Question 1

Which of the following best describes the end behavior of the function $$y = 5 - e$$^{-x}$$ as x$$ 	o \infty$$?

Accepted Answer

The function approaches $$y = 5$$ from below.

Question 2

Given $$f(x) = (3\sqrt{2x + 4}$$)^{3x}$$, which of the following is the correct expression for f$$''(x)$$?

Accepted Answer

$$f''(x) = (3\sqrt{2x + 4}$$)^{3x}$$ \left[ 9(\ln(3\sqrt{2x+4}))^2$ + \frac{9}{2x+4} + \frac{9x}{(2x+4)^2$} \right]$$

Question 3

For the function $$f(x) = e$$^{2x^2 + 3x + 4}$$, which of the following statements about concavity is correct?

Accepted Answer

The function is concave up for all real $$x.$$

Question 4

Given $$\ln(x+y) = x^3$y^3$$ + \$$ln(x^2+2) - 8$, what is the slope of the tangent line at the point (1$$,2)$?

Accepted Answer

$$m = 12$$

Question 5

A rectangular region PQRS is to be inscribed in an isosceles triangle with base $$AB = 24 ft $$and height $$CE = 10 ft. If$$ $$PQ = 2x$$ and $$PS = y$$, what values of $$x$$ and $$y$$ maximize the area of the rectangle?

Accepted Answer

$$x = 6, y = 5$$

Question 6

An axis-parallel rectangle has one corner at $$(0,0)$$ and the opposite corner at $$(a,b) on $$the line $$y = -\frac{1}{4}x + 5. $$What are the dimensions of the rectangle with maximum area?

Accepted Answer

$$a = 10, b = 2.5$$

Question 7

A 5-foot-tall actress is 12 feet from a pole and moving away at 2 ft/s. At the moment her shadow is 6 feet long and increasing at 2 ft/s, what is the height of the spotlight?

Accepted Answer

$$h = 15 ft$$

Question 8

A cylindrical pump (diameter 4 in) is used to inflate a spherical balloon. At a moment when the pump's air height is 12 in and decreasing at 4 in/min, and the balloon's diameter is 6 in, what is the rate of change of the balloon's diameter?

Accepted Answer

$$\frac{dD}{dt} = \frac{2}{9\pi} in/$$min

Question 9

Evaluate $$\$$int_1$$^e x + e$$^{-x}$$ \, dx.$$

Accepted Answer

$$\frac{e^2$ - 1}{2} + (1 - \frac{1}{e})$$

Question 10

Express the left Riemann sum for $$f(x) = 3x on$$ $$[0,5]$$ using 167 rectangles in sigma notation.

Accepted Answer

$$\sum_{i=0}^{166} 3x_i \Delta x$$, where $$x_i = 0 + i\Delta x$$, $$\Delta x = \frac{5}{167}$$

Question 11

Compute $$\$$int_0$$^5 3x \, dx$$ geometrically.

Accepted Answer

$$\frac{1}{2} \cdot 5 \cdot 15 = 37.5$$

Question 12

Given marginal cost $$c(x) = 14x\sqrt{x^2$ + 4}$$ and marginal revenue $$r(x) = 3x$$, what is the total change in profit from $$x = 3 to$$ $$x = 7$$?

Accepted Answer

$$\$$int_3$$^7 [3x - 14x\sqrt{x^2$ + 4}] \, dx$$

Question 13

Find the area between $$y = 3x - 1$$ and $$y = 5 - 2x - 3 on$$ $$(0,2).$$

Accepted Answer

$$\$$int_0$$^2 [(3x-1) - (5-2x-3)] \, dx$$

Question 14

Suppose you are tasked with maximizing the area of a rectangle inscribed under the line $$y = -\frac{1}{4}x + 5 in $$the first quadrant. What is the maximum possible area?

Accepted Answer

$$12.5$$

Question 15

If $$f(x) = e$$^{2x^2 + 3x + 4}$$, what is f$$''(x)$$?

Accepted Answer

$$f''(x) = e$$^{2x^2 + 3x + 4}$$ [4x^2$ + 6x + 2 + (4x + 3)^2$]$$

Question 16

A company’s marginal cost is $$c(x) = 14x\sqrt{x^2$ + 4}$$ and marginal revenue is $$r(x) = 3x. If $$they produce 7 units instead of 3, what is the net change in profit?

Accepted Answer

$$\$$int_3$$^7 [3x - 14x\sqrt{x^2$ + 4}] \, dx$$

Question 17

If $$\frac{d}{dx} \arctan(x) = \frac{1}{1 + x^2$}$$, what is $$\int \frac{1}{1 + x^2$} \, dx$$?

Accepted Answer

$$\arctan(x) + C$$

Business Calculus Practice Final: Step-by-Step Guidance

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