Q: Let $$h(x) = \dfrac{\tan(x) - \sin(x)}{x^2$}. $$What is $$h'(x)$$?

$$h'(x) = \dfrac{\sec^2$(x) - \cos(x)}{x^2$} - \dfrac{2(\tan(x) - \sin(x))}{x^3$}$$

Q: The pH level of a solution over time (in hours) is given by $$H(t) = \ln(0.5t + 1). $$What is the rate of change of the pH level after 8 hours?

$$H'(8) = \dfrac{0.5}{0.5 \times 8 + 1}$$

Question 1

Let $$f(x) = 5 + e$$^{x^2}$$. What is f$$'(x)$$?

Accepted Answer

$$f'(x) = 2x e$$^{x^2}$$

Question 2

Suppose $$g(x) = \ln(1 - 5x^2$ + x^3$). $$What is $$g'(x)$$?

Accepted Answer

$$g'(x) = \dfrac{-10x + 3x^2$}{1 - 5x^2$ + x^3$}$$

Question 3

A company’s profit function is $$P(x) = 3x + e$$^{x^2 + 1}$$. What is the marginal profit when x = 0$?

Accepted Answer

$$P'(0) = 3 + 2e$$

Question 4

Given $$y = \log_{10}(2x^4$ + 5x)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{8x^3$ + 5}{(2x^4$ + 5x) \ln 10}$$

Question 5

If $$f(x) = (3x + 7) \ln(2x - 1)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 3 \ln(2x - 1) + \dfrac{3x + 7}{2x - 1} \cdot 2$$

Question 6

Let $$h(x) = \dfrac{	an(x) - \sin(x)}{x^2$}. $$What is $$h'(x)$$?

Accepted Answer

$$h'(x) = \dfrac{\sec^2$(x) - \cos(x)}{x^2$} - \dfrac{2(	an(x) - \sin(x))}{x^3$}$$

Question 7

For $$f(x) = x^3$ - 3x^2$ + 2x^2$ - 8$$, find $$f''(0).$$

Accepted Answer

$$f''(0) = 2$$

Question 8

The pH level of a solution over time (in hours) is given by $$H(t) = \ln(0.5t + 1). $$What is the rate of change of the pH level after 8 hours?

Accepted Answer

$$H'(8) = \dfrac{0.5}{0.5 	imes 8 + 1}$$

Question 9

The height $$H(t) (in $$inches) of a young tree based on its age $$t (in $$months) is modeled by $$H(t) = 145(1 - e$$^{-0.03t}$$). $$What happens to the height of the tree as $$t 	o \infty$$?

Accepted Answer

The height approaches $$145$$ inches.

Question 10

Given the graph of a function $$g(x)$$, how do you determine the intervals where $$g(x) is $$increasing?

Accepted Answer

Where $$g'(x) \u003e 0$$

Question 11

Suppose $$f(x) = x^4$ - 2x^2$ + 6x + 4. $$What are the critical points?

Accepted Answer

Where $$f'(x) = 0$$

Question 12

If $$g(x) = \dfrac{1}{x^2$}$$, what is the interval where $$g(x) is $$decreasing?

Accepted Answer

$$x \u003e 0$$

Question 13

A company tracks the number of downloads $$D(t) in $$days by $$D(t) = 500 - 2t^2$. At what time is the download rate zero?

Accepted Answer

Question 14

Given $$y = \ln(x^4$ - 5)$$, what is $$y''$$?

Accepted Answer

$$y'' = \dfrac{12x^2$(x^4 + 5$$)}{$$(x^4$$ - $$5)^2$$}$

Question 15

If $$f(x)$$ has $$f'(x) \u003e 0 on$$ $$(-\infty, -3) \cup (3, \infty)$$ and $$f'(x) = 0 at$$ $$x = -3, 3$$, what can you say about $$f(x) at$$ $$x = -3$$ and $$x = 3$$?

Accepted Answer

$$x = -3$$ and $$x = 3$$ are critical points

Question 16

Which rule is used to differentiate $$y = x^2$ \sin(x)$$?

Accepted Answer

Product Rule

Question 17

If $$f(x) = \sin^2$(x)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 2\sin(x)\cos(x)$$

Question 18

Suppose $$f(x) is $$continuous and differentiable everywhere, and $$f'(x) = 0$$ only at $$x = -3, 3. $$What does this imply about the function?

Accepted Answer

The function has horizontal tangents only at $$x = -3$$ and $$x = 3$$

Question 19

If $$f(x) = \sin(2x)$$, what is $$f''(x)$$?

Accepted Answer

$$f''(x) = -4\sin(2x)$$

Business Calculus: Derivatives, Applications, and Graph Analysis Study Guide

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