inflection points. These particular inflection points are related to the standard deviation, which represents dispersal from an average when this function is used in statistics. The center of the curve, at x=0, is related to the mean of the data observed.
The diagram/picture above is an attempt to indicate how a person could
assemble the information about signs of the first and second
derivative, and patch together template pieces. The patchedtogether
curve should be continuous (no breaks) and differentiable
(smooth). This all takes some practice!
Graphing x^{3}(x1)^{4}
Here I invited students again to use a graphing calculator and try to
see what y=x^{3}(x1)^{4} "looked like". I did remark
that the "action" took place somewhere between 1 and 2. The result of
this was something like the graph shown to the right. I believe that
calculators and graphing devices are wonderful, but sometimes they
almost conceal what's going on.
Here f(x)=x^{3}(x1)^{4}. If we want to find out where
f is increasing and decreasing, we really should look at
f´(x). For this we need the product rule and the chain rule. So:
f´(x)=3x^{2}(x1)^{4}+x^{3}4(x1)^{3}.
Generally I am against "simplifying" because I view it as mostly a
chance to make lots of mistakes. But here some simplifying will reveal
structure in the derivative. So please notice the common factors, and
what you get is as follows:
f´(x)=x^{2}(x1)^{3}(3(x1)+4x)=x^{2}(x1)^{3}(7x3).
What can we tell about where the derivative is 0 and where it is positive and where it is negative? Well, the different factors allow us to deduce that the derivative is 0 at x=0 and x=1 and x=3/7.
If x is very large positive, say, then f(x) is a product of three factors, all of which are positive. And if x is very large negative, then the x^{2} is positive and the (x1)^{3} is negative and the 7x3 is negative. Therefore f(x) in that range is positive also. So we have learned (using logic from the Intermediate Value Theorem as before) that the derivative is positive on at least the intervals (,0) and (1,). There is a chance for the derivative to change signs at x=0, but the factor which controls sign change there is x^{2}: since 2 is even, there is no sign change at x=0. But there is a sign change at x=3/7 and at x=1. So now we have broken up the real line into pieces
Deriv is + Still + Now it is  Back to + here <03/71> Func increases increases The func decreases Here it decreasesSo from this I learn that f has critical points at 0 and 3/7 and 1. We can also learn that f has a local max at 3/7 and a local min at 1. This is not entirely clear from the initial graph. Actually, if we just look at the graph from .1 to 1.1, you can see some of the structure. This is shown to the right. Please notice that the vertical scale of this graph is very small. This might all be difficult to see without looking at the calculus first.
Then we considered the concavity of this function, and made some
guesses about the number and location of inflection points. We can be
sure about this if we find the second derivative.
I will use the product rule, and make the second factor, a product
itself. So:
f´(x)=x^{2}(x1)^{3}(7x3)=
x^{2}((x1)^{3}(7x3))
f´´(x)=2x((x1)^{3}(7x3))+x^{2}(3(x1)^{2}(7x3)+(x1)^{3}(7))
So let me try to "simplify" f´´(x). We will get:
x(x1)^{2}(2(x1)(7x3)+3x(7x3)+x(x1)7)=
x(x1)^{2}(2(7x^{2}10x+3)+21x^{2}9x+7x^{2}7x)=x(x1)^{2}(42x^{2}36x+6)
So the second derivative is 0 at x=0 and at x=1 and at the roots of 42x^{2}36x+6=0: those are x=(3/7)+/sqrt(2)/7 (approximately .227 and .631). The second derivative does not change sign at 1 because the factor is (x1)^{2}, an even power. It does change sign at 0 and the two other numbers which are on either side of the local max. There are three inflection points.
I needed to work fairly hard to get everything correct in the preceding example.
Second Derivative Test There is a result which allows you to tell when a critical point is a local max or local min if you have information about the second derivative at that point. It goes like this:
Inflection points as you drive! As I drive along the road, I steer to the left (that's the A region of the road). Eventually I come to a place where the road begins to bend the other way, at B. After that point, in the C region, I steer right. The bending place, where the switch from left steering to right steering occurs, is an inflection point. Please notice that from the point of view of the car, it actually doesn't matter (no gravity, we are looking from above) whether the curve is decreasing or increasing. The steering occurs as a response to how the curve is bending. When the bending changes (inflection!) we must change steering direction  if we want to stay on the road, anyway. 
Behind!!!
As of right now I am (regret, sorrow!) a bit behind the syllabus. I
haven't done horizontal asymptotes. I will try to catch up next
time. I hereby postpone the last textbook problem due tomorrow!
Friday, October 26  (Lecture #16) 

Rolle's Theorem restated
Here f is differentiable on [a,b] and f(a)=0 and f(b)=0. (f is "glued
down" at the ends.) Then somewhere in between there is at least one c
with f´(c)=0.
Below is a gallery of possible "Rolle" situations. The first, to the
left, is the simplest and silliest. The function is 0 in the whole
interval. Then any c is a valid candidate for the consequence of the
theorem. The second picture shows what happens if f is positive
somewhere. Then the absolute max must occur inside, and it must occur
at a local max inside the interval, and since f is differentiable,
f´(c)=0 there. The next is the negative situation, and the last
picture is what might happen if there f had a mixed sign behavior.
Rolle's Theorem tilted
Rolle's Theorem seems very specialized. What would happen if we tilted it?
That is, we took the coordinate axes and rotated the whole picture?
Then the graph is no longer glued down at the endpoints. This is the
important generalization that people use constantly. Of course, the
picture is relabeled and everything is more conventionally drawn.
Here is one of the two results in this course that anyone who
uses calculus will use quite frequently.
The Mean Value Theorem (MVT)
Suppose f is differentiable in [a,b]. Then there is at least one c
inside the interval so that f´(c)=(f(b)f(a))/(ba).
Discussion To the right is a tilted and relabeled picture. The
line segment joining (a,f(a)) and (b,f(b)) has slope equal to
(f(b)f(a))/(ba). The other indicated line segments are pieces of
tangent lines which are parallel to the segment joining (a,f(a)) and
(b,f(b)). Since they are parallel, their slopes must be equal. But the
slope of a tangent line at c is f´(c), and that's the equation
which appears above.
If you want an algebraic verification of MVT, then look at the end of
section 4.2. This picture is sufficient for my purposes. I want to
show you some of the ways people use this result.
Simple (?) observation #1
Suppose f is differentiable, and in some interval I we know for some
reason that f´ is always 0. What happens? Well, take two
points, x_{1} and x_{2} in I with
x_{1}<x_{2}. MVT says that
(f(x_{2})f(x_{1}))/(x_{2}x_{1})=f´(c)
But the righthand side of the equation is 0. So the quotient on the
lefthand side must then have 0 on top:
f(x_{2})f(x_{1})=0, which means
f(x_{2})=f(x_{1}) for any choices of x_{1} and
x_{2}. This means that f is constant.
If f´=0 always, then f is
constant.
Physical meaning
Well, one way to get a physical (?) interpretation of this statement
is to imagine that f(x) is the position depending on time, x. (Sorry:
you could rotate the x and get a t, if that makes you happier.) So
f(x) reports the position on a line. Then the statement above
translates to:
If velocity is always
zero, then position is constant.
Well, it sure doesn't look profound, but maybe it does make sense. By
the way, this "simple" deduction which is so clear physically wasn't
verified mathematically for about 150 years. Stupid human beings. (!)
(?) Or maybe this stuff is somewhat subtle?
Simple (?) observation #2
Suppose f is differentiable, and in some interval I we know for some
reason that f´ is always positive. What happens? Well, take two
points, x_{1} and x_{2} in I with
x_{1}<x_{2}. MVT says that
(f(x_{2})f(x_{1}))/(x_{2}x_{1})=f´(c)
Now let's see: the righthand side is now supposed to be positive. The
bottom of the lefthand side is positive (since
x_{1}<x_{2} means
x_{2}x_{1}>0). So we have a fraction with a
positive bottom equal to a positive number. So the top of the fraction
should be positive. That means f(x_{2})f(x_{1})>0
so that f(x_{1})<f(x_{2}). So what do we know:
IF f´ is positive, THEN x_{1}<x_{2}
implies f(x_{1})<f(x_{2}).
Physical meaning
Well, if derivative corresponds to velocity, and position corresponds
to the original function (with higher corresponding to more
right position on the line), then:
If velocity is always
positive, then position is moving steadily right.
Another way of thinking about this is graphical. If you wish, you
could imagine position as a function of time. Then the "progress" of
time might be the horizontal axis, and position in this diagram would
be the vertical axis. Position getting bigger would mean that the
graph of position versus time would be getting higher. So a
qualitative picture of this sort of graph is shown to the right.
Simple (?) observation #3
I will reverse the sign of the derivative. So here suppose f is
differentiable, and in some interval I we know for some reason that
f´ is always negative. What happens here? Well, take two
points, x_{1} and x_{2} in I with
x_{1}<x_{2}. MVT says that
(f(x_{2})f(x_{1}))/(x_{2}x_{1})=f´(c)
Now let's see: the righthand side is now supposed to be negative. The
bottom of the lefthand side is positive (since
x_{1}<x_{2} means
x_{2}x_{1}>0). So we have a fraction with a
positive bottom equal to a negative number. So the top of the fraction
should be negative. That means f(x_{2})f(x_{1})<0
so that f(x_{1})>f(x_{2}). So what do we know:
IF f´ is negative,
THEN x_{1}<x_{2} implies
f(x_{1})>f(x_{2}).
Physical meaning
Again, derivative corresponds to velocity, and position corresponds to
the original function (and higher still means position is more to the
right, and lower position means more to the left on the line),
then:
If velocity is always
negative, then position is moving steadily leftt.
Again, we could think graphically. Position is a function of
time, the "progress" of time is the horizontal axis, and
position is the vertical axis. Position getting
smaller would mean that the graph of position versus time would be
getting lower. A qualitative picture of this sort of grpah is
shown to the right.
Further definitions
A function f is increasing on an interval I if whenever we take
x_{1} and x_{2} in I with
x_{1}<x_{2}, then
f(x_{1})<f(x_{2}).
A function f is increasing on an interval I if whenever we take
x_{1} and x_{2} in I with
x_{1}<x_{2}, then
f(x_{1})<f(x_{2}).
Facts
MVT implies:
If f´ is positive on an interval, then f
is increasing on that interval.
If f´ is negative on an interval, then f
is decreasing on that interval.
What we can say with graphical (qualitative) evidence
This is basically a discussion of problem 24 in section 4.3. The
problem gives a graph of the derivative f´(x) of
a function f(x). Please let me repeat: the graph is the
derivative of the function. I've attempted to copy the graph in the
picture to the right.
First, we are asked to "find the critical points of f". Since f is differentiable (otherwise we wouldn't be looked at a graph of f´, darn it!) the critical points here are where f´(x)=0. Look at the graph. The curve crosses the xaxis at three values of x: 1, 0.5, and 2. These are the critical points of f.
Then we are asked to determine whether these critical points are local max or local min or neither. So let me think through this with you.
Local analysis near x=0.5 Look closely at the graph of f´ in a small interval to the left of x=0.5. The graph (inside the very light blue region) is above the xaxis. So in this interval, the derivative is positive and therefore the function is increasing. Look now at the graph of f´: in a small interval to the right of x=0.5. The graph (inside the very light green region) is below the xaxis. So in this interval, the derivative is negative and the function is decreasing.  
What should the graph of f (not the derivative!) look like? To
the left of 0.5, the function is increasing. The derivative at 0.5 is
0, so the tangent line is horizontal. To the right of 0.5, the
function is decreasing. Surely f has a local maximum at 0.5. I can't
tell the value of f(0.5), but I can tell you that value is
larger than f(x) for nearby x's, on either side of 0.5
f has a local maximum at 0.5. 
Local analysis near x=2 The situation is similar but reversed near x=2. A "blow up" of the graph of f´ near x=2 is shown to the right. In a small interval to the left of x=2, the graph of the derivative (inside the very light blue region) is below the xaxis. So in this interval, the derivative is negative and therefore the function is decreasing. The graph of f´: in a small interval to the right of x=2 is inside the very light green region, above the xaxis. So in this interval, the derivative is positive and the function is increasing.  
What should the graph of f (again, the graph of f, not of its
derivative!) look like? To the left of 2, the function is
decreasing. The derivative at 2 is 0, so the tangent line is
horizontal. To the right of 2, the function is increasing. Surely f
has a local maximum at 2. Again, we don't know the actual value of
f(2), but we do know its relative value. f(2) is
smaller than f(x) for nearby x's, on either side of 2.
f has a local minimum at 2.  
Local analysis near x=1 I saved the most interesting (or maybe most irritating!) for last. What do we see when we look very closely at the derivative near x=1? Well, to the left of 1 the derivative is positive, above the xaxis, so the function itself should be increasing in that interval. And to the right of 1, the derivative is also positive, so the function is also increasing there. That's what the graph of the derivative of f declares.  
The graph of f will show some delicate (?) behavior near 1. In a
small interval to the left of 1, we know f should be increasing (as
we walk from left to right, the graph will go up). And the same sort
of qualitative behavior will be true for the graph of f in a small
interval to the right of 1: up again. In fact, the graph is just
going up, increasing in the whole interval. The most fascinating (?)
aspect is that since f´(1)=0, we'd better draw the graph of f so
that it has
a horizontal tangent line at 1. And that's what I've tried to show to
the right. If you are not used to this sort of kink in a graph, take a
look.
f has neither a local minimum nor a local maximum at x=1.  
Maybe you'll believe this one ...
Here is a machinegenerated graph of f(x)=x^{3} for x "near" 0
(well, for x between 1.1 and 1.1). Look closely at it. The local
behavior is very much like the local behavior of the text example near
1.
The derivative is 3x^{2}. This is positive if x is not 0, so certainly the function is increasing for x<0 and is increasing for x>0. We paste these pieces together, and I think that f is increasing for all numbers. There are several things to notice "at 0" (actually, both at 0 and near 0). The tangent line is horizontal. The curvature (!!) changes. The curve to the left of 0 is concave down and, to the right of 0, it is concave up. I will spend more time on such phenomena during the next lecture.
Diary entry in progress!
The First Derivative Test
Here is how, maybe, you could detect whether a critical point is a
local minimum or a local maximum or neither.
Suppose f has one critical point in an interval.  

IF  THEN  
f´>0 to the left of the c.p.  f´<0 to the right of the c.p. 
 
f´<0 to the left of the c.p.  f´>0 to the right of the c.p. 
 
f´ has any other sign assortment (positive/positive or negative/negative)  The c.p. is neither. It is not a local max and not a local min. 
Actually I think about the local pictures and don't memorize the algebra (with only 4 brain cells, for every fact I know, I must forget two). I hope that you can see that appropriate local pictures for the last line are (positive/positive) and (negative/negative . You may prefer to think about the algebraic versions, but the key idea is that the sign of the first derivative on both sides of the critical point provides enough information to decide local max/min questions.
A simple algebraic example
Here is problem 33 of section 4.3. The general instructions are to
find critical points and the intervals where the function is
increasing and decreasing, and then apply the First Derivative Test to
each critical point. Here f(x)=x^{4}+x^{3}.
Certainly f´(x)=4x^{3}+3x^{2} and we set this equal to 0, factoring: x^{2}(4x+3)=0. The roots are 0 and 3/4. These are the only places where f´ is 0. Notice that f´ is continuous. This means if we test the sign of f´ "between" the critical points, then we know the sign in each of the intervals whose boundary is a critical point. This is because if the sign changes, then (continuity, Intermediate Value Theorem) there would have be a place where f´ is 0. This is a very useful observation if you want to be lazy and efficient.
Testing the signs
We need to look "between" (on both sides of!) 0 and 3/4. Well, I bet
that f´(10)=4·10^{3}+3·10^{2} is
positive. I bet also that
f´(10)=4·10^{3}+3·10^{2} is
negative (thousands are way bigger than hundreds, and the parity
[even/oddness] of the powers makes this easy. What about really
between 0 and 3/4? I think in class we looked at x=1/2. Then
f´(1/2)=4(1/2)^{3}+3(1/2)^{2}=(1/2)+(3/4) is
positive. So now in my head I have a picture of the real line:
What's important here is not the specific example but the
process. With a very small amount of work (compute derivative, find
critical points, take a few carefully chosen values of the derivative)
I can conclude:
If you don't believe me, then take a look at a computergenerated graph of y=x^{4}+x^{3} to the right. Here, I will make a bold statement: if I've done my critical point analysis correctly, I think I would believe more in what it told me than in a computergenerated graph. Formulas that make deceptive graphs are known.
My friend Francine drives again ...
I don't want to get into legal trouble. The "logo" of the (beloved!)
New Jersey Turnpike is shown to the right. I am using it here totally
for educational purposes. Sigh. Only education. Really. No DVD's, no
music videos, no Latin American film rights, no web things. Really,
really.
After this assurance from the cowardly and worried writer, I remind you that a previous drive of Francine on the Garden State Parkway had been analyzed earlier using the Intermediate Value Theorem.
The turnpike has a restricted number of entrances and exits. Suppose Francine
Let's model the situation the following way. Suppose f(t) describes
Francine's position in miles on the turnpike, and t is the time, in
hours, after 8 AM. Then what we know is the following:
f(0)=34.5 and f(2/3)=83.3 (40 minutes is
twothirds of an hour).
Then MVT asserts there is some time c
between 8 AM and 8:40 AM so that
f´(c)=(83.334.5)/(2/30)=48.8/(2/3)73.207. Since the speed limit on
that portion of the turnpike is 65 MPH, MVT shows that Francine was
speeding.
The Intermediate Value Theorem provides some information about
Francine's driving. It tells us that she passes through every place
(position) on the highway between Milepost 34.5 and Milepost 83.3 at
some time between 8 AM and 8:40 AM. This is somewhat indefinite.
The Mean Value Theorem gives more quantitative information,
also in a rather indefinite style (heck, we're given only outline
information!). It declares that the velocity (the derivative) must be
exactly 73.207 at some time between 8 AM and 8:40 AM.
The chunks of information have different flavors but mostly I think
that MVT information has more uses.
Semester project, honors course Why doesn't the state actually use the information it has for such things? It does have entrance/exit times and specifications of which interchanges occur, etc. Explain in detail, with citations of specific laws. Sigh. 
Quantitative information from the derivative
Here is a Math
Problem. Suppose you know the following about a
differentiable function:
f(5)=7 and, for all x, f´'s values are
between 40 and 60
(that is, 40<=f´(x)<=60).
What can you say about f(10)?
This seems to be very abstract and very silly. Let me translate it
into physical language, however.
A Physical Representation of the Math Problem:
Suppose f(t) represents the miles that have been traveled down a road
at time t (in hours, I guess AM!).
At 5 AM, you are 7 miles does the road.
Your speed is always between 40 and 60 mph.
What can be said about your position at 10 AM?
Somehow, phrased this way, the problem seems much more easily handled,
even to me, with a brain accustomed to math stuff. "Golly," I
might say, "In 5 hours you drive at least 200 [that's
5·40] miles and at most 300 [that's 5·60]
miles, so your position is between 207 and 307 miles down the
road."
We are actually using MVT reasoning on this. Look:
By MVT, [f(10)f(5)]/[105]=f´(c), for
some c between 5 and 10.
But the derivative is between 40 and 60, and f(5)=7.
So 40<=[f(10)7]/[5]<=60
Multiply by 5 and then add 7.
And 207<=f(10)<307.
The advantage of the mathematical approach is that it will apply to
all situations where we have a model and some estimate of the
derivative and knowledge of the function at one value. We then can
"predict" some estimate of the function at another value.
And even more (a possible QotD!)
I was going to ask the following. Ms. Kravitz remarked that the class had been
OVER for 5 minutes already. Sigh. Here is
what would have been the QotD, with my apology:
Suppose you know that f(0)=7 and f´(x)=sqrt(1+x^{3}). What can you tell me about f(2)?
MVT says that f´(c)=[f(2)f(0)]/[20]=[f(2)7]/2. Since
f´(c)=sqrt(1+c^{3}) and c is between 0 and 2, I
bet that
1<=f´(c)<=3. Why is this? Well, f´ is increasing (you
can take the derivative and the derivative is always positive in that
interval so the function is increasing) and thus
1=sqrt(1+0^{3})f´(0)<=f´(c)<f´(2)=sqrt(1+2^{3})=3.
Therefore 1<=[f(2)7]/2<=3. Multiply by 2 and add 7. The result is
9<=f(2)<=13.
Tuesday, October 23  (Lecture #15) 

As I remarked in class, the difficulty now (from the instructor's point of view!) is controlling the number and complexity of examples. There are many, many, many interesting examples coming from real applications.
The class interaction was rapid, and maybe I spoke too fast at times (I'm sorry). The examples below are generally similar to the ones discussed in class.
Extreme values on an interval
I wrote the textbook's definitions on the board. I remarked that I
never understood definitions until I had examples that did and did
not satisfy the definitions. So I will assume that you have a
textbook next to you (I don't want to just copy what's there). You should know the definitions. The
situation the definition deals with is an interval, I, which is the
domain of the function, and a function, f, defined on that interval.
YES!
Here is an example of a function
and an interval, where there is both an absolute maximum and an
absolute minimum.
Here I=[0,1] and f(x)=x. Then f has an
absolute maximum at x=1, where f's value is 1, and f has an absolute
minimum at x=0, where f's value is 0.
another YES!
Another example, where the interval is an open interval and the
function is still fairly familiar.
Suppose I=(Pi/4,7Pi/4) (this does not
include endpoints!) and f(x)=sin(x). This f has an absolute maximum at
x=Pi/2 where f's value is 1. It has an absolute minimum at x=3Pi/2,
where f's value is 1.
NO!
The first NO! example we had was rather subtle. I think it was
something like this.
The interval was (1,1) (an open interval,
which does not contain its endpoints. The function was something like
f(x)=x^{2}. There is an absolute minimum at (0,0), but
... what would be "your" candidate for an absolute max? Any number you
choose which is close to 1 will have a number even closer to 1 and
bigger. For example, if you were to assert that f has an absolute max
at .99998, I would tell you that f's value at .9999999998 was
bigger. If you said to me (I think Mr. Gluck mentioned this) that
.9999... (infinitely repeating) was where an abs. max was located, I
would reply that if you are sophisticated enough with decimal notation
to know about such things, this indicates a certain geometric series
whose sum is 1: the notation .9999999.... is just another "address"
for the number 1. And in this case, the domain does not include 1 and
does not include 1. So this function in this domain does not have an
absolute max.
Another NO!
Here is maybe a simpler example. Look at f(x)=tan(x) with the interval
(Pi/2,Pi/2). Certainly there will be no absolute maximum and no
absolute minimum, since tan(x)>+ as
x>(Pi/2)^{} and tan(x)> as
x>(Pi/2)^{+}. I've not supplied a graph of this function
since I hope tangent is familiar to you.
A more artificial NO!
Suppose the domain is [1,1], which is a closed interval (includes its
end points) and which is bounded. Those two attributes (closed and
bounded) turn out to be important in the theory, as is described just
below. Now define f(x) piecewise: it is 0 when x=0 and it is 1/x when
x is not 0. Part of the graph of this function is shown to the
right. And if you think this is a silly example, I will totally
agree. But why is it silly, and what is there about it which is
not "nice"? Understanding one's feelings about such things is
sometimes very helpful. I don't like this example because the function
is quite clearly not continuous at 0 (the limit as x>0 of f(x)
does not exist). I also don't like this function because it certainly
does not have an absolute maximum and does not have an absolute
minimum.
Existence of extreme values on a closed and bounded interval
In many situations mathematical models are created which need maximum
and minimums. In business, we might be interested in lowest cost (min)
or highest profit (max). In more physical situations, many people want
lowest energy or least work. Or greatest volume or ... the
applications are numerous. Before searching for max or min, we might
want to know that such things exist (apparently they don't always 
look at the examples we have for NO!).
The following result can be proved, that is, deduced from
generally accepted and more obvious statements. The process of
deducing it is rather lengthy, and takes up a substantial amount of
Math 311. I request politely that you, as student engineers and
applied science types, be willing to accept the statement as
true. Most of the proofs of this statement do not supply any
proceedures for actually getting the predicted max and min. So:
A theoretical result
Suppose I is a closed and bounded interval (so it is [a,b]), and
suppose that f is continuous. Then f has an absolute maximum and an
absolute minimum on I. So there are x_{M} and x_{m} in
I so that f(x_{m})<=f(x)<=f(x_{M}) for all x's
in I.
This result does not tell how to find x_{M} and x_{m}. It does not tell how many x_{M}'s and x_{m}'s there may be (there may be many, but there are at least one of each. For functions that are likely to be used in practice, we will outline a useful strategy. First some more definitions.
Local extreme values
f has a local maximum at x if f's value at x is the largest in
some open interval containing x. The most important and maybe
most confusing part of that sentence is "open interval". The dual
definition is: f has a local minimum at x if f's value at x is
the smallest in some open interval containing x.
YES!
Here is a very simple example which is the same function and domain as
before. The absolute min is a local min. The absolute max is a local
max. I have put magenta in the
domain and on the graph to indicate the relevant open intervals.
A shocking NO!
This function has no local min and no local max. The only points you
might want to consider are the endpoints. But notice, please, that the
endpoints are no inside an open interval of the domain. The
right endpoint has no open side to the right, and the left endpoint
has no open side to the right. This example is really quite annoying
and unexpected if you're new to this game.
A zoo of YES and NO
Below is a collection of pictures. This probably is "Too much
information" but I tried to write all of the possibilities. It
wouldn't be difficult to write some fairly simple formulas whose
graphs qualitatively have the same behavior. I wanted to show that
"local" and "absolute" have no necessary logical relationships.
Critical points
These turn out to be very important. f has a critical point at
x if either f´(x)=0 or f´(x) does not exist.
The picture below is a zoo of critical points. Again, maybe this is too much information. I do hope you get the feeling that local maxes and local mins occur at critical points. This is true.
Fermat's theorem on local extreme values
If there is one single result which has kept math people prosperous
over the last three centuries, this is it.
If f is differentiable at x and if f has a local max or a local min at x, then f´(x)=0.In this case I think it is useful for a practicing engineer to have some feeling for why this result is correct. Look at the picture to the right.
I drew what happens if you've got a local max. To get the local min
picture, just flip things over. What should you notice about this
picture? Well, the secant line from the left has a positive slope and
the secant line from the right has a negative slope. IF the derivative at the local max exists, it
will be the limit of things with different signs. The only way this
can occur is if the slope "at" the local max is 0.
Notice, though, that we need information from both sides. This
is why we are considering local maxes and
mins. If we don't have the twosided information, we won't get the
disagreement of signs which compromises (?) at a zero derivative.
Extreme Values on a closed and bounded interval
Here I describe a procedure which provides a method for computing the
absolute max and absolute min of a function in an interval in many
cases, including almost all situations which arise in practice.
Theory and practice
Mr. Theory asserts that
there must be absolute max and min. Ms. Practice says to follow the
process above, and you've got to find the max and min. The advantage
is that in most "real" cases there are only a few critical points, and
the procedure allows you to narrow the search for abs max and min to
only a few numbers.
I will admit that in practice, though, it can be confusing. You look at f(a) and f(b). You compute f´(x). Where is it 0? Where does it not exist? Take those numbers and plug them back into f. So there are two functions, f and f´, wandering around. You need to understand what's necessary.
QotD
This is problem 39 in section 4.2.
Find the maximum and minimum values of f(x)=x{4x}/{x+1} on the
interval [0,3].
Theoretical background The formula defining f makes sense away from x=1. So this function is continuous on the interval [0,3]. This is a closed and bounded interval. Mr. Theory asserts that f must have a max and a min. But Ms. Practice tells us that we only need to check a few values of f.
Endpoints
We need to check f(0)=0 and f(3)=0 (that's 3{12}/4). We also need to
check f's values at its critical points.
Critical point analysis
Step 1 f´(x)=1[{4}(x+1){1}(4x)]/[(x+1)^{2}]=
1[4]/[(x+1)^{2}]. Quotient Rule!
Step 2 When is f´ equal to 0? So
1[4]/[(x+1)^{2}]=0 means
1=[4]/[(x+1)^{2}]=0 means
(x+1)^{2}=4 means
x^{2}+2x+1=4 means
x^{2}+2x3=0 means
(x+3)(x1)=0 so x is either 3 or +1.
Step 3 3 is not relevant (outside our domain) so compute f at
x=+1: f(1)=1(4/2)=1.
Compare values
Compare the numbers 0 and 1 and 0. The max value and the min value
are there. So ("clearly") the max is 0 and the min is 1. You don't
need to even consider f(3) or compute it. The value is
irrelevant.
Rolle's Theorem
If f is continuous on[a,b], and if f(a)=0 and f(b)=0, and if f is
differentiable inside the interval, then there is at least one number
c inside the interval where f´(c)=0.
To the right is a typical picture of the situation described in Rolle's Theorem. Since the function is glued down (?) on the xaxis at a and b, either the function is always 0 (so there are lots of c's) or the absolute max and absolute min occur inside the interval. There may be more than one of each. Since the function is differentiable, these max's and min's occur at critical points where the derivative is 0, as shown.
This result looks very special. Next time I'll show you how it can be changed to apply to very general situations, and the result will be quite useful.
Friday, October 19  (Lecture #14) 

There are help sessions every week for all Math 151 students
at the MSLC (third floor of the ARC building). • Tuesday from 6:40 to 8:40; • Wednesday from 5:10 to 6:40; • Thursday from 6:00 to 7:30. 
I also said that my remarks about grades were directed at all students whose exam scores were low. More emphatically, if you had a low score, this means you (no excuses, no escapes).
Two very useful tricks
The lecture today will discuss two extremely useful "tricks" which are
constantly used, both computationally and theoretically. The relevant
textbook sections are 4.1 and 4.8. Please do not get distressed
at the numbering of the sections. The two techniques are quite related
 they use the basic idea that a tangent line is close to the curve.
Some numerical evidence
I began by presenting some numerical "evidence". Here is a table of
numbers and their square roots which was examined by students. The
table has 20 digit accuracy, far more than any real world application
I know of. But the point of what's given here is to help people think
about things. The table
has a few more entries than I wrote on the board, and since I got it
straight from the computer, it has all 20 digits.
Number  20 digit square root of the number 

4.  2.000000000000000000 
4.1  2.0248456731316586933 
4.01  2.0024984394500785728 
4.001 
2.0002499843769528199 
4.0001  2.0000249998437519531 
4.00001  2.0000024999984375020 
4.000001  2.0000002499999843750 
4.0000001  2.0000000249999998438 
I wanted people to examine the table for patterns. The first column was, of course, 4{some 0's}1. The second column, well, since the square root of 4 is 2, and the numbers in the first column were close to 4, then since square root is a continuous function, the numbers in the second column were close to 2. But what's the pattern of closeness? There are more 0's between the 2 and the first nonzero digit. But the nonzero digits seem to have some pattern. For example, consider the entry corresponding to 4.0001. That entry is 2.0000002499999843750. There are 6 zeros between "." and "2". But the structure of the nonzero digit string following is 24999998 and it sure looks like it wants to be just 25000000. What's going on?
The algebraic way
Well, we written the following a large number of times.
f(x+h)=f(x)+f´(x)h+Err(h)
and we know that the Err(h) term should >0 faster than a
multiple of h. In the case we have above, f(x)=sqrt(x) and we're
considering things that happen what x=4. Well,
f´(x)=(1/2)1/sqrt(x). At x=4, we get f(4)=2 and
f´(2)=1/4=.25 (Professor Greenfield tried several times to make
this into 1/8 but his brain was off this morning). So the equation becomes
sqrt(4+h)=2+(.25)h+Err(h)
If we just want an approximation, hey, we could forget the Err(h) term
and get
sqrt(4+h)2+(.25)h
Here I am using the weird symbol for the phrase "approximately equal" and I am deliberately
not being too precise, right now, about what this means. But
let me rewrite the table above, now with more decorations (!).
Number  20 digit square root of the number 

4.  2.000000000000000000 
4.1  2.0248456731316586933 
4.01  2.0024984394500785728 
4.001 
2.0002499843769528199 
4.0001  2.0000249998437519531 
4.00001  2.0000024999984375020 
4.000001  2.0000002499999843750 
4.0000001  2.0000000249999998438 
Here what is underlined is essentially the 2+(.25)h stuff. Look: the Err(h) term is shrinking twice as fast as the h is shrinking. So for 4.0000001, with 6 zeros, the 2+(.25)h with h=.0000001 appears in 16 places. So there is evidence (yes, I admit it, this is a "toy" problem, and all the numbers are very easy!) that the Err(h) is sort of h^{2}.
Geometric interpretation
What does 2+(.25)h mean? Look at the picture. Everything is not meant
to be totally, literally accurate, but I'd like the ideas to be
correct. The circular area on the graph is magnified. We have a point
(x,f(x)) on the graph. x is moved (I drew the picture so that h is a
small positive number, and x+h is to the right of x). In the local,
magnified picture, there's a right triangle. One leg, the "adjacent",
has length h. The other is labeled ?. The line is tangent to the graph
at (x,f(x)). But the slope of that tangent line is f´(x). The
slope of the line is the tangent of the angle that line makes with
respect to a horizontal line, such as the leg labeled h. So f´(x)
is the quotient of ? and h: f´(x)=?/h. And ? must be f´(x)h,
and this length is added on to the vertical height that (x,f(x)) has
from the xaxis. The result is f(x)+f´(x)h. Hey! Using the
tangent line to make that little approximation is exactly the
same as starting from (x,f(x)) and making the adjustment from x to x+h
by riding along the tangent line. This is why
f(x+h)f(x)+f´(x)h
is frequently called the tangent line approximation. It also
explains why, in our specific case, the approximation is always
greater than the true value, because the parabola on its side is
underneath the tangent line.
All together?
I would like you to see how all of these points of view actually
reinforce each other. As I've remarked in class, I like the pictures,
but I know that many people prefer a formula, and certainly still
others like to see convincing numbers. That's why I've tried to show
you these varied approaches.
What about the error?
I think that Mr. Poling raised a
general and extremely important question. What the heck is the
error like? Is it big or small? What can we say? Well, I first
remarked that we could see in our specific case that as we go from
left to right on the graph, the tangent lines got
flatter (the slopes themselves decreased). Each tangent line
was on top of the curve. The linear approximation was an
overestimate, and (as we will see later) since f´ is
decreasing, the second derivative is negative. The sign of the
error is more or less determined by the sign of the second
derivative. The linear approximation is larger than the true value when
f´´(x)<0 and it is smaller when f´´(x)>0.
I didn't say anything about the size (the magnitude) of the Error.
Irving
Irving is someone who remembers a formula and tries to fit every
situation to a formula. So Irving remembers f(x+h)f(x)+f´(x)h and something
about sqrt(4)=2. He decides to approximate sqrt(100) in the
following manner:
f(x+h)f(x)+f´(x)h
becomes
sqrt(100)=sqrt(4+96)sqrt(4)+(1/4)(96)=26.
Here Irving uses f(4)=4 and f´(4)=1/4 and h=96. I don't think
that 26 is a very good approximation to sqrt(100)=10. Irving may
remember the formula, but I think the formula has been used
inappropriately.
It turns out that the magnitude or size of the Error is roughly proportional to h^{2}. When h is teensyweensy (well, maybe this is too technical) then h^{2} is much teensier than h. This is what was shown above in the table. That we only get a discrepancy of 16 (from 10 to 26) in this case is not as bad as it could be.
Fifth roots
I know that 32^{1/5} is 2. What is a (useful) approximation to
(32+h)^{1/5}? Here f(x)=x^{1/5} and
f´(x)=(1/5)x^{4/5}. So the formula
f(x+h)f(x)+f´(x)h with x=32 becomes
(32+h)^{1/5}32^{1/5}+(1/5)(32)^{4/5}h and (after
arithmetic, since 32^{4/5} is 1/16) this is
(32+h)^{1/5}2+(1/80)h.
For example, 31^{1/5} is reported as 1.987340 by a computer, and the approximation (with h=1) is 31^{1/5}=(321)^{1/5}2+(1/80)(1)=1.9875000. I didn't expect that the approximation would be this good. Notice that f´´(x)=(1/5)(4/5)x^{9/5} so f´´(32)=(1/5)(4/5)32^{9/5} is negative. So just as in the square root example, the estimate is larger  an overestimate.
An implicit problem
The point (2,1) is on the curve defined by the equation
y^{2}=x^{3}3xy+3. (We can check this: plug in x=2
and y=1 and get 1^{2}=1 on the left, and
(2)^{3}3(2)(1)+3=8+6+3=1 on the right.
Probably if we change x to 1.8, there will be another point on the curve. What is the approximate ycoordinate of that point? Well, let's think: if there a curve defined by the equation near (2,1), say, y=f(x), then we know f(2)=1. We're being asked for an approximate value of f(1.8). Now this is f(2+.2), so x=2 and h=.2, and we can use f(1.8)f(2)+f´(2)(.2) as the answer if we knew f´(2).
Computing the derivative is "easy" if you realize it should be done
implicitly. We need to d/dx the whole equation
y^{2}=x^{3}3xy+3, and be careful to use the Chain
Rule and the Product Rule.
2yy´=3x^{2}3y3xy´
Let's insert x=2 and y=1:
2(1)y´=3(2)^{2}3(1)3(2)y´
this gets us 2y´=123+6y´ so that y´=9/4.
Then f(2)+f´(2)(.2) becomes 1+(9/4)(.2)=.55. So the approximate value of the ycoordinate should be .55. To the right is a graph of the curve defined by this equation. Please notice that at (2,1), the tangent line would slope down (that's the minus sign in 9/4). And if we compute carefully, the yvalue turns out to be 0.5886, so the .55 which we got with little effort is not too far off.
A more real problem... One of the early experiments attributed to Galileo is observation of a pendulum. He asserted that the period of a pendulum depended on the length of the pendulum and was independent of the size of the angular oscillation. This is really remarkable and means that maybe you can use a pendulum clock to time things accurately  only the length of the pendulum matters and not how far or wide it swings. It turns out this assertion is actually true only for small oscillations. Galileo supposedly only observed small oscillations. A mathematical model of pendulum motion (go to your physics instructor!) gives d^{2}/dt^{2} =(Constant)sin() (here is the angle between the pendulum and the vertical direction). This differential equation is too darn hard to "solve" (you will see why later) and so people want a simpler model. If is very small, then (since sin(0)=0 and sin´(0)=cos(0)=1) we can substitute for sin() and the equation becomes d^{2}/dt^{2} =(Constant). This equation is much simpler (I'll actually be able to tell you all about it in a few weeks). And the predicted independence of period turns out to be true, with this small oscillation assumption. 
What is the square root of 4?
Sigh. I'm fairly sure that the square root of 4 is 2. Really what I
wanted to discuss was the single most widely used method of taking an
approximation to the root of an equation and improving the
approximation, that is, getting closer to the root. The improvement
technique is called Newton's Method. Here is what I wrote about
Newton's Method in the diary from last year's 151. I used maybe better
notation last year. I called the old guess, G (today I wrote
x_{OLD}), and the (hoped for) improved new guess, N
(where today I wrote x_{New}).
Newton's Method
Newton's method is a way to (try to) improve a guess at a root
of f(x)=0 when f is a differentiable function. The guess, G, is
(hopefully!) improved with the following process (as you read this,
please glance at the picture to the right). First, go up from G
until you "hit" the graph of y=f(x). The point will be (G,f(G)). Then
"slide down" the tangent line of the graph at the point whose slope is
f´(G). The point that this line hits the xaxis is the new guess,
N. The picture is rather simple and shows the new guess closer to a
root of the function. This picture is rather simple, and is the way we
would like the method to work. I will discuss more horrible
possibilities later, but right now I would like to get a formula for N
in terms of G. Well, the slope is f´(G), but this slope is also
equal to OPPOSITE over ADJACENT because the slope of a line is the
tangent of the angle that the line makes with the xaxis (this is the
same key idea we've already used once before in this lecture). Here
OPPOSITE is f(G) and ADJACENT is GN. Therefore
f´(G)=f(G)/(GN) which is the same as GN=f(G)/f´(G), so
that
Newton's method N=G{f(G)/f´(G)} 

For square roots
Here f(x)=x^{2}A and f´(x)=2x, so that
G{f(G)/f´(G)} becomes
G{[G^{2}A]/2G} which is
{2G^{2}[G^{2}A]}/2G which is [G^{2}+A]/2G
which is (1/2)[G+{A/G}].
That is, "Improve the guess to a new guess, N, by taking the average
of the old guess, G, and A/G: N=(1/2)(G+[A/G])."
Here is the complete "coding" (writing of the "program") for Newton's method to compute the square root of 4. Yes, this is silly, but you should see again what the numbers look like.
> N:=x>(1/2)*(x+(4/x));
Comment This defines how to replace the old guess.
> N(3.); 2.1666666666666666667
> N(%); 2.0064102564102564103
Comment % says to use the previous answer as the input.
> N(%); 2.0000102400262144672
> N(%); 2.0000000000262144000
So just 4 iterations (repetitions of N) makes the estimate 3 close enough to 2 that there are ten 0's after the decimal point! This is pretty darn fast, and this is actually how your calculator computes square roots. There is considerable evidence suggesting that this method of computing approximations to square roots has been know for 4,000 years. Try the key words Babylonian square roots in a search engine. Human beings can be quite clever.
The picture to the right is an attempt to show you how Newton's method works geometrically for this specific function (f(x)=x^{2}4) and this specific initial guess of 3. So there's the graph, y=x^{2}4, the parent parabola pushed down 4 units. Take the initial guess at 3 (what's described now is all in magenta) and go up until you hit the parabola. Slide down the tangent line until you hit the xaxis. Then up to the parabola again, slide down the tangent line, etc. This is very silly, but look at the magenta line segments. It is rather difficult to draw them so that they can be seen, because they tend to pile up so quickly at the point where the parabola and the xaxis intersect. This is remarkably fast when it can be used, much faster than bisection.
Dividing is difficult
I mentioned that of the arithmetic things we all learned early in
grade school, certainly division is the one that I and many other
people believe is the most difficult. To find, say, 1/7, you need to
make some guesses at places in the computation, and sometimes the
guesses are just wrong. Some "backtracking" is needed. What the heck
can we do if we want to tell machines how to compute this? Here, very
briefly, is what we can do.
Suppose A is a positive number and we want to compute an approximation
to 1/A. Well, 1/A is a root of f(x)=(1/x)A. Then
f´(x)=1/x^{2}. Notice in all this, please, that
computing derivatives must be totally routine, or else everything
becomes much too hard. Then the Newton's method equation,
N=G{f(G)/f´(G)} becomes
N=G{[(1/G)A]/[1/G^{2}]
and this can be simplified compound fractions to simple fractions):
N=G+[(G)AG^{2}]=2GAG^{2}
This means to compute 1/7 we should do the following: replace an old
guess, G, by a new guess, 2G7G^{2}. Please appreciate how
easy this is compared to long division, with guessing and subtracting
and then doing it again! Here is what happens when a naive person
tries to implement this:
> INV:=x>2*x7*x^2;
Comment This defines how to replace the old guess.
> INV(.1); 0.13
> INV(%); 0.141
> INV(%); 0.1417
> INV(%); 0.14284777
> INV(%); 0.1428571422421897
> INV(%); 0.14285714285714285450
This last answer is accurate to 17 decimal digits. This is almost surely more than any moderately sane human being needs.
QotD
What is 10^{1/3}? More precisely, define a fairly simple
function f which has 10^{1/3} as a root. Then use the initial
estimate of x=2 in Newton's method twice.
Here the function that I thought of is f(x)=x^{3}10. And f´(x)=3x^{2}. The Newton's method would take a guess, x, and replace it by x(x^{3}10)/(3x^{2}).
If the initial guess is x=2, the replacement, which we hope is closer to the true value, would be 2(2^{3}10)/(3[2^{2}]). This is 2(2)/12=2+1/6=13/6. Then replace 13/6 by {13/6}({13/6}^{3}10)/(3{13/6}^{2}) The exact result is (not clearly!) 3277/1521 (yeah, I did not do this by hand!).
How about decimal computation? If the initial guess is 2, then the replacement is 2.1666666667 and that, in turn, becomes 2.1545036160. Sigh. The computer reports that 10^{1/3} is 2.1544346900 so that we seem to have gotten about three decimal places of accuracy. Three iterations gets 8 places of accuracy, and four iterations gets 15 places of accuracy. Is this exciting?
Bad things can happen The Bisection Method is much slower than Newton's Method, but the Bisection Method (assuming that the hypotheses are satisfied!) always works. Newton's Method is notorious for fouling up, sometimes especially when you don't want problems. Let me show you what can happen. Look at the picture to the right. It is a graph of a function, y=f(x). The function has one root, at x=R. Let's examine the effect of various initial guesses.

Tuesday, October 16  (Lecture #13) 

How a rectangle changes
A rectangle is changing with time. At a certain time, its length is 7
inches, and is decreasing at .6 inches per second. At the same time,
its width is 3 inches, and is increasing at .5 inches per second.
Question 1 At that time, is the area of
the rectangle increasing or decreasing? At what rate is the area
changing?
Question 2 At that time, is the length
of the diagonal of the rectangle increasing or decreasing? At what
rate is the diagonal changing?
What I might think about this
Here is how I might think about this problem. I don't know how many of
these details I would write as I analyze the problem, but almost
surely I would think what I write here.
This was the QotD.
Here's a discussion of a solution.
The geometry of this annulus is determined by the radii (that is the
plural of radius, I think). There is an inner, smaller radius which
I'll call r, and an outer, larger radius which I'll call R. The area
of the region is the difference of the area of the region bounded by the
outer circle, Pi R^{2}, and the area of the region bounded by
the inner circle, Pi r^{2}. The area, A, of the annulus,
is given by A=Pi R^{2}Pi r^{2}. The extra
minus sign, together with the two uses of the word "decreasing" in the
problem description make the solution of the problem interesting to
me. I can't easily guess whether the area is increasing or
decreasing.
If we differentiate the equation carefully, the result is dA/dt=2R(dR/dt)2r(dr/dt). At the "certain time", we get dA/dt=2*middot;6(.3)2·4(.2)=3.6+1.6=2 inches per second. So the area of the annulus is decreasing at a rate of 2 inches per second at that time.
An ant crawls on a parabola
An ant is crawling on the parabola y=x^{2}. (Fairly absurd,
but maybe slightly more realistic and useful than the first two
problems.) Suppose that the horizontal, x, coordinate of the plane is
increasing at 6 units per second when the ant crawls through (?) the
point (2,4). How fast is the y coordinate of the ant's position
changing at that time?
So x and y are functions of time and y=x^{2}. If we d/dt the
equation, the result is dy/dt=2x(dx/dt). When the ant is at (2,4),
this becomes dy/dt=2·2·6=24.
A bit more ant crawling
Suppose that the ant crawls on y=x^{2} so that the rate of
change of its first, x, coordinate is always 6 units per second. What
happens to the rate of change of the y coordinate as the x moves far
to the right? I'll try to both compute this and explain it.
Let's "think" about the situation. The picture to the right is to help with thinking. If we take a tiny piece of the parabola, and blow it up, the piece becomes almost a straight line (the parabola is locally linear). For a piece near (2,4), say (similar to the lower piece shown), the line is not too steep. For a piece "far" to the right (as the upper piece shown), the line is rather steep. Imagine the ant traveling in such a manner that the x coordinate changes steadily with a rate of 6 units per second. There will be different y changes to allow the ant to stay on the curve. In the more right, more up box, the y change will be larger than the y change needed in the lower box. So I conclude that dy/dt will grow as the ant moves more to the right.
Now let's look at the algebra. y=x^{2} again implies dy/dt=2x(dx/dt) and if dx/dt is always 6, then dy/dt=12x. Indeed, when x>, dy/dt>.
Change the crawling
What if the ant crawls so that the vertical, y, coordinate changes
steadily at 6 units per second. Again the motion of the ant along
y=x^{2} moves to the right and up. What happens to the
xcoordinate change as the ant moves steadily up?
It might be useful for you to again consider the local linearization pictures, which I hope will tell you in this case that if dy/dt is constant, dx/dt must get smaller as the ant moves to the right. And the algebra: since dy/dt=2x(dx/dt) and here dy/dt is supposed to be 6 always, we see that dx/dt=3/x. As x>, dx/dt>0.
Of course what I am doing is considering the velocity vector of the ant's motion under various hypotheses concerning components of the vector. This velocity vector is tangent to the ant's path, and this geometric fact forces the other conclusions
A conical reservoir is being filled
A right circular cone is an object which shows circular slices
when cut by a plane perpendicular to its axis of symmetry. The sides
of the cones are formed by straight lines through the vertex. The
vertex of the cone is the pointy part.
Now suppose we have a reservoir which is a right circular cone with
its vertex at the bottom. The height of the cone is 30 feet, and the
base radius is 5 feet. The cone is being steadily filled by a fluid at
the rate of .2 cubic feet per minute. How fast is the depth of the
fluid changing when the fluid is 20 feet deep in the reservoir?
Oh mechanical engineers: "A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress." I asked for some suggested fluids and got almost no answers. Sigh.
The pictures show a right circular cone with vertex at the bottom. The oblique view is prettier, I think, but a transverse "section" through the axis is probably better for analyzing this problem. A formula which certainly is needed follows: V=(Pi/3)R^{2}H is the volume of a right circular cone with height H and base radius R. If the fluid level is H and the radius of the fluidfilled volume is R, then the volume filled by the fluid is V=(Pi/3)R^{2}H. We know also that dV/dt=.2 (positive: the cone is being filled). Well, V seems to have too many letters for this course (in calc 3, functions involving several variables are dealt with). But there is geometrically forced relationship between R and H which I hope you see. Look at the transverse picture and at several similar right triangles. Ratios of corresponding sides must be equal for the two right triangles shown, so 30/H=5/R and R=H/6 (I want V in terms of H because the problem statement asks for the rate of change of the depth of the fluid, which I believe is dH/dt). Therefore V=(Pi/3)R^{2}H=(Pi/3)(H/6)^{2}H=(Pi/108)H^{3}. If we d/dt the equation V=(Pi/108)H^{3} being careful to use the Chain Rule appropriately, the result is dV/dt=(Pi/108)(3H^{2})(dH/dt)=(Pi/36)H^{2}(dH/dt). Now plug in the known dV/dt and the depth, H=20, to get .2=(Pi/36)(20)^{2}(dH/dt). Therefore the rate of change of the depth when the depth is 20 feet is .2/[(Pi/36)(20)^{2}] feet per minute.
Please notice that since dV/dt is constant (.2) we know that dH/dt=.2/[(Pi/36)H^{2}]. So we see that as H increases, the rate of change of H decreases. This is because the crosssections of the cone's volume are steadily increasing so the steady volume increase means that the height is not increasing as much. This reminds me of the last workshop a bit.
The first exam was returned.
Answers and a discussion of the grading are
available.
Friday, October 12  (Lecture #12) 

Tuesday, October 9  (Lecture #11) 

The Lambert W function
What happens if we wanted so "solve" the equation
xe^{x}=SOME NUMBER? This seems a bit
silly, but in fact in certain areas of computer science actually
occurs. What we are doing is asking if the line
y=SOME NUMBER intersects the graph we've just
drawn. Or, if we really wanted to study this systematically, we are
looking at the inverse of the graph we've just drawn. Take
y=xe^{x} and flip it over the main diagonal, y=x (which also
happens to be tangent to the curve at x=0, since the derivative,
(1+x)e^{x}, is 1 at x=0). The result, which is shown in the
middle graph below, is not the graph of a
function. Look at the vertical red line, which intersects the curve
twice. So the curve fails the vertical line test. What can we do?
Well, we can chop off a chunk of the curve. People have chosen the
chunk inside the bluegreen cloud as the part of the graph they want
to throw away. The result, in the third picture (on the right!)
is the graph of a function. It is called the Lambert W function
and frequently written W(x).
The following quote is from this
reference:
Banwell and Jayakumar (2000) showed that a Wfunction describes the relation between voltage, current and resistance in a diode, and Packel and Yuen (2004) applied the Wfunction to a ballistic projectile in the presence of air resistance. Other applications have been discovered in statistical mechanics, quantum chemistry, combinatorics, enzyme kinetics, the physiology of vision, the engineering of thin films, hydrology, and the analysis of algorithms.From the picture, I hope that you can see the domain of W is [1/e,+) and the range of W is [1,+). It is certainly one heck of a weird function.
>plot(LambertW(x),x=1/exp(1)..6,thickness=2,color=black);
Now let's discuss the collection of inverse functions that most people know about.
arcsine
What's happening in the pictures below (left to right):
The first
picture is supposed to be a portion of the graph of sine. It is 2Pi
periodic, and its range is [1,1]. The green line is the "main
diagonal", y=x, which also happens to be tangent to y=sin(x) at
(0,0). This is because the slope of the tangent line is the derivative
of sine, which is cosine, and cos(0)=1. To get the inverse function,
we interchange inputs and outputs. Geometrically we flip the graph
over the main diagonal, and get the second picture. The tangent line
is still tangent, but now, look at the red line. This demonstrates
that the flipped graph is not the graph of a function. It fails
the vertical line test to be a graph of a function. Thus we need to
cut away (!) part of the graph. The "clouds" in bluegreen (?)
demonstrate what will be cut away. And what's left is shown in the
third picture. This is the official graph of y=arcsin(x): domain
[1,1] and range [Pi/2,Pi/2]. It has arcsin(0)=0, and the tangent
lines seem always to slope up, so the derivative should be
positive. And if we are very careful, the lines tangent to sine at
+/Pi/2 are horizontal, so the lines tangent to the flipped curve will
be vertical and have no slope so there will be no derivative at
+/1. The derivative of arcsin should have domain (1,1), the interval
without endpoints.
Consider this process:
Example The derivative of arcsin(5x^{3}e^{x}) is [1/sqrt(5x^{3}e^{x})](5·3x^{2}e^{x}) using the chain rule. 
arctan
What's the picture below
supposed to show? The initial picture is y=tan(x). This function is
periodic with period Pi, and its domain does not include odd
multiples of Pi/2. The function is rather simple looking (!), always
tilted up, and has vertical asymptotes at odd multiples of
Pi/2. Flipping to get an attempted inverse function reveals lots of
problems (I omitted the red line here). The standard restriction is to
throw out the "branches" that don't intersect the horizontal axis, and
that's what I've attempted to suggest with the bluegreen
"clouds". Again, y=x is a tangent line to both arctan and tan at (0,0)
because the derivative of tangent is (sec(x))^{2}, and
sec(0)=1/cos(0)=1. Arctan is very useful. It "compresses" all of the
real numbers into the interval from Pi/2 to Pi/2, so if you have lots
of data and you don't know ahead of time how big (+ or ) the data
will be, composing it with arctan will at least control it a bit. Now
for the derivative. Notice that the derivative of arctan should >0
as x>+ or as x>, since the curve gets flatter there.
Examples The derivative of arctan(e^{(5x4)}) is
1/(1+(e^{(5x4)})^{2}·
e^{(5x4)}·5·4x^{3}: somewhat
of a mess, along with several uses of the chain rule.
The derivative of arctan([x+1]/[x1]) is 1/(1+{[x+1]/[x1]}^{2})·{[1(x1)(x+1)1]/[(x1)^{2}]} 
arcexp (?)
Should this be called arcexp? Well, it isn't. The inverse
function to an exponential function is a logarithmic
function. This logarithm function is very important. The log
functions you may deal with include log_{10} (used in the
definition of pH, and for hand calculation in "the old days") and
log_{2} (used in some computer science applications).
This
picture shows exp, the exponential function,
e^{x}. Since this function is onetoone, its inverse will be
a function: no more bluegreen clouds! The green line is the tangent
at (0,1) which has slope=1. Then we flip it and get the graph of
ln. What about the derivative?
Examples If f(x)=ln(x^{3}5x+78.6), then
f´(x)=(1/[x^{3}5x+78.6])(3x^{2}5). If f(x)=sec(ln(2x+5)), then f´(x)=sec(ln(2x+5)tan(2x+5)(1/[2x+5])2. 
a^{x}
What if you needed to find the derivative of 10^{x}? Well, if
y=10^{x}, you might do the following:
An absurdity: (sin x)^{x}
Suppose you wanted the derivative of (sin x)^{x}, a
fairly silly function. Here is a way to compute it.
Function  Derivative 

x^{n}  nx^{n1} 
CONSTANT  0 
e^{x}  e^{x} 
f(x)+g(x)  f´(x)+g´(x) 
f(x)·g(x)  f´(x)·g(x)+f(x)·g´(x) 
CONSTANT(f(x))  CONSTANT(f´(x)) 
1/f(x)  f´(x)/[f(x)]^{2} 
f(x)/g(x)  [f´(x)g(x)g´(x)f(x)]/[g(x)]^{2} 
sin(x)  cos(x) 
cos(x)  sin(x) 
tan(x)  [sec(x)]^{2}=1/[cos(x)]^{2} 
sec(x)  sec(x)tan(x) 
f(g(x))  f´(g(x))g´(x) 
arcsin(x)  1/sqrt(1x^{2}) 
arctan(x)  1/(1+x^{2}) 
ln(x)  1/x 
a^{x}  a^{x}ln(a) 
In mathematics you don't understand things. You just get used to them. 
disambiguation is defined as "clarification that follows from the removal of ambiguity".
QotD
Write an equation for the line tangent to y=arctan(x) when x=1. Also,
graph both the tangent line and y=arctan(x) together.
I need a POINT and a
SLOPE. So:
I know arctan(1)=Pi/4. So the POINT is (1,Pi/4).
Since arctan´(x)=1/(1+x^{2}), the SLOPE
is 1/(1+1^{2})=1/2.
An equation for the line is yPi/4=(1/2)(x1). A version of the
desired graph is shown to the right. The line should be tangent
to the arctan graph at the point (1,Pi/4).
Friday, October 5  (Lecture #10) 

Three chain rule examples with tabular information about a function
x  f(x)  f´(x)  f´´(x) 

1  2  0  2 
2  3  6  5 
3  7  3  4 
4  2  5  7 
A "simple" tangent line problem
Suppose we consider x^{2}+y^{2}=1, the unit circle
(center (0,0) and radius 1). What is an equation for the line tangent
to this circle when x=1/2 and the point of tangency is in the upper
half of the circle.
I just looked up "tangency" and it is a word. It means "the
state of touching". So I could have said, solving for y, that the line
is tangent to the circle at y=sqrt(3)/2, but using an infrequent word
is ... silly and therefore I did it.
More interestingly, I remarked that I knew three or four ways of computing the answer to this problem. The problem itself is, again, not profound. But knowing a variety of ways to solve it is extremely useful. Sometimes one way or the other can't be used. Enough data for the solution is: a POINT, which is (1/2,sqrt(3)/2), and a SLOPE, which is what I will concentrate on. So the answer will be (ysqrt(3)/2)=SLOPE(x1/2), and we'll compute the SLOPE.
First solution
Since x^{2}+y^{2}=1, we solve for y as a function of
x. Thus, y^{2}=1x^{2} and
y=+/sqrt(1x^{2}). We need to figure what +/ means? Here
since I specified "in the upper half of the circle" there is little to
worry about. "upper" means +, so y=sqrt(1x^{2}). The Chain
Rule now allows us to compute the derivative. So
dy/dx=(1/2)(1x^{2})^{1/2}(2x), and when x=1/2, this
is (1/2)(1(1/2)^{2})^{1/2}(2(1/2)). That number is
the SLOPE we need.
Second solution
Well, if x^{2}+y^{2}=1, I could try d/dx'ing the whole
equation. In this I will be guided by the following idea: that somehow
the equation defines y as a function of x. But, right now (forgetting
the first solution above!), I don't know the function. So y is some
unknown function of x. Let me d/dx the equation. The righthand side
is easy, since the derivative of a constant is 0. What about the
lefthand side? It is the sum of two functions, x^{2} and
y^{2}. The derivative will be the sum of two derivatives. The
derivative of x^{2} is 2x. What about y^{2}? y is some
unknown function of x. So I will use the Chain Rule, and I
can't assume anything about the function y:
d/dx(y^{2})=2y(dy/dx).
I think of y^{2} as a composition. The "outside" function is
squaring, and the "inside" function is y, the unknown function. The
result of differentiation is 2y multiplied by the derivative of the
unknown function. So put all this together.
The derivative of
x^{2}+y^{2}=1 is 2x+2y(dy/dx)=0.
Now we can solve for dy/dx: 2y(dy/dx)=2x and so
dy/dx=(2x)/(2y)=x/y.
In our specific situation, x=1/2 and y=sqrt(3)/2, so
dy/dx=(1/2)/(sqrt(3)/2))=1/sqrt(3). You can check that the previous
answer is actually the same number as this.
Key assumption
The key assumption in the second method is that the equation
defines, somehow, y as a function of x, and then that this
function is differentiable. It's possible to prove this statement
under circumstances which commonly arise in Math 151. The customary
practice is just to assume without further mention that the
assumption is true. Sometimes things can get weird if the assumption
is not valid, but this rarely happens.
Language
In the first method above, we solved for y as a function of x. People
say that y is an explicit function of x. The second method
has y defined by the equation it sits in. People say that y is defined
implictly. A dictionary definition for the word "implicit" is
"implied though not directly expressed; inherent in the nature of
something". The differentiation trick above is called implicit
differentiation and people use it whenever it is difficult or
impossible to solve for y as a function of x.
A bounding box problem
The points satisfying the equation x^{2}xy+y^{2}=3
can be sketched by a graphing device. One picture is shown to the
right. The result is actually a tilted ellipse. People interested in
various kinds of graphics applications may want to find a bounding
box for this ellipse (actually, the bounding box shown was made by
such a graphics program, independently of the program which graphed
the ellipse!). This would be the smallest rectangle which contains the
ellipse. There's a great deal of symmetry in this figure so the
computations can actually be abbreviated, but I'd like to proceed in a
rather simple fashion. How can we find the line segments of the top
and bottom of the box? Well, they are part of two lines tangent to the
ellipse which are both horizontal. So the slope of these tangent lines
is 0. I'd like to find points on the ellipse where y´=0. While I
could solve x^{2}xy+y^{2}=3 for y as a
function of x (it is a quadratic equation in y and we should know how
to find solutions of such equations) I think that an explicit solution
may be irritating to work with. Instead let me d/dx the equation.
x^{2}xy+y^{2}=3 gives
2x(1·y+xy´)+2yy´=0.
How was this done? Well, the derivative of x^{2} is 2x. The
derivative of xy: this is a product, and I'll use the product
rule. The result is 1·y+xy´. Of course the y^{2}
I'll handle just as in the circle example, and the result is
2yy´. Now let me "solve" for y´:
2x(1·y+xy´)+2yy´=0.
xy´+2yy´=2x+y
(x+2y)y´=2x+y
y´=(2x+y)/(x+2y)
This is 0 (horizontal tangent line) when 2x+y=0, or y=2x. But I want
to find points on x^{2}xy+y^{2}=3 also satisfying
y=2x. This means x^{2}x(2x)+(2x)^{2}=3 so
3x^{2}=3 so x=+/1. Why are there two solutions? Of course
because there are two points, and if x=+/1 then y=+/2 (since y=2x)
and the points of contact of the two horizontal tangent lines are
(1,2) and (1,2).
What about the line segments which are vertical? One way of thinking about them is that those values of x and y are exactly values where the dy/dx formula fails, so that there is no slope. Since dy/dx=(2x+y)/(x+2y), I know this means x+2y=0. (You see, we are getting information where implicit differentiation doesn't work!) If x+2y=0 then 2y=x and we want to know where on the ellipse x^{2}xy+y^{2}=3 we have 2y=x. This is (2y)^{2}(2y)y+y^{2}=3 or 3y^{2}=3 so y=+/1 and x=+/2. These two points are (2,1) and (2,1).
I hope that you can see the lines whose segments form the bounding box: x=2 and x=2 and y=2 and y=2.
QotD
This is part of problem 38 of section 3.8 of the text. I asked: what
is an equation of the line tangent to
y^{4}+xy=x^{3}x+2 at (1,1)?
Notice that if you plug in x=1 and y=1 into this equation, the result is 1^{4}+1·1=2 on the left and 1^{3}1+2=2 on the right. I always do this since I gave an exam problem where the point of tangency wasn't on the curve. Sigh.
A line is determined by a POINT on it and the
SLOPE of the line. Certainly (1,1) is on the line. For
the slope, we'll differentiate the equation
y^{4}+xy=x^{3}x+2 implicitly. So I d/dx the equation
and will follow all of the differentiation rules.
d(y^{4})/dx=4y^{3}(dy/dx); d(xy)/dx=1y+x(dy/dx)
(Product Rule and Chain Rule);
d(x^{3}x+2)/dx=3x^{2}1. Here is the result:
4y^{3}(dy/dx)+1y+x(dy/dx)=3x^{2}1
So I solve for dy/dx:
(4y^{3}+x)(dy/dx)=3x^{2}1y
dy/dx=(3x^{2}1y)/(4y^{3}+x)
And insert x=1 and y=1 to get dy/dx=(311)/(4+1)=1/5 at (1,1). Now we
can just write (y1)=(1/5)(x1), which is a valid equation of the
tangent line.
I would not give this as an exam question, since the numerical data is too forgiving. I mean that if the solution has incorrect powers (y instead of y^{3}, for example) the actual answer will still be correct. Such errors in solutions of test questions should signal themselves very emphatically!
Maintained by greenfie@math.rutgers.edu and last modified 10/5/2007.