2. Intro to Derivatives

Derivative of multiples Does knowing that a function g(t) is differentiable at t = 7 tell you anything about the differentiability of the function 3g at t = 7? Give reasons for your answer.

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

a. Does f'(0) exist?

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

g(x) = { 2x − x³ − 1, x ≥ 0

x − (1 / (x + 1)), x < 0

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

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

f(x) = { 2x − 1, x ≥ 0

x² + 2x + 7, x < 0

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

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

a. Let f(x) be a function satisfying |f(x)| ≤ x² for −1 ≤ x ≤ 1. Show that f is differentiable at x = 0 and find f′(0).

Let $f$ be a differentiable function such that $f\left(2\right)=5$ and $f^{\prime }\left(2\right)=3$. What is the value of the derivative of $g\left(x\right)=2f\left(x\right)$ at $x=2$?

Suppose the graph of a function $f\left(x\right)$ is given below. At which of the following $x$-values is $f\left(x\right)$ NOT differentiable?

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

f(x) = { 2x + tan x, x ≥ 0

x², x < 0

Which of the following functions is a solution to the differential equation $y^{\prime \prime }-4y=0$?

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.

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

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.

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.