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

$$f'(c) = \lim\limits_{x \to c} \dfrac{f(x) - f(c)}{x - c}$$

Question 1

Given the function $$f(x) = 4x^2$ - 1$$, what is the slope of the tangent line at $$x = -3$$?

Accepted Answer

$$-24$$

Question 2

Which of the following best describes the difference between a secant line and a tangent line to a curve $$f(x)$$?

Accepted Answer

A secant line intersects the curve at two points, representing the average rate of change, while a tangent line touches at one point, representing the instantaneous rate of change.

Question 3

Given $$f(x) = x^2$ - 10x + 2$$, calculate the slope of the tangent line at $$x = 2.$$

Accepted Answer

$$-8$$

Question 4

Given $$f(x) = x^4$ + 100$$, what is the slope of the tangent line at $$x = 0$$?

Accepted Answer

$$0$$

Question 5

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

Accepted Answer

$$f'(c) = \lim\limits_{x 	o c} \dfrac{f(x) - f(c)}{x - c}$$

Question 6

Find the equation of the tangent line to $$f(x) = 3(x^2$ - 1) at$$ $$x = 1.$$

Accepted Answer

$$y = 6x - 3$$

Question 7

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

Accepted Answer

$$y = 16x + 12$$

Question 8

Find the derivative of $$g(x) = 4x^2$ - 9x.$$

Accepted Answer

$$g'(x) = 8x - 9$$

Question 9

Use the definition of the derivative to find $$g'(x)$$ for $$g(x) = x^3$ at x = -1$.

Accepted Answer

$$g'(-1) = -3$$

Question 10

If the graph of $$f(x) is $$increasing on an interval, what can you say about $$f'(x) on $$that interval?

Accepted Answer

$$f'(x) is $$positive

Question 11

Suppose $$f(x)$$ has a horizontal tangent at $$x = 2. $$What is $$f'(2)$$?

Accepted Answer

$$f'(2) = 0$$

Question 12

Based on the graph of $$f(x)$$, where would the derivative $$f'(x) be $$below the $$x-$$axis?

Accepted Answer

Where $$f(x) is $$decreasing

Question 13

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

Accepted Answer

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

Question 14

Given a graph of $$f(x)$$ with a sharp corner at $$x = 1$$, what happens to $$f'(x) at$$ $$x = 1$$?

Accepted Answer

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

Question 15

If $$f(x) is a $$position function for a car, what does $$f'(x)$$ represent in a real-world context?

Accepted Answer

The car's velocity

Question 16

Given $$f(x) = x^2$, use the definition of the derivative to find f$$'(x)$$.

Accepted Answer

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

Question 17

If $$f(x) is $$decreasing on $$(-\infty, 0)$$ and increasing on $$(0, \infty)$$, what can you say about $$f'(x) at$$ $$x = 0 if$$ $$f(x) is $$smooth?

Accepted Answer

$$f'(0) = 0$$

Question 18

A function $$f(x)$$ has a jump discontinuity at $$x = 2. $$What is true about $$f'(2)$$?

Accepted Answer

$$f'(2)$$ does not exist

Question 19

If $$f(x) is $$constant on an interval, what is $$f'(x) on $$that interval?

Accepted Answer

$$f'(x) = 0$$

Question 20

Given $$f(x) = x^3$ - 3x^2$ + 2x$$, at what $$x-$$values does $$f'(x)$$ equal zero?

Accepted Answer

$$x = 1$$ and $$x = 2$$

Slopes of Tangent Lines and Derivatives: Calculus Study Guide

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