Math 105 Exam II: Derivatives and Antiderivatives, Exams of Calculus

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Exam II, page 1. February 10, 2006

1. The derivative of a certain function f is f′^ (x) = (cos x −

3 /2)(x − 3 .5) on the interval [− 1 , 5]. a. Sketch the graph of f′^ (x) here. The vertical window is [-2,2].

Note that the following questions are about f(x), not f′(x) b. What are the exact values of the x coordinates where f has horizontal tangents (i.e. stationary points)?

c. On which intervals is f increasing? (use exact values for your endpoints)

d. By tracing, estimate to one decimal place the x coordinates of the inflection points of f.

e. On which intervals is f concave down? (Use estimates as in (d)).

2. Suppose g(x) = ex^ + 4x^3 + 2 sin x.

a. Find the “most general” antiderivative of g.

b. Find the specific antiderivative of g which has 6 as its y-intercept.

Exam II, page 3. February 10, 2006

4. Suppose the line tangent to the graph of f(x) at x = 4 has equation y−5 = 3(x−4) while y = 7x− 34 is the equation of the line tangent to g(x) at x = 4.

a. What is the equation of the line tangent to the graph of q(x) = f(x)g(x) at x = 4?

b. What is the equation of the line tangent to the graph of s(x) = 10f(x) + 20g(x) at x = 4?

5A. Note that both the numerator and denominator approach 0 as h approaches 0 in the expression

sin 3h/ 5 h. Make a table with h = 0.1,0.01, and 0.001 which suggest what lim h→ 0

sin 3h 5 h

is. So what is the limit??

5B. What does the answer to 6A suggest lim h→ 0

sin ah bh

is in general, for any numbers a and b (b 6 = 0).

Exam II, page 4. February 10, 2006

6A. Use the definition of the derivative to show that the derivative of f(x) = 2x^3 − 5 x is f′^ (x) = 6x^2 − 5. Note that (A + B)^3 = A^3 + 3A^2 B + 3AB^2 + B^3.

6B. What is the average rate of change of f from x = 3 to x = 3.2?

6C. What’s the instantaneous rate of change of f at x = 3?