Q: Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Suppose a company’s revenue $$R(x) in $$thousands of dollars is given by $$R(x) = 50x - 0.5x^2$, where x$ is the number of units sold (in thousands). What is the marginal revenue when x = 20$?

Accepted Answer

$$R'(20) = 30$$

Question 2

Which of the following best describes the meaning of the derivative $$f'(a) in a $$business context?

Accepted Answer

The instantaneous rate of change of $$f$$ with respect to $$x at$$ $$x = a$$

Question 3

A manufacturer’s cost function is $$C(x) = 1000 + 20x + 0.1x^2$. What is the marginal cost when producing 50 units?

Accepted Answer

$$C'(50) = 30$$

Question 4

If $$f(x) = 3x^2$ - 4x + 7$$, what is the equation of the tangent line to $$f at$$ $$x = 2$$?

Accepted Answer

$$y = 8x - 5$$

Question 5

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$For what value of $$x is $$the profit maximized?

Accepted Answer

$$x = 10$$

Question 6

Given $$f(x) = \sqrt{x}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 7

If the demand function for a product is $$p = 100 - 2x$$, where $$p is $$the price and $$x is $$the quantity sold, what is the revenue function $$R(x)$$?

Accepted Answer

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

Question 8

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 9

A company’s cost function is $$C(x) = 500 + 15x. $$What is the average cost per unit when producing 50 units?

Accepted Answer

$$C_{avg} = 25$$

Question 10

If $$f(x) = x^3$ - 6x^2$ + 9x + 5$$, at what value of $$x$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 3$$

Question 11

Suppose the marginal cost for producing $$x$$ units is $$MC(x) = 4x + 10. $$What is the total cost function $$C(x) if $$the fixed cost is $$200$$?

Accepted Answer

$$C(x) = 2x^2$ + 10x + 200$$

Question 12

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

Accepted Answer

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

Question 13

A company’s revenue function is $$R(x) = 120x - x^2$. What is the maximum revenue, and at what quantity x$ does it occur?

Accepted Answer

Maximum revenue is $$3600 at$$ $$x = 60$$

Question 14

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

Accepted Answer

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

Question 15

A business finds that the cost to produce $$x$$ units is $$C(x) = 2000 + 30x. If $$each unit sells for $$p = 50$$, what is the break-even quantity?

Accepted Answer

$$x = 100$$

Business Calculus Exam Coverage Overview

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