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MAT 135: Calculus Assessment 2

This is a 50-minute timed exam assessing your mastery of derivatives and their applications. You

may use a calculator; but all other implements and notes are prohibited. Please write your solutions

out neatly and in numerical order on separate sheets of paper.

Section I: Conceptual Knowledge 24 points

(4 points each) Below is the graph ofy, the derivative of a function y = f(x). To repeat: The graph yousee below is NOT f(x) but rather its derivative, f(x) (also denoted dy/dx).

1 0 1 2 3 4 520

10

0

10

20

30

40Graph of dy/dx

1 0 1 2 3 4 520

10

0

10

20

30

40

x

dy/dx

Based only on the graph ofdy/dx above, answer each of the following and give a one-sentence explanationof your reasoning for each. WARNING: Remember that the graph above IS NOT THE GRAPH OF f,the original function. Any answers to these questions that assume that the graph above is f, the original

function, will be given no credit.

1. Suppose that x is measured in dollars and f, the original function, is measured in hours. At whatrate is f, the original function, changing when x = 3, and what are the units of this quantity?

2. What are the critical numbers off, the original function?

3. On what interval(s) is f, the original function, increasing? Decreasing?

4. Of the critical numbers you found, which give local maximum values on f, the original function? Which

give local minimum values?

5. On what interval(s) is f, the original function, concave up? Concave down? And what (if anything)are the inflection points of f?

6. Where (if anywhere) does f, the original function, attain an absolute maximum value on the interval[1, 4]? If there is no absolute maximum value on this interval, or if there is not enough informationto answer the question, say so and explain.

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Section II: Calculations 26 points

7. For each of the following, calculate y. Show all work and simplify completely.

(a) (5 points) y =ex

2

x2

(b) (4 points) y = (x

4

2x

2

+ 5)

3

(c) (5 points) y = ln(x4 + 1)

8. (6 points) Find an equation of the tangent line to the curve traced by x2 + 4xy + y2 = 13 at the point(2, 1).

9. (6 points) Find the critical numbers ofy = x2 ln x.

Section II I: Problems to Solve 50 points

10. (16 points) A rectangular swimming pool is to be built with an area of 180 square feet. The ownerwants 5-foot wide decks along either side and 10-foot wide decks at the two ends. Find the dimensions

of the smallest piece of property on which the pool can be built satisfying these conditions1

.

11. The number (N) of people who have heard a rumor spread by mass media at time t is given by:

N(t) = 200000

1 e0.1054t

where t is measured in days.

(a) (10 points) At what rate is the rumor spreading after 2 days? Show all work and put the correctunits on your answer.

(b) (6 points) Show mathematically that N is always increasing.

(c) (18 points) Find the interval on which N is concave up and the interval on which N is concavedown, and locate the inflection point. Then explain what this inflection point means in terms of

the number of people hearing the rumor.

1Problem taken from Single Variable Calculus, Third Edition by Hughes-Hallett, et al.

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