Question 1

Given the function $$f(x) = 4x on $$the interval $$x = 1 to$$ $$x = 16$$, what is the average rate of change?

Accepted Answer

$$4$$

Question 2

Joanna spends $$70 to rent a table at the Flea Market. It costs 0.50 to $$make a small bag of popcorn and she sells it for $$3. What is the profit function P(q$$)$$, where q$ is the number of bags sold?

Accepted Answer

$$P(q) = 3q - (70 + 0.5q)$$

Question 3

A population of bacteria grows exponentially at 6%. If there were 100 bacteria initially present, what is the function for the population at time $$t$$ hours?

Accepted Answer

$$P(t) = 100 \cdot e$$^{0.06t}$$

Question 4

Given the table below, what is $$p'(2) if$$ $$p(x) = f(x) \cdot g(x)$$?

Accepted Answer

$$p'(2) = f'(2) \cdot g(2) + f(2) \cdot g'(2) = 4 \cdot 1 + 2 \cdot 2 = 4 + 4 = 8$$

Question 5

If $$A(t) = 12000 \cdot e$$^{0.0516t}$$, after how many years will the investment reach 20$$,000$$?

Accepted Answer

$$t = \frac{\ln(\frac{20000}{12000})}{0.0516}$$

Question 6

Given $$f(x) = -x^3$ + 3x^2$ - 2$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = -3x^2$ + 6x$$

Question 7

For the function $$f(x) = x^4$ - 2x^3$, find the point(s) of inflection.

Accepted Answer

$$x = 0$$ 	ext{ and } x = 3$

Question 8

Given the cost function $$C(q) = 200 + 300 \cdot \ln q$$, what is the marginal cost function?

Accepted Answer

$$C'(q) = \frac{300}{q}$$

Question 9

If the demand curve is $$p = 10(3 - q)$$, what is the revenue function $$R(q)$$?

Accepted Answer

$$R(q) = 10(3 - q)q$$

Question 10

Given $$f(x) = 8x^3$ - x^2$ and g(x$$) = \ln x$$, what is the derivative of p(x) = f(x$$) \cdot g(x)$$?

Accepted Answer

$$p'(x) = (8x^3$ - x^2$) \cdot \frac{1}{x} + (24x^2$ - 2x) \cdot \ln x$$

Question 11

For the function $$f(x) = x e$$^{-3x}$$, use the first and second derivative tests to determine if a critical value is an extrema.

Accepted Answer

A critical value is an extrema if $$f'(x) = 0$$ and $$f''(x) is $$positive (minimum) or negative (maximum) at that point.

Question 12

Given the graph of $$f(x) ($$first image), which of the following statements is true about the critical points?

Accepted Answer

The function has two local extrema and one inflection point.

Question 13

Given the table (second image), what is $$q'(1) if$$ $$q(x) = \frac{f(x)}{g(x)}$$?

Accepted Answer

$$q'(1) = \frac{f'(1) \cdot g(1) - f(1) \cdot g'(1)}{[g(1)]^2} = \frac{1 \cdot 2 - 3 \cdot 4}{2^2$} = \frac{2 - 12}{4} = -2.5$$

Question 14

For the function $$f(x) = 2x^3$ - 3x^2$ - 36x on $$the interval $$[-3, 4]$$, what are the global maximum and minimum values?

Accepted Answer

Global maximum and minimum occur at critical points and endpoints.

Question 15

If the marginal cost function is $$MC = 5q^2$ - 2q + 1$$ and fixed costs are $$200$$, what is the total cost function $$C(q)$$?

Accepted Answer

$$C(q) = \int (5q^2$ - 2q + 1) dq + 200 = \frac{5}{3}q^3$ - q^2$ + q + 200$$

Question 16

Given $$f(x) = x + 2$$ and $$g(x) = x$$ for $$-1 \leq x \leq 2$$, what is the area between the two curves?

Accepted Answer

$$\int_{-1}^{2} [(x + 2) - x] dx = \int_{-1}^{2} 2 dx = 2 \cdot (2 - (-1)) = 6$$

Question 17

If $$f(x) = \ln(2x - 5)^3$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = \frac{6}{2x - 5}$$

Question 18

Evaluate the definite integral $$\int_{1}^{3} (3x^2$ - 1) dx$$ using the Fundamental Theorem of Calculus.

Accepted Answer

$$[x^3$ - x]_{1}^{3} = (27 - 3) - (1 - 1) = 24$$

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