Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x_0$$?

Accepted Answer

It is the instantaneous rate of change of $$f(x) at$$ $$x_0$$, given by $$f'(x_0$) = \lim_{h 	o 0} \frac{f(x_0$ + h) - f(x_0$)}{h}.$$

Question 2

Given $$f(x) = x^2$ + 2x$$, what is the slope of the tangent line to the graph at $$x = 4$$?

Accepted Answer

$$12$$

Question 3

Which of the following is NOT a correct step in computing the derivative using the limit definition?

Accepted Answer

Multiply $$f(x + h) by$$ $$h$$ and then take the limit as $$h 	o 0.$$

Question 4

If $$y = \frac{1}{x}$$, what is $$\frac{dy}{dx} at$$ $$x = 3$$?

Accepted Answer

$$-\frac{1}{9}$$

Question 5

Which of the following statements about differentiability is TRUE?

Accepted Answer

A function is differentiable at a point only if it is continuous at that point.

Question 6

Find $$\frac{d}{dx}$$ where $$y = 2x^2$.

Accepted Answer

$$4x$$

Question 7

Given $$y = \frac{2}{x+1}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$-\frac{2}{(x+1)^2$}$$

Question 8

A function $$f(x)$$ has a vertical asymptote at $$x = a. $$Which of the following is most likely TRUE about the derivative at $$x = a$$?

Accepted Answer

The derivative does not exist at $$x = a.$$

Question 9

Suppose the position of a car is given by $$s(t) = t^2$ + 2t. $$What is the instantaneous velocity at $$t = 3$$ seconds?

Accepted Answer

$$8 m/s$

Question 10

Which of the following best describes the graphical interpretation of the derivative at a point?

Accepted Answer

It is the slope of the tangent line to the graph at that point.

Question 11

If $$f(x) = x^2$ - 4x + 1$$, what is the equation of the tangent line at $$x = 3$$?

Accepted Answer

$$y = 2x - 5$$

Question 12

Which of the following is a correct interpretation of $$\lim_{h 	o 0} \frac{f(x + h) - f(x)}{h}$$?

Accepted Answer

It gives the instantaneous rate of change of $$f(x) at$$ $$x.$$

Question 13

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

Accepted Answer

Any of the above.

Question 14

If $$f(x) = x^2$, use the limit definition to find f$$'(x)$$. Which is correct?

Accepted Answer

$$f'(x) = 2x$$

Question 15

Suppose $$f(x) is $$continuous at $$x = a$$ but not differentiable at $$x = a. $$Which of the following could be true?

Accepted Answer

The graph of $$f(x)$$ has a cusp at $$x = a.$$

Question 16

If $$f(x) = x^2$ + 2x$$, what is the equation of the tangent line at $$x = 4$$?

Accepted Answer

$$y = 12x - 32$$

Question 17

Which of the following is the correct process for finding the derivative using the limit definition?

Accepted Answer

Evaluate $$f(x + h)$$, subtract $$f(x)$$, divide by $$h$$, and take the limit as $$h 	o 0.$$

Question 18

A real-world example: The height $$h (in $$meters) of a ball thrown upward is given by $$h(t) = -4.9t^2$ + 20t + 1. $$What is the velocity of the ball at $$t = 2$$ seconds?

Accepted Answer

$$0.4 m/s$

Calculus Study Notes: Derivatives and Their Applications (Sections 3.1 & 3.2)

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