Question 1

Given the function $$y = x^2$ + 9x + 8$$, use the limit definition of the derivative to find the slope of the curve at $$x = 1. $$What is the value of the derivative at this point?

Accepted Answer

$$7$$

Question 2

Consider the piecewise function $$f(x) = \begin{cases} 3x & x \u003c 0 \ 2x + 8 & x \geq 0 \end{cases}. Is$$ $$f(x)$$ differentiable at $$x = 0$$?

Accepted Answer

No, because the left and right derivatives at $$x = 0$$ are not equal.

Question 3

For the function $$f(x) = 5x + 9$$, use the limit definition of the derivative to find $$f'(2). $$What is the value?

Accepted Answer

$$5$$

Question 4

Given $$f(x) = \frac{8}{x}$$, use the limit definition to find $$f'(1). $$What is the value?

Accepted Answer

$$-8$$

Question 5

Which of the following statements is true about the function $$y = |x| at$$ $$x = 0$$?

Accepted Answer

It is continuous but not differentiable at $$x = 0.$$

Question 6

Find the derivative of $$y = 11 - 4x^2$.

Accepted Answer

$$y' = -8x$$

Question 7

Find the derivative of $$y = x^4$ + 6x^2$ + 3.$$

Accepted Answer

$$y' = 4x^3$ + 12x$$

Question 8

Find the derivative of $$y = 2x^2$ + 8x + 2x^3$.

Accepted Answer

$$y' = 4x + 8 + 6x^2$

Question 9

Find the derivative of $$y = \frac{1}{7x^2$} - \frac{1}{7x}.$$

Accepted Answer

$$y' = -\frac{2}{7x^3$} + \frac{1}{7x^2$}$$

Question 10

Given $$y = (x-1)^3$, find y$$'$$ using the chain rule.

Accepted Answer

$$y' = 3(x-1)^2$

Question 11

Suppose $$u(x)$$ and $$v(x)$$ are differentiable functions. If $$u(1) = 9$$, $$u'(1) = -5$$, $$v(1) = -3$$, and $$v'(1) = -3$$, what is $$\left.\frac{d}{dx}(uv)\right|_{x=1}$$?

Accepted Answer

$$-97$$

Question 12

A ball is dropped from the top of a building with a height of $$s = 400 - 16t^2$ meters after t$ seconds. How long does it take the ball to reach the ground?

Accepted Answer

$$5$$ seconds

Question 13

Given the velocity function $$v(t)$$ shown in the graph (see study materials), at what time $$t is $$the body's acceleration equal to zero?

Accepted Answer

$$t = 2$$ 	ext{ and } t = 6$

Question 14

Find the equation of the tangent to the curve $$y = \frac{18}{x^2$ + 2} at $$the point $$(1,6).$$

Accepted Answer

$$y = -4x + 10$$

Question 15

The area $$A of a $$circular oil spill changes with the radius. If $$A = \pi r^2$, at what rate does the area change with respect to the radius when r$$ = \pi$$?

Accepted Answer

$$2\pi^2$

Question 16

Given $$y = \frac{x-1}{x^2$}$$, find $$y'$$ using the quotient rule.

Accepted Answer

$$y' = \frac{2x - 3}{x^3$}$$

Question 17

For the function $$y = \frac{x^2$ - 2x - 2}{x^2$ - 2x + 2}$$, find $$y'$$ using the quotient rule.

Accepted Answer

$$y' = \frac{-4x^2$ + 8x}{(x^2$ - 2x + 2)^2$}$$

Question 18

A rock is thrown vertically upward from the surface of an airless planet. Its height is $$s = 120t - 10t^2$ meters in t$ seconds. How high does the rock go?

Accepted Answer

$$360$$ meters

Question 19

Suppose $$u(x)$$ and $$v(x)$$ are differentiable functions. If $$u(1) = 4$$, $$u'(1) = -7$$, $$v(1) = -2$$, and $$v'(1) = -2$$, what is $$\left.\frac{d}{dx}\left(\frac{v}{u}\right)\right|_{x=1}$$?

Accepted Answer

$$\frac{7}{8}$$

Differentiation: Limits, Derivatives, and Applications in Calculus

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