Question 1

A car moves along a straight road such that its position at time $$t is $$given by $$s(t) = 3t^2$ - 2t + 5. $$What is the car's acceleration at $$t = 4$$ seconds?

Accepted Answer

$$a(4) = 6$$

Question 2

Which of the following correctly expresses the derivative of $$y = \arcsin(x)$$ with respect to $$x$$?

Accepted Answer

$$\frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2$}}$$

Question 3

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

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 1}$$

Question 4

A medication is administered such that its concentration in the bloodstream at time $$t (in $$hours) is $$C(t) = 5e$$^{-0.2t}$$. What is the rate of change of the concentration at t = 3$ hours?

Accepted Answer

$$C'(3) = -1e$$^{-0.6}$$

Question 5

Which of the following is the correct derivative of $$y = 	an$$^{-1}$$(2x)$$ with respect to $$x$$?

Accepted Answer

$$\frac{2}{1 + 4x^2$}$$

Question 6

A ball is thrown upward and its height at time $$t is$$ $$s(t) = 20t - 5t^2$. What is the maximum height reached by the ball?

Accepted Answer

Question 7

Which of the following is the correct application of the product rule for $$f(x) = x^2$ \sin(x)$$?

Accepted Answer

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

Question 8

If $$f(x) = \sin(x)$$ and $$g(x) = \cos(x)$$, what is $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)$$?

Accepted Answer

$$\frac{\cos(x)\cos(x) - \sin(x)(-\sin(x))}{\cos^2$(x)}$$

Question 9

Which of the following is the correct derivative of $$y = \sec$$^{-1}$$(x)$$ for $$x \u003e 1$$?

Accepted Answer

$$\frac{1}{|x|\sqrt{x^2$-1}}$$

Question 10

A population of bacteria grows according to $$P(t) = 100e$$^{0.03t}$$. What is the instantaneous rate of growth at t = 10$ hours?

Accepted Answer

$$3e^{0.3}$$

Question 11

Which of the following is the correct chain rule application for $$y = \sin(3x^2$)$$?

Accepted Answer

$$y' = 6x \cos(3x^2$)$$

Question 12

If $$f(x) = \ln(x)$$, what is the value of $$f'(e)$$?

Accepted Answer

$$1$$

Question 13

Which of the following is the correct derivative of $$y = \cos$$^{-1}$$(x)$$?

Accepted Answer

$$-\frac{1}{\sqrt{1-x^2$}}$$

Question 14

A particle moves such that its position is $$s(t) = t^3$ - 6t^2$ + 9t + 2. At $$what time $$t is $$the velocity zero?

Accepted Answer

$$t = 1$$ and $$t = 3$$

Question 15

Which of the following is the correct derivative of $$y = e$$^{2x}$$?

Accepted Answer

$$2e^{2x}$$

Question 16

If $$f(x) = 	an(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\sec^2$(x)$$

Question 17

A function $$f$$ has an inverse $$f^{-1}$$ and $$f'(x)$$ exists and is never zero. If $$f(2) = 5$$ and $$f'(2) = 4$$, what is $$(f^{-1}$$)'(5)$$?

Accepted Answer

$$\frac{1}{4}$$

Question 18

Which of the following is the correct derivative of $$y = \log_2$(x)$$?

Accepted Answer

$$\frac{1}{x \ln 2}$$

Question 19

If $$y = \cot(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$-\csc^2$(x)$$

Calculus I: Derivatives, Trigonometric Functions, and the Unit Circle

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