Q: A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. What is the rate of growth of the population at t = 10$ hours?

$$P'(10) = 0.05 \times 100e$$^{0.05 \times 10}$$

Question 1

A company’s profit function is given by $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold. What is the maximum profit, and how many units must be sold to achieve it?

Accepted Answer

Maximum profit is $$P(10) = 100$$, achieved at $$x = 10$$ units.

Question 2

Evaluate the following limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 3

Which of the following statements best describes the continuity of the function $$f(x) = \frac{1}{x-3} at$$ $$x = 3$$?

Accepted Answer

The function is not continuous at $$x = 3$$ because $$f(3) is $$undefined.

Question 4

Given the cost function $$C(x) = 500 + 20x$$, where $$x is $$the number of units produced, what is the marginal cost when $$x = 100$$?

Accepted Answer

$$20$$

Question 5

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

Accepted Answer

$$2e^{2x}$$

Question 6

A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. What is the rate of growth of the population at t = 10$ hours?

Accepted Answer

$$P'(10) = 0.05 	imes 100e$$^{0.05 	imes 10}$$

Question 7

Find the equation of the tangent line to the curve $$y = x^3$ - 2x at $$the point where $$x = 1.$$

Accepted Answer

$$y = x + 0$$

Question 8

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 9

A company’s demand function is $$p = 100 - 2x. $$Find the elasticity of demand $$E at$$ $$x = 20.$$

Accepted Answer

$$E = \frac{2x}{100 - 2x}$$, so $$E = \frac{40}{60} = \frac{2}{3}$$

Question 10

Evaluate the definite integral $$\$$int_0$$^2 (3x^2$ + 2x) dx.$$

Accepted Answer

$$20$$

Question 11

Which of the following is the correct application of the substitution method for $$\int x e$$^{x^2}$$ dx$$?

Accepted Answer

Let $$u = x^2$, then du = 2x dx$, so $$\int x $$e^{x^2} dx$$ = \frac{1}{2} \int $$e^{u} du$

Question 12

Suppose the revenue function for a product is $$R(x) = 50x - x^2$. What is the maximum revenue, and at what value of x$ does it occur?

Accepted Answer

Maximum revenue is $$R(25) = 625$$, at $$x = 25$$

Question 13

Find the area between the curves $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$.

Accepted Answer

$$\$$int_0$$^2 (2x - x^2$) dx = \left[ x^2$ - \frac{1}{3}x^3$ \right]_0^2$ = 4 - \frac{8}{3} = \frac{4}{3}$$

Question 14

A chemical decays according to $$Q(t) = Q_0$ e$$^{-kt}$$. If the half-life is 5 hours, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{5}$$

Question 15

Given $$f(x, y) = x^2$ y + 3y^2$, what is the partial derivative $$\frac{\partial f}{\partial y}$$?

Accepted Answer

$$x^2 + 6y$

Question 16

Evaluate the improper integral $$\$$int_1$$^{\infty} \frac{1}{x^2$} dx.$$

Accepted Answer

$$1$$

Question 17

A business invests continuously at a rate of $$R(t) = 2000e$$^{0.03t}$$ dollars per year for 10 years. What is the accumulated future value at the end of 10 years? (Assume continuous compounding at 3% per year.)

Accepted Answer

$$\$$int_0$$^{10} 2000e$$^{0.03t}$$ dt = \left[ \frac{2000}{0.03} e$$^{0.03t}$$ \right]_0$$^{10}$$

Question 18

If $$f(x) = \sqrt{x}$$, use the definition of the derivative to find $$f'(4).$$

Accepted Answer

$$f'(4) = \frac{1}{4}$$

Business Calculus Syllabus and Study Guide Overview

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