Q: A population grows according to $P(t) = 100e^{0.05t}$. What is the instantaneous rate of change of the population at $t = 10$?

$P'(10) = 5 \times 100e^{0.5} = 500e^{0.5}$

Question 1

Given the function $f(x) = e^{2x}$, what is the second derivative $f''(x)$?

Accepted Answer

$f''(x) = 4e^{2x}$

Question 2

Suppose $f(x) = \ln(x^2 + 1)$. On which intervals is $f(x)$ increasing?

Accepted Answer

$(-\infty, 0)$ and $(0, \infty)$

Question 3

Given $f(x) = x^3 - 3x^2 + 2$, find the intervals where the function is concave up.

Accepted Answer

$(-\infty, 1) \cup (2, \infty)$

Question 4

A company models its profit by $P(x) = -2x^2 + 8x + 5$, where $x$ is the number of units sold. What is the maximum profit, and at how many units sold does it occur?

Accepted Answer

Maximum profit is $13$ at $x = 2$

Question 5

Given $f(x) = \sqrt{x}$, find the value of $x$ where the tangent line is horizontal.

Accepted Answer

There is no value of $x$ where the tangent is horizontal.

Question 6

For the function $f(x) = \ln(x)$, what is the domain and where is the function increasing?

Accepted Answer

Domain: $(0, \infty)$; Increasing on $(0, \infty)$

Question 7

A population grows according to $P(t) = 100e^{0.05t}$. What is the instantaneous rate of change of the population at $t = 10$?

Accepted Answer

$P'(10) = 5 	imes 100e^{0.5} = 500e^{0.5}$

Question 8

Given $f(x) = x^2 - 4x + 3$, find the absolute minimum on the interval $[0, 5]$.

Accepted Answer

Minimum is $f(2) = -1$

Question 9

If $f(x) = \ln(x)$ and $g(x) = e^x$, what is the derivative of $h(x) = f(x)g(x)$?

Accepted Answer

$h'(x) = e^x + \frac{e^x}{x}$

Question 10

A medication's concentration in the bloodstream is modeled by $C(t) = 50e^{-0.2t}$. At what time $t$ does the concentration decrease at a rate of $-5$ mg/hour?

Accepted Answer

$t = 5$

Question 11

Given $f(x) = x^4 - 4x^3 + 6x^2$, find all local extrema and classify them as maxima or minima.

Accepted Answer

Local minimum at $x = 2$, local maximum at $x = 1$

Question 12

A business finds its cost function is $C(x) = 100 + 20x - x^2$. What is the marginal cost when $x = 5$?

Accepted Answer

$C'(5) = 10$

Question 13

Given $f(x) = \sqrt{x}$, what is the second derivative $f''(x)$?

Accepted Answer

$f''(x) = -\frac{1}{4}x^{-3/2}$

Question 14

A company’s revenue is $R(x) = 50x - x^2$. Find the production level $x$ that maximizes revenue.

Accepted Answer

$x = 25$

Question 15

Given $f(x) = \ln(x)$, find the equation of the tangent line at $x = 1$.

Accepted Answer

$y = x - 1$

Question 16

If $f(x) = x^2$ and $g(x) = \sqrt{x}$, what is the derivative of $h(x) = f(x)g(x)$?

Accepted Answer

$h'(x) = 2x\sqrt{x} + \frac{1}{2}x^2x^{-1/2}$

Question 17

A company’s profit function is $P(x) = 10x - x^2$. Find the maximum profit and the number of units that yield it.

Accepted Answer

Maximum profit is $25$ at $x = 5$

Question 18

Given $f(x) = x^3 - 6x^2 + 9x$, find the inflection point(s).

Accepted Answer

$x = 2$

Question 19

A company’s demand function is $p(x) = 100 - 2x$. Find the price and quantity that maximize revenue.

Accepted Answer

Price $p = 50$, quantity $x = 25$

Business Calculus: Sample Problems on Derivatives, Graphing, and Optimization

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