## Hints for the review problems for the fist semitest in Math 135, section F2, summer 2006

Below are some hints for the review problems for the first "semiexam". I encourage you also to get together with other students or to exchange e-mail with them.

1. All of the properties given provide restrictions on the graph to be drawn. The strictest restriction (not supposed to be a pun, darn it!) is provided in e) where you are told that, away from 2 and 6, limits are the values of the functions at the points. This declares that there aren't any "holes" or other misbehavior at those other points: no jumps, breaks, etc. So restriction e) is an effort to make drawing the graph less random.

2. We've done a number of problems exactly like this in class and others are done in the text. Compute f(x+h) and f(x), then look at the fraction {f(x+h)-f(x)}/h which results. If you try to just "plug in" 0 for h you'll wind up with a division by 0 expression. Therefore you need to transform the quotient (which is actually a compound fraction) algebraically and somehow get another expression, equal to the original, for which the limit h-->0 can be evaluated easily. Combine fractions, do some algebra, and things will work out.

4. The derivative is the slope of the tangent line. The problem should be almost no work! You don't need to verify the assertion about what f´(x) is. In fact, we are officially not very sophisticated computationally yet, and this verification would be a horrible mess using what we know so far. I hope that later you will see the result is not difficult with additional ideas. So for right now: find a point on the tangent line and find the slope of the tangent line, and then write an equation for the tangent line.

6. This is really a pre-calc problem. Find an equation for the line AB (you'll need to get m and b for this line). Finad an equation for the line CD (you'll need to get m and b for this line). Then you'll have two equations for x and y: solve these two linear equations in two unknowns by, say, subtracting one from the other (that will eliminate the y variable). We did this at least once in class.

9. c and d: this is difficult. You don't know very much about h(x). There are three alternatives: compute the limit, assert the limit does not exist, or declare that you can't determine the result from the information given. In c) knowing just that h(x) is somewhere between -304 and +304 doesn't provide you with much information. Can you come up with two examples satisfying the 304 restrictions, one where the limit shown does exist and one where it does? If you can, then maybe the third alternative ("...you can't determine the result from the information given") is correct. In d) the function being considered is 6+(x-5)h(x). What happens as x-->5? Well,the 6 sure doesn't change, and the x-5 approaches What? so the product of, say, 304 and x-5 approaches What? ... and then ....

10. The problem a) and b) and c) should all follow from the discussion Thursday about sin(h)/h. The idea in there is that sine of a very small angle, measure in radians, divided by that angle (again measured in radians), will approach 1 as the angle measure goes to 0. In b) what can you tell me about how sine looks like near 2pi compared to how it looks near 0?
d) is slightly different. It has a lot of what test writers call distracters in it. What happens inside the sine function is not too important here. The output of the sine mess is some stuff between -1 and +1. It is being multiplied by x-2 as x-->2. It sure looks like it is being squeezed between |x-2| and -|x-2| as x-->2. Indeed.