Q: A bank account earns interest at a continuous annual rate of 5%. If $$2,000 is deposited, what will be the value of the account after 10 years? Use the formula A$$ = $$Pe^{rt}$.

$$A = 2000e$$^{0.05 \times 10}$$

Q: A city’s population is modeled by $$P(t) = 100,000e$$^{0.02t}$$, where t$ is in years. What is the instantaneous rate of change of the population at t = 5$?

$$P'(5) = 100,000 \times 0.02 \times e$$^{0.1}$$

Question 1

A bank account earns interest at a continuous annual rate of 5%. If $$2,000 is deposited, what will be the value of the account after 10 years? Use the formula A$$ = $$Pe^{rt}$.

Accepted Answer

$$A = 2000e$$^{0.05 	imes 10}$$

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

$$\frac{e$$^{2x}$$(2x-1)}{x^2$}$$

Question 4

A radioactive substance decays at a rate proportional to its mass. If the half-life is 20 years, what is the decay rate $$k$$? Use $$k = \frac{\ln 2}{T}.$$

Accepted Answer

$$k = \frac{\ln 2}{20}$$

Question 5

Given $$f(x) = \ln(5x-2)$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x \u003e \frac{2}{5}$$

Question 6

Which property of logarithms allows you to write $$\ln(AB) = \ln A + \ln B$$?

Accepted Answer

Product property

Question 7

Suppose the population of a bacteria culture doubles every 6 hours. What is the exponential growth rate $$r$$? Use $$r = \frac{\ln 2}{T}.$$

Accepted Answer

$$r = \frac{\ln 2}{6}$$

Question 8

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 9

A forensic scientist uses Newton’s Law of Cooling to estimate time of death. If the room temperature is $$20^\circC$$, and the body temperature drops from $$37^\circC to$$ $$35^\circC in 2 $$hours, which equation models the temperature $$T(t) at $$time $$t$$?

Accepted Answer

$$T(t) = 20 + 17e$$^{-kt}$$

Question 10

Which of the following is the correct derivative of $$f(x) = 5^x$$?

Accepted Answer

$$f'(x) = 5^x \ln 5$$

Question 11

If $$y = \log_3$(x^2 + 1$$)$$, what is $$\frac{dy}{dx}$?

Accepted Answer

$$\frac{2x}{(x^2$ + 1) \ln 3}$$

Question 12

A city’s population is modeled by $$P(t) = 100,000e$$^{0.02t}$$, where t$ is in years. What is the instantaneous rate of change of the population at t = 5$?

Accepted Answer

$$P'(5) = 100,000 	imes 0.02 	imes e$$^{0.1}$$

Question 13

If $$y = \ln(3x + 2)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{3}{3x + 2}$$

Question 14

A medication decays in the bloodstream at a rate of 8% per hour. What is the half-life of the medication? Use $$T = \frac{\ln 2}{k}.$$

Accepted Answer

$$T = \frac{\ln 2}{0.08}$$

Question 15

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

Accepted Answer

$$2 \ln x + 2$$

Question 16

A city’s population follows the logistic model $$P(t) = \frac{L}{1 + Ce$$^{-rt}$$}. $$What does the parameter $$L$$ represent?

Accepted Answer

The upper limit of the population

Question 17

If $$y = \log_{10} x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{x \ln 10}$$

Question 18

A sample of carbon-14 has lost 60% of its original amount. If the half-life is 5730 years, how old is the sample? Use $$N = N_0$ e$$^{-kt}$$ and k$$ = \frac{\ln 2}{5730}$$.

Accepted Answer

$$t = \frac{\ln 0.4}{-\frac{\ln 2}{5730}}$$

Question 19

If $$y = 3$$^{2x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2 \ln 3 \cdot 3$$^{2x}$$

Chapter 2: Exponential and Logarithmic Functions – Calculus and Its Applications

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