Question 1

Given the graph in Image 1, which of the following statements is TRUE about the critical points (green dots)?

Accepted Answer

The middle critical point is a local minimum.

Question 2

Using the table in Image 2, what is the value of $$p'(2) if$$ $$p(x) = f(x) \cdot g(x)$$?

Accepted Answer

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

Question 3

For the function $$f(x) = x^4$ - 2x^3$, what is the point of inflection?

Accepted Answer

$$x = 3$$

Question 4

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

Accepted Answer

$$MC(q) = 300 \cdot \frac{1}{q}$$

Question 5

If a population of bacteria grows continuously at a rate of 6% per hour and starts with 100 bacteria, what is the function for the population at time $$t$$ hours?

Accepted Answer

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

Question 6

Using the table in Image 2, what is the value of $$q'(1) if$$ $$q(x) = \frac{f(x)}{g(x)}$$?

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 7

Given the revenue function $$R(q) = 5q - 0.001q^2$ and cost function C(q) = 300 + 2.5q$, what production level maximizes profit?

Accepted Answer

$$q = 1500$$

Question 8

For the function $$f(x) = -x^3$ + 3x^2$ - 2$$, what is the sign of the first derivative at $$x = -1$$?

Accepted Answer

Negative

Question 9

If $$A(t) = P_0$ \cdot e$$^{rt}$$ represents continuously compounded interest, what is the value of t$ when A(t) = 20$$,000$$, P_0$ = 12,000$$, and $$r = 0.0516$$?

Accepted Answer

$$t = \frac{\ln(\frac{20,000}{12,000})}{0.0516}$$

Question 10

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

Accepted Answer

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

Question 11

For the function $$f(x) = 2x^3$ - 3x^2$ - 36x$$, what are the critical values?

Accepted Answer

$$x = 0, 3, -6$$

Question 12

If $$f(x) = x e$$^{-3x}$$, use the first derivative test to determine if x = 0$ is a local maximum, minimum, or neither.

Accepted Answer

Local minimum

Question 13

Given the table in Image 2, what is the value of $$h'(4) if$$ $$h(x) = g(f(x))$$?

Accepted Answer

$$h'(4) = g'(f(4)) \cdot f'(4) = g'(4) \cdot 3 = 1 \cdot 3 = 3$$

Question 14

For the function $$f(x) = x^4$ - 2x^3$, what is the second derivative f$$''(x)$$?

Accepted Answer

$$f''(x) = 12x^2$ - 12x$$

Question 15

Given the graph in Image 3, which interval is the function increasing?

Accepted Answer

From the local minimum to the rightmost point

Question 16

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

Accepted Answer

$$p'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$

Question 17

For the function $$f(x) = -x^3$ + 3x^2$ - 2$$, what is the sign of the second derivative at $$x = 3$$?

Accepted Answer

Positive

Question 18

If $$f(x) = 3x^2$ - 1$$, what is the value of the definite integral $$\$$int_1$$^3 f(x) dx$$?

Accepted Answer

$$\$$int_1$$^3 (3x^2$ - 1) dx = [x^3$ - x]_1^3$ = (27 - 3) - (1 - 1) = 24$$

Final Exam Review: Calculus and Financial-Accounting Applications

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