Q: Suppose the value of a car depreciates exponentially at a rate of 15% per year. If the car is worth $20,000 now, what is its value after 5 years?

$$20,000 \cdot e$$^{-0.15 \times 5}$$

Q: A savings account earns interest compounded continuously at a rate of 6% per year. If you deposit $1,000, how much will you have after 10 years?

$$1,000 \cdot e$$^{0.06 \times 10}$$

Question 1

Which of the following best describes the function $$f(x) = e^x$$?

Accepted Answer

It is an exponential function with a base of $$e.$$

Question 2

What is the derivative of $$f(x) = \ln(x)$$ with respect to $$x$$?

Accepted Answer

$$\frac{1}{x}$$

Question 3

A population of bacteria doubles every 3 hours. If the initial population is 200, which equation models the population $$P(t)$$ after $$t$$ hours?

Accepted Answer

$$P(t) = 200 \cdot e$$^{\frac{\ln 2}$${3} t}$$

Question 4

Which of the following is the correct derivative of $$f(x) = a^x$$ where $$a \u003e 0$$ and $$a 
eq 1$$?

Accepted Answer

$$a^x \ln(a)$$

Question 5

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

Accepted Answer

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

Question 6

A radioactive substance decays according to $$N(t) = N_0$ e$$^{-kt}$$. If the half-life is 10 years, what is the value of k$?

Accepted Answer

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

Question 7

Which of the following statements is true about the natural logarithm function $$f(x) = \ln(x)$$?

Accepted Answer

It is only defined for $$x \u003e 0.$$

Question 8

Suppose the value of a car depreciates exponentially at a rate of 15% per year. If the car is worth $20,000 now, what is its value after 5 years?

Accepted Answer

$$20,000 \cdot e$$^{-0.15 	imes 5}$$

Question 9

If $$y = \log_a(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{x \ln(a)}$$

Question 10

A medication is eliminated from the bloodstream at a rate proportional to the amount present. If $$C(t) = C_0$ e$$^{-kt}$$, and after 4 hours only 25% of the medication remains, what is k$?

Accepted Answer

$$k = \frac{\ln 4}{4}$$

Question 11

Which of the following is the correct derivative of $$f(x) = \ln(ax)$$ where $$a \u003e 0$$?

Accepted Answer

$$\frac{1}{x}$$

Question 12

If $$f(x) = e$$^{2x}$$, what is f$$''(x)$$?

Accepted Answer

$$4e^{2x}$$

Question 13

A savings account earns interest compounded continuously at a rate of 6% per year. If you deposit $1,000, how much will you have after 10 years?

Accepted Answer

$$1,000 \cdot e$$^{0.06 	imes 10}$$

Question 14

Which of the following is the correct derivative of $$f(x) = \log_{10}(x)$$?

Accepted Answer

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

Question 15

If $$f(x) = e$$^{x^2}$$, what is f$$'(x)$$?

Accepted Answer

$$2x e$$^{x^2}$$

Unit 3: Exponential and Logarithmic Functions – Study Guide

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