Chapter 11, Problem 107RE

To determine

To identify true or false:

  $f\text{'}\left(z\right)$ Gives the slope of tangent line at the point $\left(z,f\text{'}\left(z\right)\right)$

Answer to Problem 107RE

  $f\text{'}\left(z\right)$ Gives the slope of tangent line at the point $\left(z,f\text{'}\left(z\right)\right)$ is true

Explanation of Solution

Given information:

Function $=f$

Tangent point $=\left(z,f\left(z\right)\right)$

The graph slope $f$ at the point of $\left(x,f\left(x\right)\right)$ which is represented by the derivative of this function $f$ at point $x$ .

Know that:

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

The derivative of function $f$ at the point is the slope of tangent line to graph at the point.

Hence, $f\text{'}\left(z\right)$ gives the tangent line slope to graph of $f$ at that point.

  $f\text{'}\left(z\right)$ Gives the slope of tangent line at the point $\left(z,f\text{'}\left(z\right)\right)$ is true