Question 1

Given $$\$$int_3$$^5 f(x)\,dx = 7$$ and $$\$$int_3$$^5 g(x)\,dx = 1$$, find $$\$$int_3$$^5 [4f(x) - 2g(x)]\,dx.$$

Accepted Answer

$$30$$

Question 2

If the marginal cost function is $$C'(x) = 10x - 7$$ and the fixed cost is $$8$$, what is the total cost function $$C(x)$$?

Accepted Answer

$$C(x) = 5x^2$ - 7x + 8$$

Question 3

Refer to the first image (rectangles under a line). Are the rectangles left rectangles, right rectangles, or neither? Would the area be overestimated or underestimated?

Accepted Answer

Left rectangles; area is underestimated

Question 4

Refer to the second image (rectangles under a curve). Are the rectangles left rectangles, right rectangles, or neither? Would the area be overestimated or underestimated?

Accepted Answer

Right rectangles; area is overestimated

Question 5

Set up a definite integral that represents the shaded area in the third image (area under $$y = g(x)$$ from $$x=1 to$$ $$x=2).$$

Accepted Answer

$$\$$int_1$$^2 g(x)\,dx$$

Question 6

Set up a definite integral that represents the shaded area in the fourth image (area under $$y = h(x)$$ from $$x=0 to$$ $$x=2).$$

Accepted Answer

$$\$$int_0$$^2 h(x)\,dx$$

Question 7

Given the areas in the fifth image, what is $$\int_a^b f(x)\,dx$$?

Accepted Answer

$$-1$$

Question 8

Given the areas in the fifth image, what is $$\int_c^a f(x)\,dx$$?

Accepted Answer

$$-1$$

Question 9

Find the average value of the function $$y = 6 - 3x^2$ over the interval $$[-4, 6]$$.

Accepted Answer

$$-22$$

Question 10

Find the equation of the curve that passes through $$(8, 30) if $$the slope at any point $$x is$$ $$\frac{dy}{dx} = 3x.$$

Accepted Answer

$$y = \frac{3}{2}x^2$ + 18$$

Question 11

Given $$u = 7x + 12$$, what is $$du$$?

Accepted Answer

$$du = 7dx$$

Question 12

Evaluate $$\int (2x^5$ - 7x^3$ + 5)\,dx.$$

Accepted Answer

$$\frac{1}{3}x^6$ - \frac{7}{4}x^4$ + 5x + C$$

Question 13

Suppose the Gini Index for a country in 2017 was $$0.286$$ and is projected to be $$0.444 in 2030. If $$this projection is correct, the Gini Index will ______ and income will be ______ equally divided in 2030 than in 2017.

Accepted Answer

increase; less

Question 14

Suppose the Gini Index for a country in 2017 was $$0.788$$ and is projected to be $$0.172 in 2030. If $$this projection is correct, the Gini Index will ______ and income will be ______ equally divided in 2030 than in 2017.

Accepted Answer

decrease; more

Question 15

A population of bacteria grows at a rate of $$P'(t) = 10e^t. $$How much does the population increase from $$t=2 to$$ $$t=7$$? (Round your answer to two decimal places.)

Accepted Answer

$$10(e^7$$ - $$e^2$$) \approx 10892.44$$

Question 16

Find the total income produced by a continuous income stream in the first nine years if the rate of flow is $$f(t) = 500.$$

Accepted Answer

$$4500$$

Question 17

Evaluate $$\$$int_0$$^5 e$$^{2x}$$\,dx ($$round to two decimal places).

Accepted Answer

$$11012.73$$

Question 18

Find the area bounded by the parabolas $$y = 6x - x^2$ and y$$ = $$x^2 - 2x$ (see the last image). Round your answer to three decimal places.

Accepted Answer

$$21.333$$

Business Calculus Exam 4 Review: Step-by-Step Guidance

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