Question 1

What is the derivative of $$f(x) = 40e$$^{-2x}$$ with respect to x$?

Accepted Answer

$$-80e^{-2x}$$

Question 2

Given $$g(x) = x^3$ e$$^{-4x}$$, which of the following best describes the intervals where g(x$$)$$ is increasing?

Accepted Answer

Where $$g'(x) \u003e 0$$, i.e., for $$0 \u003c x \u003c \frac{3}{4}$$

Question 3

Find the second derivative of $$g(x) = \ln|5x + 1|.$$

Accepted Answer

$$g''(x) = -\frac{25}{(5x+1)^2$}$$

Question 4

For $$f(x) = 2x^3$ - 15x^2$ + 24x + 6$$, at which $$x-$$value does the function have an inflection point?

Accepted Answer

$$x = 3$$

Question 5

Given $$f(x) = \frac{1}{4}x^4$ - \frac{7}{3}x^3$ + 5x^2$ - 2 on$$ $$[-1,4]$$, which value of $$x$$ gives the absolute maximum?

Accepted Answer

$$x = 3$$

Question 6

A farmer wants to maximize the area of a rectangular pen with one side against a stream (no fencing needed). If fencing costs $$10$$ dollars per meter and the budget is $$1500$$ dollars, what is the maximum area that can be enclosed?

Accepted Answer

$$562.5 m$^2$$

Question 7

If $$C(x) = 300 + 1.2x is $$the cost and $$p = D(x) = 200 - 0.2x is $$the price per unit, what quantity $$x$$ maximizes profit?

Accepted Answer

$$x = 100$$

Question 8

What is the derivative of $$f(x) = \ln|t^3$ + 4t^2$ + 2|$$ with respect to $$t$$?

Accepted Answer

$$\frac{3t^2$ + 8t + 4}{t^3$ + 4t^2$ + 2}$$

Question 9

Find the derivative of $$f(x) = \sqrt{x^4$ + 5}$$ with respect to $$x.$$

Accepted Answer

$$\frac{4x^3$}{2\sqrt{x^4$ + 5}}$$

Question 10

For $$f(x) = x^3$ - 9x^2$ + 4x + 6$$, what is $$f'''(x)$$?

Accepted Answer

$$6$$

Question 11

Which of the following is the correct derivative for $$f(x) = (5x + 3)^4$ e$$^{x^3}$$?

Accepted Answer

$$4(5x + 3)^3$ e$$^{x^3}$$ \cdot 5 + (5x + 3)^4$ e$$^{x^3}$$ \cdot 3x^2$

Question 12

For $$f(x) = \frac{2}{3}x^3$ + x^2$ - 12x + 10$$, which $$x-$$value is a local minimum?

Accepted Answer

$$x = -3$$

Question 13

If $$f(x) = 2x^3$ - 15x^2$ + 24x + 6$$, on which interval is the function concave up?

Accepted Answer

$$x \u003e 3$$

Question 14

A company’s revenue function is $$R(x) = x(200 - 0.2x). $$What is the marginal revenue when $$x = 100$$?

Accepted Answer

$$160$$

Question 15

If $$f(x) = x^3$ - 9x^2$ + 4x + 6$$, what is $$f''(x)$$?

Accepted Answer

$$6x - 18$$

Question 16

A business wants to maximize profit given $$C(x) = 300 + 1.2x$$ and $$p = 200 - 0.2x. $$What is the maximum profit?

Accepted Answer

$$7,880$$

Question 17

If $$f(x) = \frac{1}{4}x^4$ - \frac{7}{3}x^3$ + 5x^2$ - 2$$, what is $$f'(x)$$?

Accepted Answer

Question 18

A farmer builds a pen with one side against a stream. If the length parallel to the stream is $$x$$ meters and the width is $$y$$ meters, what values of $$x$$ and $$y$$ maximize the area given $$10(x + 2y) = 1500$$?

Accepted Answer

$$x = 75$$, $$y = 37.5$$

Question 19

Which of the following is the correct formula for the quadratic formula?

Accepted Answer

$$x = \frac{-B \pm \sqrt{B^2$ - 4AC}}{2A}$$

Business Calculus Exam 2 Review – Step-by-Step Guidance

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