Q: A company manufactures x lawn mowers, and the total profit function is $$P(x) = 20x - 0.02x^2$ - 320. $$What is the marginal profit when 100 mowers are produced?

$$P'(100) = 20 - 0.04 \times 100 = 16$$

Q: Given $$R(t) = 20,000(0.86)^t$$, what is the rate of depreciation after 2 years?

$$R'(2) = 20,000 \times \ln(0.86) \times (0.86)^2$ \approx -3,958$$

Q: A company’s weekly cost function is $$C(x) = 10,000 + 90x - 0.05x^2$. What is the marginal cost at a production level of 500 tanks?

$$C'(500) = 90 - 0.1 \times 500 = 40$$

Q: A company’s cost function is $$C(x) = 60,000 + 300x. $$What is the average cost per unit if 500 frames are produced?

$$\frac{C(500)}{500} = \frac{60,000 + 300 \times 500}{500} = 420$$

Question 1

A company manufactures x lawn mowers, and the total profit function is $$P(x) = 20x - 0.02x^2$ - 320. $$What is the marginal profit when 100 mowers are produced?

Accepted Answer

$$P'(100) = 20 - 0.04 	imes 100 = 16$$

Question 2

A family bought a house for $$99,000 and sold it 15 years later for 195$$,000. If interest is compounded continuously, what is the annual nominal rate of interest earned?

Accepted Answer

$$r = \frac{1}{15} \ln\left(\frac{195,000}{99,000}\right) \approx 0.0476$$

Question 3

Given $$R(t) = 20,000(0.86)^t$$, what is the rate of depreciation after 2 years?

Accepted Answer

$$R'(2) = 20,000 	imes \ln(0.86) 	imes (0.86)^2$ \approx -3,958$$

Question 4

A hospital model relates systolic blood pressure to age: $$P(x) = 40 + 25 \ln(x+1). $$What is the rate of change of pressure at age 30?

Accepted Answer

$$P'(30) = \frac{25}{31} \approx 0.806$$

Question 5

A company’s weekly cost function is $$C(x) = 10,000 + 90x - 0.05x^2$. What is the marginal cost at a production level of 500 tanks?

Accepted Answer

$$C'(500) = 90 - 0.1 	imes 500 = 40$$

Question 6

Find the intervals where $$f(x) = 3x^2$ - 12x + 8 is $$increasing.

Accepted Answer

$$x \u003e 2$$

Question 7

For $$f(x) = x^3$ - 3x^2$ + 7x + 2$$, at what $$x-$$value does the inflection point occur?

Accepted Answer

$$x = 1$$

Question 8

A company manufactures $$x$$ HDTVs per month. The marginal profit is $$P'(x) = 165 - 0.1x. If $$production increases from 1,500 to 1,600 units, what is the change in monthly profit?

Accepted Answer

$$\int_{1500}^{1600} (165 - 0.1x) dx = [165x - 0.05x^2$]_{1500}^{1600} = 16,500 - 12,000 = 4,500$$

Question 9

Find the absolute maximum of $$f(x) = -x^2$ + 8x on$$ $$[0,5].$$

Accepted Answer

$$f(4) = -16 + 32 = 16$$

Question 10

Find the particular antiderivative of $$R'(x) = 500 - 0.4x$$ given $$R(0) = 0.$$

Accepted Answer

$$R(x) = 500x - 0.2x^2$

Question 11

Find the area bounded by $$y = x^2$ - 4$$ and $$y = 0$$ over $$[0,3].$$

Accepted Answer

$$\$$int_0$$^3 |x^2$ - 4| dx = \$$int_0$$^2 (4 - x^2$) dx + \$$int_2$$^3 (x^2$ - 4) dx = 8.333$$

Question 12

The rate of change of income from a vending machine is $$I'(t) = 1200 - 80t. $$What is the total income produced in the first 5 years?

Accepted Answer

$$\$$int_0$$^5 (1200 - 80t) dt = [1200t - 40t^2$]_0^5$ = 6000 - 1000 = 5000$$

Question 13

Find $$\int x \ln x dx$$ using integration by parts.

Accepted Answer

$$\int x \ln x dx = \frac{1}{2}x^2$ \ln x - \frac{1}{4}x^2$ + C$$

Question 14

A company’s cost function is $$C(x) = 60,000 + 300x. $$What is the average cost per unit if 500 frames are produced?

Accepted Answer

$$\frac{C(500)}{500} = \frac{60,000 + 300 	imes 500}{500} = 420$$

Question 15

A company sells $$x$$ smartphones per week. The price-demand equation is $$p = 500 - 0.4x. $$What price maximizes weekly revenue?

Accepted Answer

$$p = 250$$

Question 16

Find the indefinite integral $$\int (x^2$ - 3) - 6(8x) dx.$$

Accepted Answer

$$\frac{1}{3}x^3$ - 3x - 48x^2$ + C$$

Question 17

The weekly marginal cost of producing $$x$$ pairs of tennis shoes is $$C'(x) = 12 + \frac{2000}{x}. If $$fixed costs are $$2,000$$, what is the cost function?

Accepted Answer

$$C(x) = 12x + 2000 \ln x + 2000$$

Question 18

A manufacturer sells sunglasses for $$21 a $$pair. The price-demand equation is $$x = 9500 - 250p. If $$the price increases, will revenue increase or decrease?

Accepted Answer

Revenue will decrease because elasticity $$E(p) \u003e 1 at$$ $$p = 21$$

Question 19

Find the consumers’ surplus at price $$\bar{p} = 120$$ for $$p = D(x) = 200 - 0.02x.$$

Accepted Answer

$$\$$int_0$$^{4000} (200 - 0.02x) dx - 120 	imes 4000 = 160,000 - 480,000 = -320,000$$

Question 20

Find the producers’ surplus at price $$\bar{p} = 55$$ for $$p = S(x) = 15 + 0.1x + 0.003x^2$.

Accepted Answer

$$55 	imes 20 - \$$int_0$$^{20} (15 + 0.1x + 0.003x^2$) dx = 1,100 - [15x + 0.05x^2$ + 0.001x^3$]_0$$^{20}$$ = 1,100 - (300 + 20 + 8) = 772$$

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