Question 1

Given the cost function $$C(x) = 12x + 6000$$ and the ticket price of $$60$$, what is the profit function $$P(x)$$ for selling $$x$$ premium tickets?

Accepted Answer

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

Question 2

A vendor sells t-shirts with a demand function $$p = 40 - 0.04x$$ and cost function $$C(x) = 3x + 200. $$What is the profit from selling 400 t-shirts?

Accepted Answer

$$P(400) = 8200$$

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

$$7$$

Question 5

Given the function $$f(x) = x^3$ - x^2$ + 2x + 1$$, what is the average rate of change from $$x = 2 to$$ $$x = 8$$?

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

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

For the function $$f(x) = 6x^4$ + 4x^5$ + 3$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 24x^3$ + 20x^4$

Question 10

A pharmaceutical company’s revenue is $$R(x, y) = 200x + 200y - 4x^2$ - 3y^2$ - 8xy. $$What is the marginal revenue with respect to $$x$$?

Accepted Answer

$$R_x(x, y) = 200 - 8x - 8y$$

Question 11

Refer to Figure (1). At which $$x-$$values is the function discontinuous, and why?

Accepted Answer

$$x = -7 ($$undefined), $$x = -1 ($$limit does not exist)

Question 12

Refer to Figure (4). What is $$\lim\limits_{x 	o 4^-} f(x)$$?

Accepted Answer

$$\infty$$

Question 13

Given $$f(x, y) = 5x^2$y^3$$ - $$4x^3y^2$ + 6x - 8\ln y$$, what is $$f_x(x, y)$$?

Accepted Answer

$$f_x(x, y) = 10xy^3$ - 12x^2$y^2 + 6$

Question 14

A company’s cost function is $$C(x) = 80x^2$ + 3x + 1. $$What is the marginal cost when $$x = 8$$?

Accepted Answer

$$C'(8) = 128 + 3 = 131$$

Question 15

Given $$f(x, y) = 3x^2$y - 6x + 4y^3$ - 2x^3$ + 4y$$, what is $$f_{yy}(x, y)$$?

Accepted Answer

$$f_{yy}(x, y) = 24y$$

Question 16

A store’s HDTV sales are modeled by $$S(x, y) = 200 - 10x + 5y^2$. What does S_y(5$$, 4)$$ represent?

Accepted Answer

The increase in sales for each additional thousand dollars spent on advertising when $$x = 5$$ and $$y = 4$$

Business Calculus Midterm Review: Limits and Continuity from Graphs

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