2. Intro to Derivatives

Determine if the graph of the function $f\left(x\right)$is continuous and/or differentiable at $x=2$.

Determine if the graph of the function $f\left(x\right)$is continuous and/or differentiable at $x=1$.

Determine if the graph of the function $f\left(x\right)$is continuous and/or differentiable at $x=1$.

Determine if the function$f\left(x\right)$ is continuous and/or differentiable at $x=3$.

Determine if the function$f\left(x\right)$ is continuous and/or differentiable at $x=2$.

Determine where the function $f\left(x\right)$ is not differentiable.

$f\left(x\right)=\frac{3}{x+2}$

Determine if the function$f\left(x\right)$ is continuous and/or differentiable at $x=3$.

If f is differentiable at a, must f be continuous at a?

If f is continuous at a, must f be differentiable at a?

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>

b. Find the values of x in (0, 3) at which f is not differentiable.

Use the graph of g in the figure to do the following. <IMAGE>

b. Find the values of x in (-2,2) at which g is not differentiable.

Determining the unknown constant Let f(x) = {2x² if x≤1 ax-2 if x>1. Determine a value of a (if possible) for which f' is continuous at x=1.

a. Graph the function

ƒ(x) = { x², -1 ≤ x < 0

{ -x², 0 ≤ x ≤ 1.

b. Is ƒ continuous at x = 0?

c. Is ƒ differentiable at x = 0?

Give reasons for your answers.

a. Graph the function

ƒ(x) = { x, -1 ≤ x < 0

{ tan x, 0 ≤ x ≤ π/4.

b. Is ƒ continuous at x = 0?

c. Is ƒ differentiable at x = 0?

Give reasons for your answers.

For what value or values of the constant m, if any, is

ƒ(x) = { sin 2x, x ≤ 0

{ mx, x > 0

a. continuous at x = 0?

b. differentiable at x = 0?

Give reasons for your answers.