Question 1

Suppose the position of a car at time $$t (in $$seconds) is given by $$s(t) = -9t^2$ + 15t + 25. $$What is the instantaneous velocity of the car at $$t = 2$$ seconds?

Accepted Answer

$$-21$$

Question 2

Given $$f(x) = (x^2$ + 2x - 9$$)^{5/4}$$, use the chain rule to find f$$'(x)$$. Which of the following is correct?

Accepted Answer

$$\frac{5}{4}(x^2$ + 2x - 9$$)^{1/4}$$(2x + 2)$$

Question 3

If $$y = \sin(x$$^{75}$$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$75x^{74}$$\$$cos(x^{75}$$)$$

Question 4

A ball is thrown vertically and its position is given by $$s(t) = -9t^2$ + 15t + 25. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = \frac{15}{18}$$

Question 5

Which of the following correctly applies the chain rule to $$f(x) = 	an(10x$$^{2}$$ - 23x + 1)$$?

Accepted Answer

$$\sec^2$(10x^{2} - 23x + 1$$) \cdot (20x - 23)$

Question 6

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

Accepted Answer

$$\cos^2$(x) - \sin^2$(x)$$

Question 7

The average velocity of a particle moving along a line from $$t = a to$$ $$t = b is $$given by which formula?

Accepted Answer

$$\frac{s(b) - s(a)}{b - a}$$

Question 8

Given $$f(x) = e$$^{x^2}$$, what is f$$'(x)$$?

Accepted Answer

$$2x e$$^{x^2}$$

Question 9

If $$f(x) = \ln(x^3$)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{3}{x}$$

Question 10

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

Accepted Answer

$$-1e^{-0.6}$$

Question 11

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

Accepted Answer

$$2\sin(x)\cos(x)$$

Question 12

Which of the following is the correct derivative of $$f(x) = \ln(5x^2$ + 1)$$?

Accepted Answer

$$\frac{10x}{5x^2$ + 1}$$

Question 13

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 14

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

Accepted Answer

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

Question 15

Which of the following is the correct application of the product rule to $$f(x) = x^2$ e^x$$?

Accepted Answer

$$2x e^x + x^2$ e^x$$

Question 16

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

Accepted Answer

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

Question 17

A chemical reaction's temperature (in $$^ C) at $$time $$t (in $$minutes) is $$T(t) = 20 + 5\sin(\pi t/10). $$What is the rate of change of temperature at $$t = 5$$ minutes?

Accepted Answer

$$\frac{\pi}{2}$$

Question 18

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

Accepted Answer

$$\sec(x)	an(x)$$

Question 19

Suppose $$f(x) = x^3$ - 9x^2$ + 15x + 25$$ models the height (in meters) of a projectile at time $$x (in $$seconds). What is the acceleration at any time $$x$$?

Accepted Answer

$$6x - 18$$

Derivatives as Rate of Change and the Chain Rule

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