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^2$x^{-1/2}$

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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