Q: The Irwins want to make an initial investment $P$ that will grow to $40,000$ in 20 years, compounded continuously at 3.5%. What should the initial investment be?

$P = 40000 e^{-0.035 \times 20} \approx 20,000$

Question 1

The population of San Estaban in 2015 was 45,000 and it grows continuously at a rate of 3% per year. Which of the following functions models the population $A(t)$ years after 2015?

Accepted Answer

$A(t) = 45000 e^{0.03t}$

Question 2

Using the model $A(t) = 45000 e^{0.03t}$, what will the population of San Estaban be in 2025?

Accepted Answer

$A(10) = 45000 e^{0.3} \approx 60,616$

Question 3

Maria invests $20,000 in an IRA account whose value after $t$ years is modeled by $V(t) = 20000 e^{0.054t}$. What is the value of the account after 7 years?

Accepted Answer

$V(7) = 20000 e^{0.378} \approx 24,600$

Question 4

Suppose a radioactive sample of carbon-14 has lost 22.3% of its carbon-14. Given the half-life of carbon-14 is 5730 years, how old is the sample? (Use $A = A_0 e^{kt}$, $k = \frac{\ln(0.5)}{5730}$)

Accepted Answer

$t = \frac{\ln(0.777)}{\ln(0.5)/5730} \approx 1,900$ years

Question 5

If the amount of iodine-125 in a cancer treatment sample decreased by 2% while in storage, and the half-life is 60.1 days, how long was the sample in storage? (Use $A = A_0 e^{kt}$, $k = \frac{\ln(0.5)}{60.1}$)

Accepted Answer

$t = \frac{\ln(0.98)}{\ln(0.5)/60.1} \approx 1.73$ days

Question 6

Given the function $f(x) = 2^x - e^x$, what is $f'(x)$?

Accepted Answer

$f'(x) = 2^x \ln 2 - e^x$

Question 7

Find the critical points of $h(x) = x^3 - 6x^2 - 15x + 1$.

Accepted Answer

$x = -1$ and $x = 5$

Question 8

Determine the vertical asymptote(s) of $f(x) = \frac{x+2}{x^2 + 6x + 8}$.

Accepted Answer

$x = -4$ and $x = -2$

Question 9

Determine the horizontal asymptote of $f(x) = \frac{5x^4 + 2x^3 + x}{x^4 + 8}$.

Accepted Answer

$y = 5$

Question 10

Given the demand function $p(x) = 35 - 0.01x$, where $x$ is in millions of barrels per day and $p$ is in dollars per barrel, what is the price elasticity of demand $E(x)$?

Accepted Answer

$E(x) = -\frac{x}{35 - 0.01x}$

Question 11

For the function $f(x) = \ln(x^2 + 6)$, what is $f'(x)$?

Accepted Answer

$f'(x) = \frac{2x}{x^2 + 6}$

Question 12

A bank account grows continuously at 5.9% per year. If $P(0) = 1000$, what is the balance after 2 years?

Accepted Answer

$P(2) = 1000 e^{0.118} \approx 1,125$

Question 13

The Irwins want to make an initial investment $P$ that will grow to $40,000$ in 20 years, compounded continuously at 3.5%. What should the initial investment be?

Accepted Answer

$P = 40000 e^{-0.035 	imes 20} \approx 20,000$

Question 14

Given $f(x) = 3x^4 + 4x^3$ on $[-2, 0]$, what is the absolute maximum value?

Accepted Answer

$f(-2) = -16$

Question 15

If $f(x) = \log_2(3x-1)$, what is $f'(x)$?

Accepted Answer

$f'(x) = \frac{3}{(3x-1) \ln 2}$

Question 16

For the function $f(x) = x^3 + 3x^2 - 2x + 4$ on $[-2, 0]$, what is the absolute minimum value?

Accepted Answer

$f(-2) = 0$

Question 17

Suppose the demand function for oil is $p(x) = 35 - 0.01x$. At what price is the demand unit elastic?

Accepted Answer

$p = 17.5$

Question 18

If $y = 4^x \cdot \log_5 x$, what is $\frac{dy}{dx}$?

Accepted Answer

$y' = 4^x \log_5 x \ln 4 + 4^x \frac{1}{x \ln 5}$

Business Calculus Study Guide: Exponential Growth, Derivatives, and Extrema

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