Q: A company’s profit $$P(x)$$, in thousands of dollars, is given by $$P(x) = 100x - 5x^2$, where x$ is the number of units sold (in thousands). What is the marginal profit when x = 4$?

$$P'(4) = 100 - 10 \times 4 = 60$$

Q: A medication’s concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of concentration at t = 5$ hours?

$$C'(5) = 50 \times (-0.2) e$$^{-0.2 \times 5}$$ = -10 e$$^{-1}$$

Q: A population grows according to $$P(t) = 1000e$$^{0.05t}$$. What is the instantaneous rate of growth at t = 10$?

$$P'(10) = 1000 \times 0.05 \times e$$^{0.5}$$ = 50e$$^{0.5}$$

Question 1

A company’s profit $$P(x)$$, in thousands of dollars, is given by $$P(x) = 100x - 5x^2$, where x$ is the number of units sold (in thousands). What is the marginal profit when x = 4$?

Accepted Answer

$$P'(4) = 100 - 10 	imes 4 = 60$$

Question 2

Evaluate the limit: $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 3

Which of the following best describes the meaning of the derivative $$f'(a) at$$ $$x = a$$?

Accepted Answer

The slope of the tangent line to $$f(x) at$$ $$x = a$$

Question 4

Find the derivative of $$f(x) = x^3$ e^x$$ using the product rule.

Accepted Answer

$$f'(x) = 3x^2$ e^x + x^3$ e^x$$

Question 5

A medication’s concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of concentration at t = 5$ hours?

Accepted Answer

$$C'(5) = 50 	imes (-0.2) e$$^{-0.2 	imes 5}$$ = -10 e$$^{-1}$$

Question 6

Which rule is most appropriate for finding $$\frac{d}{dx} \left( \frac{x^2$ + 1}{x - 3} \right)$$?

Accepted Answer

Quotient rule

Question 7

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

Accepted Answer

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

Question 8

A business finds that the demand for its product is given by $$p = 200 - 4q$$, where $$p is $$price and $$q is $$quantity. What is the elasticity of demand when $$q = 30$$?

Accepted Answer

$$E = \frac{dq}{dp} \cdot \frac{p}{q} = -\frac{1}{4} \cdot \frac{80}{30} = -\frac{2}{3}$$

Question 9

Which of the following is NOT a critical point of $$f(x)$$?

Accepted Answer

A point where $$f(x) is $$not defined

Question 10

Find the antiderivative of $$f(x) = 3x^2$.

Accepted Answer

$$F(x) = x^3$ + C$$

Question 11

Evaluate the definite integral $$\$$int_0$$^2 (4x - 1) \, dx.$$

Accepted Answer

$$\left[2x^2$ - x \right]_0^2$ = (8 - 2) - (0 - 0) = 6$$

Question 12

Which method is most appropriate for evaluating $$\int x e$$^{x^2}$$ dx$$?

Accepted Answer

Substitution

Question 13

A population grows according to $$P(t) = 1000e$$^{0.05t}$$. What is the instantaneous rate of growth at t = 10$?

Accepted Answer

$$P'(10) = 1000 	imes 0.05 	imes e$$^{0.5}$$ = 50e$$^{0.5}$$

Question 14

Which of the following statements about the Fundamental Theorem of Calculus is correct?

Accepted Answer

It relates the derivative of the antiderivative to the original function

Question 15

If $$f(x)$$ has a horizontal asymptote at $$y = 3$$, which of the following is true?

Accepted Answer

$$\lim_{x 	o \infty} f(x) = 3$$

Question 16

A company’s cost function is $$C(x) = 500 + 20x. $$What is the marginal cost?

Accepted Answer

$$C'(x) = 20$$

Question 17

Find $$\int \frac{1}{x} dx.$$

Accepted Answer

$$\ln|x| + C$$

Question 18

Which of the following is an indeterminate form when evaluating limits?

Accepted Answer

$$\frac{0}{0}$$

Question 19

A function $$f(x)$$ has a local maximum at $$x = 2. $$Which of the following must be true?

Accepted Answer

$$f'(2) = 0$$ and $$f''(2) \u003c 0$$

Business Calculus: Limits, Derivatives, and Integrals – Course Outline and Key Concepts

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