Q: Which of the following statements about the continuity of a function is correct?

A function is continuous at $$x = a if$$ $$\lim_{x \to a} f(x)$$ exists and equals $$f(a).$$

Q: A population of bacteria grows according to the exponential model $$P(t) = 500e$$^{0.03t}$$, where t$ is in hours. What is the rate of change of the population at t = 10$ hours?

$$P'(10) = 500 \times 0.03 \times e$$^{0.03 \times 10}$$ = 15 \times e$$^{0.3}$$

Q: If $$f(x) = x^3$ - 4x^2$ + 6x$$, what is the value of $$f''(2)$$?

$$f''(2) = 6 \times 2 - 8 = 12 - 8 = 4$$

Q: A business wants to minimize the cost function $$C(x) = 2x^2$ - 12x + 30. $$What is the minimum cost and at what value of $$x$$ does it occur?

Minimum at $$x = 3$$, cost $$C(3) = 2 \times 9 - 12 \times 3 + 30 = 18 - 36 + 30 = 12$$

Q: A company’s profit function is $$P(x) = 20x - 0.5x^2$. What is the maximum profit and at what value of x$ does it occur?

Maximum at $$x = 20$$, $$P(20) = 20 \times 20 - 0.5 \times 400 = 400 - 200 = 200$$

Question 1

A company models its revenue with the function $$R(x) = 100x - 0.5x^2$, where x$ is the number of units sold. What is the marginal revenue when 20 units are sold?

Accepted Answer

$$R'(x) = 100 - x$$,\ R'(20) = 100 - 20 = 80$

Question 2

Which of the following statements about the continuity of a function is correct?

Accepted Answer

A function is continuous at $$x = a if$$ $$\lim_{x 	o a} f(x)$$ exists and equals $$f(a).$$

Question 3

A population of bacteria grows according to the exponential model $$P(t) = 500e$$^{0.03t}$$, where t$ is in hours. What is the rate of change of the population at t = 10$ hours?

Accepted Answer

$$P'(10) = 500 	imes 0.03 	imes e$$^{0.03 	imes 10}$$ = 15 	imes e$$^{0.3}$$

Question 4

If $$f(x) = x^3$ - 4x^2$ + 6x$$, what is the value of $$f''(2)$$?

Accepted Answer

$$f''(2) = 6 	imes 2 - 8 = 12 - 8 = 4$$

Question 5

A business wants to minimize the cost function $$C(x) = 2x^2$ - 12x + 30. $$What is the minimum cost and at what value of $$x$$ does it occur?

Accepted Answer

Minimum at $$x = 3$$, cost $$C(3) = 2 	imes 9 - 12 	imes 3 + 30 = 18 - 36 + 30 = 12$$

Question 6

Evaluate the definite integral $$\int_{0}^{2} (3x^2$ - 4x + 1)\ dx.$$

Accepted Answer

$$\left[ x^3$ - 2x^2$ + x \right]_0^2$ = (8 - 8 + 2) - (0 - 0 + 0) = 2$$

Question 7

A manufacturer finds that the marginal cost function is $$MC(x) = 5e$$^{0.1x}$$. What is the total cost function C(x$$)$$ if the fixed cost is 200$?

Accepted Answer

$$C(x) = 5 \int e$$^{0.1x}$$ dx + 200 = 50e$$^{0.1x}$$ + 200$$

Question 8

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 1}$$

Question 9

A company’s profit function is $$P(x) = 20x - 0.5x^2$. What is the maximum profit and at what value of x$ does it occur?

Accepted Answer

Maximum at $$x = 20$$, $$P(20) = 20 	imes 20 - 0.5 	imes 400 = 400 - 200 = 200$$

Question 10

Find the partial derivative $$\frac{\partial f}{\partial x}$$ for $$f(x, y) = 3x^2$y + 2y^3$.

Accepted Answer

$$\frac{\partial f}{\partial x} = 6xy$$

Question 11

A business uses the function $$f(x) = e$$^{2x}$$ to model sales growth. What is the second derivative f$$''(x)$$?

Accepted Answer

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

Question 12

If $$f(x) = x^2$ + 3x$$, use the definition of the derivative to find $$f'(2).$$

Accepted Answer

$$f'(2) = \lim_{h 	o 0} \frac{(2+h)^2$ + 3(2+h) - (2^2$ + 3 	imes 2)}{h} = \lim_{h 	o 0} \frac{4 + 4h + h^2$ + 6 + 3h - 4 - 6}{h} = \lim_{h 	o 0} \frac{7h + h^2$}{h} = 7$$

Question 13

A company’s cost function is $$C(x) = 1000 + 20x. $$What is the average rate of change of cost as production increases from $$x = 10 to$$ $$x = 30$$?

Accepted Answer

$$\frac{C(30) - C(10)}{30 - 10} = \frac{1000 + 600 - (1000 + 200)}{20} = \frac{1400 - 1200}{20} = 10$$

Question 14

Find the indefinite integral $$\int (4x - 5)\ dx.$$

Accepted Answer

$$2x^2 - 5x + C$

Question 15

A company’s revenue function is $$R(x) = 50x. $$What is the marginal revenue and what does it represent?

Accepted Answer

Marginal revenue is $$R'(x) = 50$$, representing the additional revenue from selling one more unit.

Question 16

If $$f(x) = \ln(x)$$, what is the derivative $$f'(x)$$?

Accepted Answer

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

Question 17

A company’s demand function is $$p(x) = 100 - 2x. $$What is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = p(x) 	imes x = (100 - 2x)x = 100x - 2x^2$

Question 18

Use substitution to evaluate $$\int x e$$^{x^2}$$ dx.$$

Accepted Answer

Let $$u = x^2$, du = 2x dx$, so x dx$$ = \frac{du}{2}$$. Thus, $$\int x $$e^{x^2} dx$$ = \frac{1}{2} \int $$e^{u} du$$ = \frac{1}{2} $$e^{x^2} + C$

Business Calculus: Course Overview, Competencies, and Key Topics

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