Question 1

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

Accepted Answer

The instantaneous rate of change of $$f(x)$$ with respect to $$x at$$ $$x = a$$

Question 2

Given $$f(x) = x^2$ + 2x - 3$$, use the definition of the derivative to find $$f'(x). $$Which of the following is correct?

Accepted Answer

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

Question 3

Which of the following notations does NOT represent the derivative of $$y = f(x)$$ with respect to $$x$$?

Accepted Answer

$$\int f(x) dx$$

Question 4

Suppose $$f(x) = \sqrt{x} + 1. $$Using the definition of the derivative, what is $$f'(x)$$?

Accepted Answer

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

Question 5

A function $$f(x) is $$differentiable at $$x = a if$$:

Accepted Answer

The left-hand and right-hand derivatives at $$x = a$$ are equal

Question 6

Given $$f(x) = 3x - 5$$, what is the value of the derivative $$f'(x)$$?

Accepted Answer

$$f'(x) = 3$$

Question 7

If the graph of $$f(x)$$ has a local maximum at $$x = 2$$, what can you say about $$f'(2)$$?

Accepted Answer

$$f'(2) = 0$$

Question 8

Given $$f(x) = \frac{1}{x-4}$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$f'(x) = -\frac{1}{(x-4)^2$}$$

Question 9

A tangent line to the curve $$y = f(x) at$$ $$x = a$$ passes through the point $$(a, f(a))$$ and has slope $$f'(a). $$What is the equation of the tangent line?

Accepted Answer

$$y = f'(a)(x - a) + f(a)$$

Question 10

Suppose $$f(x) is $$continuous at $$x = 1$$ but the left-hand and right-hand derivatives at $$x = 1$$ are not equal. What can you conclude?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 1$$

Question 11

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

Accepted Answer

$$y = 4x - 2$$

Question 12

If $$f(3) = 5$$ and $$f'(3) = -4$$, what is the equation of the normal line to $$f(x) at$$ $$x = 3$$?

Accepted Answer

$$y - 5 = \frac{1}{4}(x - 3)$$

Question 13

Which of the following statements about the relationship between the graphs of $$f$$ and $$f' is $$TRUE?

Accepted Answer

Where $$f$$ has a local maximum or minimum, $$f'$$ crosses the $$x-$$axis

Question 14

A function $$f(x) is $$defined as $$f(x) = x^2$ for x  1$. Is f(x$$)$$ differentiable at x = 1$?

Accepted Answer

No, because the left-hand and right-hand derivatives at $$x = 1$$ are not equal

Question 15

If $$f(x) is $$differentiable at $$x = a$$, which of the following must also be true?

Accepted Answer

$$f(x) is $$continuous at $$x = a$$

Question 16

Given $$g(x) = \frac{2}{x}$$, what is the equation of the normal line to $$g(x) at$$ $$x = -2$$?

Accepted Answer

$$y + 1 = \frac{1}{2}(x + 2)$$

Question 17

Which of the following is the difference quotient for $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\frac{f(a+h) - f(a)}{h}$$

Question 18

If the slope of the tangent line to $$f(x) at$$ $$x = 0 is$$ $$4$$, what is $$f'(0)$$?

Accepted Answer

$$4$$

Question 19

Suppose you are given the graph of $$f(x)$$ and asked to sketch $$f'(x). $$Which of the following is a correct approach?

Accepted Answer

Estimate the slope of the tangent to $$f(x) at $$various points and plot these slopes as $$y-$$values for $$f'(x)$$

Derivatives: Definition, Notation, and Applications (Section 2.1a)

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