Question 1

Which of the following is the correct definition of the derivative of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h}$$

Question 2

Given $$f(x) = \sqrt{2x}$$, what is the slope of the tangent line to $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$\frac{1}{2}$$

Question 3

For $$f(x) = x^3$ - x$$, at which $$x-$$values does the tangent line to the graph become horizontal?

Accepted Answer

$$x = \pm \frac{1}{\sqrt{3}}$$

Question 4

Which of the following statements is TRUE about the differentiability of $$f(x) = |x| at$$ $$x = 0$$?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 0$$ because the left and right limits of the difference quotient are not equal.

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

A car's position at time $$t is $$given by $$s(t) = t^2$. What does the derivative s$$'(t)$$ represent in this context?

Accepted Answer

The instantaneous velocity of the car at time $$t$$

Question 8

Which of the following is an alternative but equivalent definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}$$

Question 9

Suppose $$f(x) is $$differentiable at $$x = 2$$ and $$f(2) = 5$$, $$f'(2) = -3. $$What is the equation of the tangent line to $$y = f(x) at$$ $$x = 2$$?

Accepted Answer

$$y = -3(x - 2) + 5$$

Question 10

If the slope of the tangent line to $$y = f(x) at$$ $$x = a is$$ $$m$$, what is the slope of the normal line at that point?

Accepted Answer

$$-\frac{1}{m}$$

Question 11

A function $$f(x) is $$defined as $$f(x) = |x|. $$What is $$\lim_{h \rightarrow 0} \frac{|h|}{h}$$?

Accepted Answer

The limit does not exist

Question 12

Which of the following notations does NOT represent the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(a)$$

Question 13

If $$f(x) is $$not differentiable at $$x = a$$, which of the following could be a possible reason?

Accepted Answer

The graph of $$f(x)$$ has a sharp corner at $$x = a$$

Question 14

Let $$f(x) = \sin x. $$What is the value of $$f'(\frac{\pi}{2})$$?

Accepted Answer

$$0$$

Question 15

Suppose $$f(x) is $$differentiable everywhere. Which of the following statements is always true?

Accepted Answer

$$f(x) is $$continuous everywhere

Question 16

A biologist models the growth of a bacteria population with $$P(t) = 100e$$^{0.05t}$$. What does P$$'(t)$$ represent?

Accepted Answer

The instantaneous rate of change of the population at time $$t$$

Question 17

If $$f(x) is $$differentiable at $$x = a$$, which of the following is the best interpretation of $$f'(a) in a $$real-world context?

Accepted Answer

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

Derivatives: Definition, Interpretation, and Applications

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