$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

Question 1

Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

Accepted Answer

$$f'(a) = \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Question 2

Given $$f(x) = -7x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = -7$$

Question 3

For $$f(x) = x^2$ + 6x$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

$$y = 10x - 4$$

Question 4

Which of the following is NOT a reason a function might fail to be differentiable at a point?

Accepted Answer

The function is continuous and smooth at that point.

Question 5

If $$f(x) is $$differentiable at $$x = a$$, which statement must be true?

Accepted Answer

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

Question 6

Which of the following is a correct notation for the derivative of $$f$$ evaluated at $$x = a$$?

Accepted Answer

$$\left. \frac{df}{dx} \right|_{x=a}$$

Question 7

Suppose $$f(x) is $$not continuous at $$x = 3. $$What can you conclude about $$f'(3)$$?

Accepted Answer

$$f'(3)$$ does not exist

Question 8

Given $$f(x) = x^2$ + 6x$$, find $$f'(3).$$

Accepted Answer

$$f'(3) = 12$$

Question 9

A function $$f(x)$$ has a sharp corner at $$x = 1. $$What does this imply about $$f'(1)$$?

Accepted Answer

$$f'(1)$$ does not exist

Question 10

Which of the following best describes the relationship between differentiability and continuity?

Accepted Answer

Differentiability implies continuity

Question 11

If $$f(x) is a $$quadratic function, what is the general form of its derivative?

Accepted Answer

$$f'(x) is a $$linear function

Question 12

Given $$f(x) = x^2$ + 6x$$, use the limit definition to find $$f'(x).$$

Accepted Answer

$$f'(x) = 2x + 6$$

Question 13

Which of the following functions is NOT differentiable at $$x = 0$$?

Accepted Answer

$$f(x) = |x|$$

Question 14

A car's position is given by $$s(t) = t^2$ + 6t. $$What is the car's instantaneous velocity at $$t = 2$$ seconds?

Accepted Answer

$$v(2) = 10$$

Question 15

Which of the following is the correct limit definition of the derivative of $$f at$$ $$x$$?

Accepted Answer

$$f'(x) = \lim_{h 	o 0} \frac{f(x + h) - f(x)}{h}$$

Question 16

If $$f(x)$$ has a vertical tangent at $$x = a$$, what can you say about $$f'(a)$$?

Accepted Answer

$$f'(a)$$ does not exist

Question 17

Suppose $$f(x) is $$continuous everywhere but not differentiable at $$x = 2. $$Which of the following could be true?

Accepted Answer

$$f(x)$$ has a corner at $$x = 2$$

Question 18

Which of the following is a correct way to write the equation of the tangent line to $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 19

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

Accepted Answer

The graph is smooth and has no corners or vertical tangents at $$x = a$$

The Derivative as a Function 3.2

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