Question 1

A company’s profit function is given by $$P(x) = 2x^2$ - 12x + 20$$, where $$x is $$the number of units sold. What is the minimum profit, and at how many units sold does it occur?

Accepted Answer

Minimum profit is $$-8 at$$ $$x = 3$$

Question 2

Which of the following is the correct derivative of $$f(x) = x^3$ + 2x^2$ - 5x + 7$$?

Accepted Answer

$$f'(x) = 3x^2$ + 4x - 5$$

Question 3

A business’s revenue function is $$R(x) = 100x - x^2$. What is the marginal revenue when x = 20$?

Accepted Answer

$$MR = 60$$

Question 4

Given $$f(x) = 3x^2$ - 4x + 1$$, use the definition of the derivative to find $$f'(x).$$

Accepted Answer

$$f'(x) = 6x - 4$$

Question 5

If $$f(x) = x^2$ and g(x) = 2x + 1$, what is the derivative of h(x) = f(x)g(x$$)$$ using the product rule?

Accepted Answer

$$h'(x) = 2x(2x + 1) + x^2$(2)$$

Question 6

A manufacturer’s cost function is $$C(x) = 5x^3$ - 2x^2$ + 4x. $$Find the rate of change of cost when $$x = 2.$$

Accepted Answer

$$C'(2) = 56$$

Question 7

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

Accepted Answer

If $$f''(x) \u003e 0$$, the function is concave up at $$x$$

Question 8

Evaluate the definite integral $$\$$int_0$$^2 (3x^2$ - 2x + 1)dx to $$find the total quantity produced over 2 hours if the rate of production is $$3x^2 - 2x + 1$ units per hour.

Accepted Answer

Total quantity is $$8$$

Question 9

A company wants to maximize its profit given by $$P(x) = -2x^2$ + 40x - 100. $$What is the optimal number of units to produce?

Accepted Answer

$$x = 10$$

Question 10

Which of the following is the correct application of the quotient rule for $$f(x) = \frac{x^2$}{x+1}$$?

Accepted Answer

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

Question 11

If $$f(x) = \ln(x)$$, what is $$\frac{d}{dx}f(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 12

A business’s demand function is $$p(x) = 50 - 2x. $$What is the elasticity of demand at $$x = 10$$?

Accepted Answer

Elasticity is $$-0.4$$

Question 13

Which of the following is the correct antiderivative of $$f(x) = 4x^3$ - 2x$$?

Accepted Answer

$$F(x) = x^4$ - x^2$ + C$$

Question 14

A company’s cost function is $$C(x) = 10x + 500. $$What is the average cost per unit when producing 50 units?

Accepted Answer

Average cost is $$20$$

Question 15

Which of the following is the correct application of the chain rule for $$f(x) = (2x + 1)^5$?

Accepted Answer

$$f'(x) = 5(2x + 1)^4$ \cdot 2$$

Question 16

Evaluate $$\$$int_0$$^1 (6x^2$ - 4x + 2)dx to $$find the total change over the interval.

Accepted Answer

Total change is $$4/3$$

Question 17

Which of the following is the correct formula for the area between two curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b [f(x) - g(x)]dx$$

Question 18

If $$f(x) = e$$^{2x}$$, what is $$\frac{d}{dx}f(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 19

A company’s revenue is given by $$R(x) = 120x - x^2$. What is the maximum revenue and at what value of x$ does it occur?

Accepted Answer

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

Business Calculus Study Guide: Functions, Limits, Derivatives, Integration, and Applications

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