Question 1

Suppose a company wants to maximize its weekly revenue from selling smartphones, where the price-demand equation is $$p = 500 - 0.4x$$ and $$x is $$the number of phones sold per week. What is the number of phones that should be produced to maximize weekly revenue?

Accepted Answer

$$625$$

Question 2

A car rental agency rents 200 cars per day at $$30 per day. For each 1 $$increase in price, 5 fewer cars are rented. What price should the cars be rented at to produce the maximum income?

Accepted Answer

$$35$$

Question 3

Find the indefinite integral: $$\int (2 - 4z + z^2$)\,dz$$

Accepted Answer

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

Question 4

A manufacturer can produce sunglasses at a cost of $$5$$ apiece and estimates that if they are sold for $$x$$ dollars apiece, consumers will buy $$100(20-x)$$ sunglasses a day. At what price should the manufacturer sell the sunglasses to maximize profit?

Accepted Answer

$$12.50$$

Question 5

Evaluate the definite integral: $$\$$int_1$$^3 \frac{1}{x}\,dx$$

Accepted Answer

$$\ln(3) - \ln(1)$$

Question 6

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side, using $$480 ft of $$fencing. What dimensions maximize the total enclosed area?

Accepted Answer

$$80 ft by$$ $$120 ft$

Question 7

Given $$f'(x) = 5x^4$ - x^3$ + 2x^2$ - e^x$$ and $$f(0) = -4$$, find the particular antiderivative $$f(x).$$

Accepted Answer

$$x^5$$ - \frac{1}{4}$$x^4$$ + \frac{2}{3}$$x^3 - e^x - 3$

Question 8

Which of the following is the correct application of the Fundamental Theorem of Calculus for a continuous function $$f on$$ $$[a, b]$$ with antiderivative $$F$$?

Accepted Answer

$$\int_a^b f(x)\,dx = F(b) - F(a)$$

Question 9

A company has the marginal cost function $$C'(x) = 4x^3$ - 6x^2$ and a fixed cost of 5000$. What is the total cost function C(x$$)$$?

Accepted Answer

$$x^4$$ - $$2x^3 + 5000$

Question 10

Find the average value of $$f(x) = x - 3x^2$ over the interval $$[-1, 2]$$.

Accepted Answer

$$-\frac{5}{2}$$

Question 11

A fence is to be built to enclose a rectangular area of $$450$$ square feet. The fence along three sides costs $$3$$ per foot, and the material for the fourth side costs $$9$$ per foot. What are the dimensions of the rectangle that will allow for the most economical fence to be built?

Accepted Answer

$$15 ft by$$ $$30 ft$

Question 12

Which of the following is the correct indefinite integral for $$\int 4t$$^{-1}$$\,dt$$?

Accepted Answer

$$4\ln|t| + C$$

Question 13

A defense contractor's labor use rate for assembling missile control systems is $$L'(x) = 2400x$$^{-1/2}$$. How many labor hours are required to assemble from the 16th through the 25th control units?

Accepted Answer

$$4800$$ hours

Question 14

Given the marginal average cost function $$\overline{C}'(x) = -\frac{5000}{x^2$}$$ and $$\overline{C}(100) = 250$$, what is the average cost function?

Accepted Answer

$$\overline{C}(x) = \frac{5000}{x} + 200$$

Question 15

A farmer has $$1800 ft of $$fencing and wants to fence off a rectangular field that borders a straight river (no fence needed along the river). What are the dimensions of the field that has the largest area?

Accepted Answer

$$450 ft by$$ $$900 ft$

Question 16

Which of the following statements about absolute extrema is TRUE?

Accepted Answer

Absolute extrema must occur at critical numbers or at endpoints.

Question 17

Find the general solution to the differential equation $$\frac{dy}{dx} = 3x^2$ + 4x + 2.$$

Accepted Answer

$$y = x^3$ + 2x^2$ + 2x + C$$

Question 18

A university student center sells $$1,600$$ cups of coffee per day at $$2.40$$ per cup. For every $$0.05$$ reduction in price, $$50$$ more cups are sold. What price should be charged to maximize revenue?

Accepted Answer

$$2.00$$

Question 19

Which of the following is the correct indefinite integral for $$\int e^x\,dx$$?

Accepted Answer

$$e^x + C$$

Question 20

Suppose $$f(x) is $$continuous on $$[a, b]$$ and has only one critical number $$c in$$ $$(a, b)$$ with $$f''(c) \u003e 0. $$What can you conclude about $$f(c)$$?

Accepted Answer

$$f(c) is $$the absolute minimum of $$f on$$ $$[a, b]$$

Business Calculus Exam 4 Study Guide – Step-by-Step Guidance

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