Q: A company’s profit is given by $$P(x) = R(x) - C(x)$$, where $$R(x) is $$revenue and $$C(x) is $$cost. If $$R(x) = 50x$$ and $$C(x) = 10x + 200$$, what value of $$x$$ maximizes profit?

Profit increases with $$x$$ and is maximized as $$x \to \infty$$

Question 1

Suppose the graph of $$f(x) is $$given below. On which interval is $$f'(x)$$ negative and $$f''(x)$$ positive?

Accepted Answer

$$(-2, 0)$$

Question 2

Which of the following statements about the function $$f(x) = x^3$ - 3x^2$ + 2 is $$TRUE?

Accepted Answer

It has an inflection point at $$x = 1.$$

Question 3

Which of the following is the correct horizontal asymptote for $$f(x) = \frac{2x^2$ - 3x + 1}{x^2$ + 1}$$?

Accepted Answer

$$y = 2$$

Question 4

Solve for $$x$$: $$5e^{2x} = 20$

Accepted Answer

$$x = \frac{1}{2} \ln(4)$$

Question 5

A company’s profit is given by $$P(x) = R(x) - C(x)$$, where $$R(x) is $$revenue and $$C(x) is $$cost. If $$R(x) = 50x$$ and $$C(x) = 10x + 200$$, what value of $$x$$ maximizes profit?

Accepted Answer

Profit increases with $$x$$ and is maximized as $$x 	o \infty$$

Question 6

Which of the following is the correct derivative of $$f(x) = e$$^{3x^2}$$?

Accepted Answer

$$f'(x) = 6x e$$^{3x^2}$$

Question 7

Use logarithmic differentiation to find the derivative of $$y = x^x.$$

Accepted Answer

$$y' = x^x (\ln x + 1)$$

Question 8

A radioactive substance decays according to $$Q(t) = Q_0$ e$$^{-kt}$$. If the half-life is 10 days, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{10}$$

Question 9

Which of the following is TRUE about the inflection points of a function $$f(x)$$?

Accepted Answer

They occur where $$f''(x)$$ changes sign.

Question 10

A rectangular garden is to be built in a corner where two brick walls meet. If 40 meters of fencing is available for the other two sides, what is the maximum area that can be enclosed?

Accepted Answer

Question 11

Given $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 12

Which of the following is the correct vertical asymptote for $$f(x) = \frac{2x^2$ - 3x + 1}{x^2$ + 1}$$?

Accepted Answer

None; there is no vertical asymptote.

Question 13

If $$f(x) = x^4$ - 4x^3$ + 6x^2$, at which x$-value does f(x$$)$$ have a local minimum?

Accepted Answer

$$x = 2$$

Question 14

Which property of logarithms is used to simplify $$\ln(ab^2$)$$?

Accepted Answer

$$\ln(ab^2$) = \ln a + 2\ln b$$

Question 15

If $$y = e$$^{2x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2e^{2x}$$

Question 16

A population of bacteria doubles every 3 hours. If the initial population is 100, what is the population after 9 hours?

Accepted Answer

$$800$$

Question 17

Which of the following is the correct equation for the tangent line to $$y = x^2$ at x = 1$?

Accepted Answer

$$y = 2x - 1$$

Question 18

If $$f(x) = \ln(x)$$, what is the domain of $$f(x)$$?

Accepted Answer

$$(0, \infty)$$

Question 19

Which of the following is the correct solution to $$e^{2x} = 7$?

Accepted Answer

$$x = \frac{1}{2} \ln 7$$

Question 20

If $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, what is the $$x-$$coordinate of the inflection point?

Accepted Answer

$$x = 2$$

Calculus II Exam Study Guide: Differentiation, Applications, Logarithms, Exponentials, and Optimization

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