Apentix/AP Calculus AB/Unit 2: Differentiation: Definition and Basic Derivative Rules/Question

AP Calculus ABmediummcq1 pt

Let $f(x)=\begin{cases} x^2, & x<1 \\ x, & x\ge 1 \end{cases}$ . Which of the following statements about $f$ at $x=1$ is true?

A.

f

is not continuous at

x=1

B.

f

is continuous at

x=1

but not differentiable there

C.

f

is differentiable at

x=1

and

f'(1)=2

D.

f

is differentiable at

x=1

and

f'(1)=1

Explanation

Core Concept

The left-hand derivative is $2$ and the right-hand derivative is $1$ . Since the one-sided slopes disagree, $f'(1)$ does not exist even though $f$ is continuous.

Correct Answer

B $f$ is continuous at $x=1$ but not differentiable there

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