Q: Suppose you invest $P = 2000$ dollars at an annual interest rate of $r = 5\%$ compounded quarterly for $t = 3$ years. What is the total amount on deposit at the end of 3 years?

$A = 2000 \left(1 + \frac{0.05}{4}\right)^{4 \times 3} = 2000 \times (1.0125)^{12} \approx 2268.24$

Q: A bank offers an account with continuous compounding at an annual rate of $4\%$. If you deposit $P = 1000$ dollars, how much will be in the account after $5$ years?

$A = 1000 \times e^{0.04 \times 5} \approx 1000 \times e^{0.2} \approx 1221.40$

Question 1

Which of the following best describes the domain and range of the exponential function $f(x) = a^x$ where $a > 0$ and $a 
eq 1$?

Accepted Answer

Domain: $(-\infty, \infty)$; Range: $(0, \infty)$

Question 2

Suppose you invest $P = 2000$ dollars at an annual interest rate of $r = 5\%$ compounded quarterly for $t = 3$ years. What is the total amount on deposit at the end of 3 years?

Accepted Answer

$A = 2000 \left(1 + \frac{0.05}{4}ight)^{4 	imes 3} = 2000 	imes (1.0125)^{12} \approx 2268.24$

Question 3

If $f(x) = 2^x$, which transformation results in the function $g(x) = 2^{x+2}$?

Accepted Answer

Shift the graph of $f(x)$ 2 units to the left

Question 4

Which of the following is the correct formula for the amount $A$ after $t$ years if $P$ dollars is invested at an annual rate $r$ compounded continuously?

Accepted Answer

$A = Pe^{rt}$

Question 5

Given the exponential equation $9^{x} = 27^{x-1}$, what is the value of $x$?

Accepted Answer

$x = -2$

Question 6

The number $e$ is defined as $\lim_{m 	o \infty} \left(1 + \frac{1}{m}ight)^m$. Which of the following best approximates the value of $e$?

Accepted Answer

$2.718$

Question 7

If $f(x) = a^x$ and $a > 1$, which statement about the graph of $f(x)$ is true?

Accepted Answer

The graph increases as $x$ increases and has a horizontal asymptote at $y = 0$

Question 8

A principal of $P$ dollars is invested at an annual rate $r$ compounded $m$ times per year for $t$ years. Which formula gives the total amount on deposit?

Accepted Answer

$A = P \left(1 + \frac{r}{m}\right)^{mt}$

Question 9

Suppose the U.S. GDP in 1940 was 103 billion dollars and in 2019 was 21,429 billion dollars. If the exponential model is $f(x) = 103(1.0644)^x$, where $x$ is years since 1940, in which year will the GDP double its 2019 value?

Accepted Answer

Approximately in the year 2030

Question 10

Which of the following statements is true about the function $f(x) = \left(\frac{1}{2}\right)^x$?

Accepted Answer

The function decreases as $x$ increases and has a horizontal asymptote at $y = 0$

Question 11

If $f(x) = e^x$ and $g(x) = e^{-x} + 3$, what is the range of $g(x)$?

Accepted Answer

$(-\infty, 3)$

Question 12

A bank offers an account with continuous compounding at an annual rate of $4\%$. If you deposit $P = 1000$ dollars, how much will be in the account after $5$ years?

Accepted Answer

$A = 1000 	imes e^{0.04 	imes 5} \approx 1000 	imes e^{0.2} \approx 1221.40$

Question 13

Which property of exponential functions allows us to solve equations like $a^u = a^v$ by equating exponents?

Accepted Answer

The one-to-one property

Question 14

If $f(x) = 2^x$ and $k(x) = -2^x$, how does the graph of $k(x)$ relate to $f(x)$?

Accepted Answer

It is a reflection of $f(x)$ across the $x$-axis

Question 15

Which of the following is the correct formula for simple interest earned on a principal $P$ at rate $r$ for $t$ years?

Accepted Answer

$I = Prt$

Question 16

If $f(x) = a^x$ and $a$ is such that $0 < a < 1$, which of the following describes the behavior of $f(x)$ as $x 	o \infty$?

Accepted Answer

$f(x) 	o 0$

Question 17

A company’s revenue grows exponentially according to $R(t) = 5000 \cdot (1.08)^t$, where $t$ is years since 2020. In which year will the revenue first exceed $10,000$?

Accepted Answer

2029

Question 18

Which of the following is NOT a property of the exponential function $f(x) = a^x$ for $a > 0$, $a 
eq 1$?

Accepted Answer

It is periodic with period $2\pi$

Exponential Functions and Applications in Business Calculus

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