Math 1210 Quiz 6 February 21st, 2014

Math 1210
Quiz 6
February 21st, 2014
Answer the following two (2) questions. The value of every question is
indicated at the beginning of it. You may use scratch paper, but you can
only turn in this sheet. Please write your answer in the space provided. You
have 20 minutes.
1. (15 points) Consider the function f (x) =
cos x
1+sin x
over the interval (0, 2π).
1. (2 point) What is the domain of f ?
2. (4 points) Where is f increasing (resp. decreasing)? Namely, for which x is f 0 (x)
positive (resp. negative)?
3. (4 points) What are, if any, the inflection points?
4. (3 points) Where is f concave up (resp. down)? Namely, for which x is f 00 (x)
positive (resp. negative)?
5. (2 points) Are there any extreme points?
2. (10 points) Out of all the rectangles with diagonal D, what are the dimensions of the
one with the largest area?
1. (10 points) Find the area of a rectangle with diagonal D as a function of one single
variable A(x). What is the domain of this function?
2. (2.5 points) Find A0 (x) and determine at which point it is 0.
3. (2.5 points) Show that the point you found in (2) is indeed a maximum (in this
case, it might be easier to look at the sign of A0 (x) than to compute A00 (x)).
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