L3.2

Multiple Choice

For what value(s) of x is the function continuous but not differentiable?

• x = -1

• x = 0

• x = 2

• x = 3; x = -2

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Multiple Choice

What is the correct representation of a piecewise function that is defined as $$f(x) = x^2$$ for $$x \leq 1$$ and $$f(x) = 2x + 1$$ for $$x > 1$$ ?

• $$f(x) = \begin{cases} x^2 & \text{if } x \leq 1 \ 2x + 1 & \text{if } x > 1 \end{cases}$$

• $$f(x) = \begin{cases} x^2 & \text{if } x < 1 \ 2x + 1 & \text{if } x \geq 1 \end{cases}$$

• $$f(x) = \begin{cases} 2x + 1 & \text{if } x \leq 1 \ x^2 & \text{if } x > 1 \end{cases}$$

• $$f(x) = \begin{cases} 2x + 1 & \text{if } x < 1 \ x^2 & \text{if } x \geq 1 \end{cases}$$

Multiple Choice

Find the limit of the function as x approaches 2⁺.

• 1
• -1
• 5
• DNE

Multiple Choice

For the function f below, compute the right-hand derivative D_+f(0) = lim (x→0+) (f(h)−f(0))/h and the left-hand derivative D_-f(0) = lim (x→0−) (f(h)−f(0))/h. Does f'(0) exist?

• Yes, it exists

• No, it does not exist

• It is undefined

• It exists but is not continuous

Open Ended

For the function f(x) = { x^2 + 2x, x < 0; ax + b, x ≥ 0 }, find all real numbers a and b such that f'(0) exists.

Multiple Choice

What is the equation of the line tangent to the curve y = x² at x = 1?

• y = 2x + 1

• y = 2x - 1

• y = -2x + 1

• y = x - 1

Multiple Choice

Is the above function differentiable at

x = 0? Why?

• Yes. The derivative would be equal to zero since there is a vertical tangent.

• No. The derivative would not exist at x = 0 since there is a vertical tangent.

• No. The derivative does not exist since $$\lim_{x\rightarrow0}\ x^{\frac{1}{3}}$$ does not exist.

• Yes since $$\lim_{h\rightarrow0}\ \frac{\left(x+h\right)^{\frac{1}{3}}-x^{\frac{1}{3}}}{h}$$ exists.

Multiple Choice

If f '(x) = 0, what does that mean with respect to the tangent line?

• Tangent line is vertical

• Tangent line has a positive slope

• Tangent line is horizontal

• Tangent line has a negative slope

Categorize

Options (6)

f(x) has 2 relative extrema

f(x) has 3 P.O.I.

f is concave up on (2,4)

x=5 is a local minimum

f is increasing on [1,5]

f is never concave down

Based on this f'(x), categorize each statement as true or false.

True

False

Poll

How confident do you feel about this topic now?

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• Somewhat confident
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