Question 1

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

Accepted Answer

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

Question 2

For the function $$f(x) = x^2$ + 3x - 1$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 2x + 3$$

Question 3

A company models its profit with $$f(x) = x^2$. What is the equation of the tangent line to the graph at x = 1$?

Accepted Answer

$$y = 2x - 1$$

Question 4

For the function $$f(x) = \frac{1}{x+1}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 5

For which value(s) of $$x is $$the function $$f(x) = |x|$$ not differentiable?

Accepted Answer

$$x = 0$$, because the function has a corner there.

Question 6

Suppose $$f(x) = \sqrt{x-2}. $$For which values of $$x is$$ $$f'(x)$$ defined?

Accepted Answer

$$x \geq 2$$

Question 7

A business uses the function $$f(x) = x^3$ - x to $$model revenue. What is the slope of the tangent line at $$x = 1$$?

Accepted Answer

$$2$$

Question 8

For the function $$f(x) = 3x$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

$$y = 3x - 3$$

Question 9

Which of the following is NOT a reason why a function might fail to be differentiable at $$x = a$$?

Accepted Answer

The function is increasing at $$x = a.$$

Question 10

For the function $$f(x) = \frac{1}{x-2}$$, for which value is the derivative NOT defined?

Accepted Answer

$$x = 2$$

Question 11

Given $$f(x) = x^2$, what is the instantaneous rate of change at x = -2$?

Accepted Answer

$$-4$$

Question 12

If $$f(x) = \sqrt{x+1}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 13

A function $$f(x) is $$continuous everywhere but has a sharp corner at $$x = 3. $$What can you conclude?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 3.$$

Question 14

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

Accepted Answer

$$y = 2x - 2$$

Question 15

For $$f(x) = \sqrt{x-2}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 16

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

Accepted Answer

$$y = 1x + 1$$

Question 17

For $$f(x) = \frac{1}{x+1}$$, what is the slope of the tangent line at $$x = 1$$?

Accepted Answer

$$-\frac{1}{4}$$

Question 18

Which of the following statements is TRUE?

Accepted Answer

If a function is differentiable at $$x = a$$, then it is also continuous at $$x = a.$$

Question 19

For $$f(x) = x^3$, what is f$$'(x)$$?

Accepted Answer

$$3x^2$$

Question 20

For $$f(x) = \frac{1}{x-2}$$, what is $$f'(x)$$?

Accepted Answer

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

Lecture 4: The Definition of the Derivative and Differentiability

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