$$\lim_{x \to -1^-} f(x) \neq \lim_{x \to -1^+} f(x)$$

Question 1

Which of the following statements is TRUE regarding the continuity of the function $$f(x) at$$ $$x = -1 as $$shown in the graph?

Accepted Answer

The function is not continuous at $$x = -1$$ because there is a jump discontinuity.

Question 2

At which of the following points is the function $$f(x)$$ NOT differentiable?

Accepted Answer

$$x = 1$$

Question 3

Given the graph, what is the value of $$f'(0)$$?

Accepted Answer

$$f'(0) = 0$$

Question 4

Which of the following best describes the behavior of $$f(x) at$$ $$x = 2$$?

Accepted Answer

The function is continuous and differentiable at $$x = 2.$$

Question 5

Suppose $$f(x)$$ models the height of a roller coaster over time. At which point does the roller coaster change direction most sharply?

Accepted Answer

$$x = 1$$

Question 6

If $$f(x) is $$not differentiable at $$x = 1$$, which of the following is the most likely reason based on the graph?

Accepted Answer

There is a sharp corner (cusp) at $$x = 1.$$

Question 7

Which of the following points is a local minimum of $$f(x) as $$shown in the graph?

Accepted Answer

$$x = 2$$

Question 8

If $$f(x)$$ represents the position of a particle, what does $$f'(0) = 0$$ indicate about the particle at $$x = 0$$?

Accepted Answer

The particle is momentarily at rest at $$x = 0.$$

Question 9

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

Accepted Answer

The function is continuous at $$x = 0.$$

Question 10

If $$f(x) is $$not continuous at $$x = -1$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o -1^-} f(x) 
eq \lim_{x 	o -1^+} f(x)$$

Analyzing Continuity, Differentiability, and Derivatives from a Graph

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