Question 1

Let $$f(x) = 5\sin(x) - 4x on$$ $$[1,3]. $$What is the absolute maximum value of $$f(x) on $$this interval?

Accepted Answer

$$-3.804 at$$ $$x = 5/4$$

Question 2

Which of the following is the correct process for finding the absolute extrema of a continuous function $$f(x) on a $$closed interval $$[a,b]$$?

Accepted Answer

Find the critical points of $$f(x) in$$ $$[a,b]$$, evaluate $$f(x) at $$the critical points and endpoints, and compare the values.

Question 3

A company manufactures and sells smartphones per week. The weekly price-demand and cost equations are $$p = 500 - 0.04x$$ and $$C(x) = 26000 + 20x. $$What is the production level $$x$$ that maximizes weekly profit?

Accepted Answer

$$6,000$$ phones

Question 4

Which of the following is the correct statement of the Fundamental Theorem of Calculus?

Accepted Answer

If $$f is $$continuous on $$[a,b]$$ and $$F is $$any antiderivative of $$f$$, then $$\int_a^b f(x) dx = F(b) - F(a).$$

Question 5

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

Accepted Answer

$$x^3$$ - \frac{1}{4}$$x^4 + C$

Question 6

A manufacturer wants to build a rectangular area of 450 square feet. The fence along 3 sides is to be made of material that costs twice as much as the material used for the fourth side. What are the dimensions of the rectangle that will minimize the cost?

Accepted Answer

Width $$= 15 ft$$, Length $$= 30 ft$$

Question 7

Given the marginal cost function $$C'(x) = 6x^2$ - 4x$$ and a fixed cost of $$5000$$, what is the total cost function $$C(x)$$?

Accepted Answer

$$C(x) = 2x^3$ - 2x^2$ + 5000$$

Question 8

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

Accepted Answer

$$-\frac{11}{6}$$

Question 9

A manufacturer estimates that after $$t$$ months, the rate of sales of HDD will decline at $$S'(t) = -350e$$^{-0.1t}$$ units per month. If monthly sales are currently 2790$ HDD per month, how long will the company continue to manufacture HDD if it stops when sales reach 900$ HDD per month?

Accepted Answer

$$t = 15$$ months

Question 10

Which of the following best describes the Second Derivative Test for absolute extrema?

Accepted Answer

If $$f''(x) \u003e 0 at a $$critical number $$x = a$$, $$f(a) is a $$local minimum; if $$f''(x) \u003c 0$$, $$f(a) is a $$local maximum.

Question 11

Evaluate the definite integral $$\$$int_1$$^2 4x \ln(x) dx.$$

Accepted Answer

$$2(2\ln 2 - 1)$$

Question 12

A company’s marginal cost function is $$C'(x) = 400 - 0.5x. $$What is the total increase in cost to raise production from 200 to 800 units?

Accepted Answer

$$120,000$$

Question 13

Find the particular antiderivative of $$f(x)$$ that satisfies the given condition: $$f'(x) = 5x^4$ + 3x^2$ + 2x^3$ - x$$, $$f(1) = -4.$$

Accepted Answer

$$f(x) = x^5$ + x^3$ + 0.5x^4$ - 0.5x^2$ - 4.5$$

Question 14

Which of the following is the correct formula for the average value of a continuous function $$f(x) on$$ $$[a,b]$$?

Accepted Answer

$$\frac{1}{b-a} \int_a^b f(x) dx$$

Question 15

A rectangular playground is to be fenced on all sides and divided by another fence parallel to one side. If the total fencing available is $$600 ft$$, what are the dimensions of the playground that maximize the area?

Accepted Answer

Width $$= 100 ft$$, Length $$= 200 ft$$

Question 16

Find the indefinite integral $$\int (\sqrt{2x} + 5e^x) dx.$$

Accepted Answer

$$\frac{2}{3}x$$^{3/2}$$ + 5e^x + C$$

Question 17

A farmer has $$1000 ft of $$fencing and wants to fence off a rectangular field that borders a straight river. No fence is needed along the river. What are the dimensions of the field that has the largest area?

Accepted Answer

Width $$= 250 ft$$, Length $$= 500 ft$$

Question 18

Which of the following is a correct property of definite integrals?

Accepted Answer

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

Calculus Study Guide: Absolute Extrema, Optimization, Indefinite Integrals, and Applications

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