Question 1

Given the function $$y = (3x^2$ + 4x)^5$, use the chain rule to find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{dy}{dx} = 5(3x^2$ + 4x)^4$ \cdot (6x + 4)$$

Question 2

A ball is thrown off a 40 ft. cliff with an initial velocity of $$-30 ft/s. $$Its position is given by $$s(t) = -16t^2$ - 30t + 40. $$What is the instantaneous velocity when $$t = 2$$?

Accepted Answer

$$-94 ft/s$$

Question 3

Which of the following statements is TRUE about differentiability?

Accepted Answer

If $$f is $$differentiable at $$a$$, then $$f is $$continuous at $$a.$$

Question 4

The position of a particle is given by $$s(t) = t^3$ - 6t^2$ + 9t. $$What is the acceleration function $$a(t)$$?

Accepted Answer

$$a(t) = 6t - 12$$

Question 5

Which of the following best describes the OUTER and INNER functions for $$y = \sin(2x^2$ + 1)$$?

Accepted Answer

OUTER: $$\sin(u)$$, INNER: $$u = 2x^2$ + 1$$

Question 6

A function $$f(x) is $$not differentiable at $$x = 2 if$$:

Accepted Answer

The left and right limits of the derivative at $$x = 2$$ are not equal.

Question 7

If $$f(x) = |x|$$, which of the following is true?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 0$$

Question 8

A car's position is given by $$s(t) = 4t^2$ + 2t. $$What is the average velocity between $$t = 1$$ and $$t = 3$$?

Accepted Answer

$$18$$ units/time

Question 9

Which of the following is the correct application of the chain rule for $$y = \cos(5x^2$)$$?

Accepted Answer

$$\frac{dy}{dx} = -\sin(5x^2$) \cdot 10x$$

Question 10

A rocket is launched vertically and its position is $$s(t) = -16t^2$ + 64t + 80. $$When does the rocket hit the ground?

Accepted Answer

$$t = 10$$ seconds

Question 11

If $$f(x) = x^3$ - 2x$$, what is the slope of the tangent line at $$x = 2$$?

Accepted Answer

$$8 - 2 = 6$$

Question 12

Which of the following is the correct definition of acceleration?

Accepted Answer

The rate at which velocity is changing.

Question 13

Given $$y = e$$^{3x^2}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$6x e$$^{3x^2}$$

Question 14

A particle moves along a line so that its position at time $$t is$$ $$s(t) = t^2$ - 4t + 3. At $$what time does the particle change direction?

Accepted Answer

$$t = 2$$

Question 15

If $$f(x) = (2x + 1)^4$, what is f$$'(x)$$?

Accepted Answer

$$8(2x + 1)^3$

Question 16

Which of the following is NOT a requirement for differentiability at a point $$x = a$$?

Accepted Answer

The function must have a vertical tangent at $$x = a.$$

Calculus Chapter 3: Derivatives and Their Applications

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