Q: A company’s cost function is $$C(x) = 500 + 20x + 0.1x^2$, where x$ is the number of units produced. What is the marginal cost when x = 100$?

$$C'(100) = 20 + 0.2 \times 100 = 30$$

Q: A company’s revenue function is $$R(x) = 50x - x^2$. What is the marginal revenue when x = 10$?

$$R'(10) = 50 - 2 \times 10 = 30$$

Question 1

Which of the following best describes the concept of the derivative at a point $$x = a$$ for a function $$f(x)$$?

Accepted Answer

It is the instantaneous rate of change of $$f(x) at$$ $$x = a$$, given by $$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}.$$

Question 2

A company’s cost function is $$C(x) = 500 + 20x + 0.1x^2$, where x$ is the number of units produced. What is the marginal cost when x = 100$?

Accepted Answer

$$C'(100) = 20 + 0.2 	imes 100 = 30$$

Question 3

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

Accepted Answer

4

Question 4

Suppose the demand function for a product is $$p = 100 - 2x$$, where $$p is $$the price and $$x is $$the quantity sold. What is the elasticity of demand when $$x = 30$$?

Accepted Answer

$$E = -\frac{p}{x} \cdot \frac{dx}{dp} = -\frac{40}{30} \cdot \left(-\frac{1}{2}\right) = \frac{2}{3}$$

Question 5

Which of the following is the correct antiderivative of $$f(x) = 6x^2$ - 4x + 7$$?

Accepted Answer

$$F(x) = 2x^3$ - 2x^2$ + 7x + C$$

Question 6

A business wants to maximize its profit, given by $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold. What is the optimal number of units to sell for maximum profit?

Accepted Answer

$$x = 10$$

Question 7

Which of the following statements about the definite integral $$\int_a^b f(x)\,dx is $$TRUE?

Accepted Answer

It represents the total accumulated change of $$f(x)$$ from $$x = a to$$ $$x = b.$$

Question 8

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

Accepted Answer

$$6xy$$

Question 9

A population of bacteria grows according to $$P(t) = 500e$$^{0.2t}$$, where t$ is in hours. What is the rate of growth at t = 5$ hours?

Accepted Answer

$$P'(5) = 500 \cdot 0.2 \cdot e$$^{1}$$ = 100e$$

Question 10

Evaluate $$\$$int_0$$^2 (3x^2$ + 2x)\,dx.$$

Accepted Answer

14

Question 11

Which of the following functions is continuous everywhere?

Accepted Answer

$$f(x) = e^x$$

Question 12

Suppose $$f(x) = x^3$ - 6x^2$ + 9x + 1. At $$what value of $$x$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 3$$

Question 13

If $$y = e$$^{2x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2e^{2x}$$

Question 14

A company’s revenue function is $$R(x) = 50x - x^2$. What is the marginal revenue when x = 10$?

Accepted Answer

$$R'(10) = 50 - 2 	imes 10 = 30$$

Question 15

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 16

Evaluate $$\$$int_1$$^e \frac{1}{x}\,dx.$$

Accepted Answer

1

Question 17

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

Accepted Answer

$$x^2$$ + $$3y^2$$

Question 18

A pharmaceutical company models the rate at which a drug leaves the bloodstream as $$C(t) = 100e$$^{-0.3t}$$, where C$ is the concentration in mg/L and t$ is in hours. What is the instantaneous rate of change at t = 2$ hours?

Accepted Answer

$$C'(2) = 100 \cdot (-0.3) \cdot e$$^{-0.6}$$ = -30e$$^{-0.6}$$

Question 19

Suppose $$f(x) = x^2$ - 4x + 3. $$For what value of $$x$$ does $$f(x)$$ attain its minimum?

Accepted Answer

$$x = 2$$

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