Question 1

Evaluate the following definite integral, which models the total accumulated profit over a time period: $$\$$int_0$$^2 (6e$$^{3x}$$)\,dx.$$

Accepted Answer

Question 2

Given $$y = x^2$ e$$^{3x}$$, find $$\frac{dy}{dx}$$.

Accepted Answer

$$2x e$$^{3x}$$ + 3x^2$ e$$^{3x}$$

Question 3

Find the critical values, if any, for $$f(x) = xe$$^{-x^2}$$.

Accepted Answer

$$x = 0, x = \frac{1}{\sqrt{2}}$$

Question 4

A company’s marginal profit function is $$P'(x) = 5x^2$ - 10x + 6. If$$ $$P(2) = 20$$, find the profit function $$P(x).$$

Accepted Answer

$$P(x) = \frac{5}{3}x^3$ - 5x^2$ + 6x + 4$$

Question 5

The height $$h (in $$meters) of a plant grows so that the rate of change is given by $$\frac{dh}{dt} = 3t^2$ + 2t$$, where $$t is in $$weeks. If $$h(0) = 5$$, what is $$h(2)$$?

Accepted Answer

$$h(2) = 21$$

Question 6

Evaluate $$\int (5x^3$ + 8x^2$ - x)\,dx.$$

Accepted Answer

$$\frac{5}{4}x^4$ + \frac{8}{3}x^3$ - \frac{1}{2}x^2$ + C$$

Question 7

A company’s marginal cost function is $$C'(x) = 160 - e$$^{-0.2x}$$, where x$ is the number of items produced. If the fixed cost is 500$, find the total cost function C(x$$)$$.

Accepted Answer

$$C(x) = 160x + 5e$$^{-0.2x}$$ + 500$$

Question 8

Find the area bounded by the curves $$y = 5 - x$$ and $$y = x^2$.

Accepted Answer

$$\int_{-1}^{2} [(5-x) - x^2$] dx$$

Question 9

Given $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, find all local maxima and minima.

Accepted Answer

Local max at $$x = 1$$, local min at $$x = 3$$

Question 10

A retailer sells calculators for $$p$$ dollars each. The demand function is $$p = 50 - 2x$$, where $$x is $$the number of calculators sold. What price maximizes revenue?

Accepted Answer

$$p = 25$$

Question 11

Given $$f''(x) = -6x + 12$$, determine where the original function $$f(x) is $$concave up.

Accepted Answer

$$x \u003e 2$$

Question 12

A company’s marginal revenue function is $$R'(x) = 100 - 4x. If $$fixed revenue is $$0$$, what is the total revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 100x - 2x^2$

Question 13

Find the absolute maximum and minimum values of $$f(x) = x^3$ - 3x^2$ + 1 on $$the interval $$[0, 3].$$

Accepted Answer

Max at $$x=3$$, min at $$x=2$$

Question 14

If $$y = \ln(x^2$ + 5y^2$)$$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

Question 15

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$How many units should be produced to maximize profit?

Accepted Answer

$$x = 10$$

Question 16

Evaluate $$\lim_{x 	o 0} \frac{e$$^{2x}$$ - 1}{x}.$$

Accepted Answer

$$2$$

Question 17

Given $$f(x) = x^2$ e$$^{-x}$$, find all critical points.

Accepted Answer

$$x = 0, x = 2$$

Question 18

A model for advertising is given by $$M(q) = 600 \ln(q+500)$$, where $$q is $$the number of units sold. What does $$M'(q)$$ represent?

Accepted Answer

The marginal effect of selling one more unit on advertising impact

Question 19

Given $$f(x) = 40x$$^{0.5}$$, where f(x$$)$$ is in millions of dollars of finished product, and x$ is in billions of dollars spent on labor, find the marginal productivity f$$'(x)$$ at x = 10$.

Accepted Answer

$$f'(10) = 2$$

Business Calculus: First Exam Review Study Notes

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