2. Intro to Derivatives

One-Sided Derivatives

Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.

One-Sided Derivatives

Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

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

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

c. neither continuous nor differentiable?

Give reasons for your answers.

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.

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

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

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 $x=3$.

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.

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.

b. Show that

f(x) = { x² sin(1/x), x ≠ 0

0, x = 0

is differentiable at x = 0 and find f′(0).

Find the value of a that makes the following function differentiable for all x-values.

g(x) = { ax, if x < 0

x² − 3x, if x ≥ 0