Chapter 2.2, Problem 1C

(a)

To determine

Whether given statement is true or false.

  $f\text{'}(x)=\left(\frac{f(x+h)-f(x)}{h}\right)$

(b)

To determine

Whether given statement is true or false.

  $f\text{'}(x)=\underset{x\to 0}{\lim }\left(\frac{f(x+h)-f(x)}{h}\right)$

(c)

To determine

Whether given statement is true or false.

  $f\text{'}(x)=\underset{z\to 0}{\lim }\left(\frac{f(z)-f(x)}{z-x}\right)$

(d)

To determine

Whether given statement is true or false.

‘If $f(x)=x^3$ then $f(x+h)=x^3+h$ ’

(e)

To determine

Whether given statement is true or false.

If $f(x)=x^3$ then $f\text{'}(x)=\underset{h\to 0}{\lim }\left(\frac{f(x^3+h)-f(x)}{h}\right)$

(f)

To determine

Whether given statement is true or false.

‘A function $f(x)$ is differentiable at $x=c$ if and only if $f{\text{'}}_{+}(c)\text{ and }f{\text{'}}_{-}(c)$ both exist’

(g)

To determine

Whether given statement is true or false.

‘If $f(x)$ is continuous at $x=c$ , then $f(x)$ is differentiable at $x=c$ ’

(h)

To determine

Whether given statement is true or false.

‘If $f(x)$ is not continuous at $x=c$ , then $f(x)$ is not differentiable at $x=c$ ’