Question 1

Which of the following statements best describes the behavior of the function $$f(x) = b^x$$ for $$b \u003e 1$$?

Accepted Answer

The function is increasing for all $$x.$$

Question 2

Given the function $$f(x) = 0.5^x$$, which of the following best describes its long-term behavior as $$x 	o \infty$$?

Accepted Answer

The function approaches zero.

Question 3

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

Accepted Answer

$$3^x \ln 3$$

Question 4

If $$y = \log_b x$$, what is the domain of this function?

Accepted Answer

$$(0, \infty)$$

Question 5

Evaluate $$\int 2^x dx.$$

Accepted Answer

$$\frac{2^x}{\ln 2} + C$$

Question 6

Which of the following is the correct inverse of $$f(x) = b^x$$ for $$b \u003e 0$$, $$b 
eq 1$$?

Accepted Answer

$$f^{-1}(x$$) = \log_b x$$

Question 7

A radioactive substance decays according to $$m(t) = 1000 e$$^{-t/850}$$. When does the mass reach 1 gram?

Accepted Answer

$$t = 850 \ln(1000)$$

Question 8

Which of the following is the correct derivative of $$f(x) = \log_5$(2x+1)$$?

Accepted Answer

$$\frac{1}{(2x+1) \ln 5} \cdot 2$$

Question 9

Which of the following integrals is equal to $$\int b^x dx$$ for $$b \u003e 0$$, $$b 
eq 1$$?

Accepted Answer

$$\frac{b^x}{\ln b} + C$$

Question 10

Which of the following statements is true about the function $$y = \log_b x$$ for $$b \u003e 1$$?

Accepted Answer

The function is increasing for $$x \u003e 0.$$

Question 11

Given $$f(x) = x$$^{\sin x}$$ for x$$ \u003e 0$$, which method is most appropriate to find f$$'(x)$$?

Accepted Answer

Logarithmic differentiation

Question 12

Which of the following is the correct derivative of $$f(x) = b$$^{u(x)}$$ where u(x$$)$$ is differentiable?

Accepted Answer

$$b^{u(x)}$$ \ln b \cdot u'(x)$$

Question 13

Which of the following is the correct derivative of $$f(x) = \log_b(u(x))$$ where $$u(x) is $$differentiable?

Accepted Answer

$$\frac{1}{u(x) \ln b} u'(x)$$

Question 14

Which of the following is the correct indefinite integral of $$\int x^p dx$$ for $$p 
eq -1$$?

Accepted Answer

$$\frac{x$$^{p+1}$$}{p+1} + C$$

Question 15

Which of the following is true about the range of $$f(x) = b^x$$ for $$b \u003e 0$$, $$b 
eq 1$$?

Accepted Answer

$$(0, \infty)$$

Question 16

Suppose $$f(x) = x^\pi$$ and $$g(x) = \pi^x. $$Which statement is correct?

Accepted Answer

$$f(x) is a $$power function and $$g(x) is an $$exponential function.

Question 17

Which of the following is the correct derivative of $$f(x) = x^3$ \cdot 3^x$$?

Accepted Answer

$$3x^2$$ \cdot 3^x + $$x^3$$ \cdot 3^x \ln 3$$

Question 18

Which of the following is the correct value of $$\log_b 1$$ for any $$b \u003e 0$$, $$b 
eq 1$$?

Accepted Answer

$$0$$

Question 19

Which of the following is the correct change of base formula for logarithms?

Accepted Answer

$$\log_b x = \frac{\ln x}{\ln b}$$

Logarithmic and Exponential Functions with Other Bases

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