Question 1

Given the cost function $$C(x) = 12x + 6000$$ and a ticket price of $$60,\ what is the profit function P(x$$)$$ in terms of the number of premium tickets x$ sold?

Accepted Answer

$$P(x) = 60x - (12x + 6000)$$

Question 2

A vendor's t-shirt demand function is $$p = 40 - 0.04x. If $$each shirt costs $$3 to $$produce and fixed costs are $$200$$, what is the profit from selling $$400 t-$$shirts?

Accepted Answer

$$8200$$

Question 3

For the candy vendor, the demand function is $$p = 60 - 2x ($$cents per piece), and cost is $$C(x) = 5 + 2.5x ($$dollars, $$x in $$hundreds). What is the revenue function $$R(x) in $$dollars?

Accepted Answer

$$R(x) = x(60 - 2x)/100$$

Question 4

Evaluate $$\lim_{x 	o 3} \frac{x^2$ - 6}{x - 3}.$$

Accepted Answer

$$7$$

Question 5

Find the average rate of change of $$f(x) = x^3$ + 2x + 1 on $$the interval $$[2, 8].$$

Accepted Answer

$$52$$

Question 6

A company’s revenue is $$R(x) = 2000 \ln(0.04x + 1). $$What is the marginal revenue function $$R'(x)$$?

Accepted Answer

$$R'(x) = \frac{80}{0.04x + 1}$$

Question 7

Given the graph in Figure (1), at which $$x-$$values is $$f$$ discontinuous?

Accepted Answer

$$x = -7$$\ and\ x = -1$

Question 8

For the function $$f(x) = 3x^4$ + 4x^3$ + 3x^2$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 12x^3$ + 12x^2$ + 6x$$

Question 9

If $$p = 1000 + \frac{1000}{q} is $$the price-demand function, what is the revenue function $$R(q)$$?

Accepted Answer

$$R(q) = 1000q + 1000$$

Question 10

Given $$f(x, y) = 200 - 10x + 5y^2$, what is S_y(5$$, 4)$$ and its interpretation?

Accepted Answer

$$40$$,\ meaning\ sales\ increase\ by\ 40\ units\ for\ each\ additional\ thousand\ dollars\ spent\ on\ advertising$

Question 11

For the function $$f(x, y) = x^2$ y^3$ - 6x + 4y$$, what is the partial derivative $$f_x(x, y)$$?

Accepted Answer

$$2x y^3$ - 6$$

Question 12

A company’s cost function is $$C(x) = 80x^2$ + 3x + 1. $$What is the average cost function $$AC(x)$$?

Accepted Answer

$$AC(x) = 80x + 3 + \frac{1}{x}$$

Question 13

Given the price-demand function $$p = 60e$$^{-0.025q}$$, what is the revenue function R(q$$)$$?

Accepted Answer

$$R(q) = 60q e$$^{-0.025q}$$

Question 14

For the function $$f(x) = 2x^2$ - 5x + 7$$, what is the second derivative $$f''(x)$$?

Accepted Answer

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

Question 15

A manufacturing company’s production function is $$U(x, y) = 500x^2$ + 800y^3$ + 3x^4$ - 11x - 15y. $$What is the marginal productivity of labor function $$U_x(x, y)$$?

Accepted Answer

$$U_x(x, y) = 1000x + 12x^3$ - 11$$

Limits and Continuity from Graphs – Business Calculus Study Guidance

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