Q: The height $$H(t) (in $$inches) of a young tree is modeled by $$H(t) = 145(1 - e$$^{-0.03t}$$). $$What is the rate at which the height is changing after 48 months?

$$H'(48) = 145 \times 0.03 \times e$$^{-0.03 \times 48}$$

Q: A solution's pH level 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?

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

Question 1

Which of the following is the correct derivative of $$f(x) = 5 + e$$^{x^2}$$?

Accepted Answer

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

Question 2

Given $$p(x) = \ln(1 - 5x^2$ + x^3$)$$, what is $$p'(x)$$?

Accepted Answer

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

Question 3

A company tracks the number of downloads $$D(t) of $$its new app, where $$D(t) = 500 - 2t^2$. What is the instantaneous rate of change of downloads at t = 5$ days?

Accepted Answer

$$D'(5) = -20$$

Question 4

If $$y = 3 + e$$^{4x+1}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = 4e$$^{4x+1}$$

Question 5

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

Accepted Answer

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

Question 6

For $$g(x) = \sin^2$(2x)$$, what is $$g'(x)$$?

Accepted Answer

$$g'(x) = 4\sin(2x)\cos(2x)$$

Question 7

The height $$H(t) (in $$inches) of a young tree is modeled by $$H(t) = 145(1 - e$$^{-0.03t}$$). $$What is the rate at which the height is changing after 48 months?

Accepted Answer

$$H'(48) = 145 	imes 0.03 	imes e$$^{-0.03 	imes 48}$$

Question 8

Which of the following intervals describes where a function is concave up?

Accepted Answer

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

Question 9

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

Accepted Answer

$$f''(0) = 4$$

Question 10

If $$y = \ln(x^4$ - 5)$$, what is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$\frac{d^2$y}{dx^2$} = \frac{4x^2$(x^4 - 9$$)}{$$(x^4$$ - $$5)^2$$}$

Question 11

A solution's pH level 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

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

Question 12

Which rule is used to find the derivative of $$f(x) = x^2$ \sin(x)$$?

Accepted Answer

Product Rule

Question 13

Given $$f(x) = \frac{	an(x) - \sin(x)}{x^2$}$$, which rule(s) must be used to find $$f'(x)$$?

Accepted Answer

Quotient Rule and derivatives of trigonometric functions

Question 14

If $$f'(x) = 0 at$$ $$x = a$$, what does this indicate about the function $$f(x) at$$ $$x = a$$?

Accepted Answer

There may be a local maximum or minimum at $$x = a$$

Question 15

Which of the following is the correct second derivative of $$f(x) = -9x^4$ + 3x^2$ - 2$$?

Accepted Answer

$$f''(x) = -108x^2$ + 6$$

Question 16

Suppose a function $$f(x) is $$continuous and differentiable everywhere, has a relative minimum at $$x = -3$$, a relative maximum at $$x = 0$$, and $$f'(x) \u003e 0 on$$ $$(-3,0)$$ and $$(0,8). $$Which of the following statements is true?

Accepted Answer

The function is increasing on $$(-3,0)$$ and $$(0,8)$$

Question 17

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

Accepted Answer

$$\frac{dy}{dx} = \frac{8x^3$ + 5}{2x^4$ + 5x}$$

Question 18

A company observes that the number of downloads is increasing at a rate of $$D'(t) = 3t^2$ + 2t. $$How many total downloads have occurred when the download rate is 20 downloads per day?

Accepted Answer

Solve $$3t^2 + 2t = 20$ for t$, then compute D(t$$)$$

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

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