Chapter 11.3, Problem 75E

(a)

To determine

To find: The method to approximate the tangent line at $\left(x,f\left(x\right)\right)$ with the help of secant line.

Answer to Problem 75E

Take the points of the secant line very close to the point $x$ and decrease the distance between points. So, it will change into tangent line.

Explanation of Solution

Calculation:

The secant line is the joining of two points on the curve and tangent line touches the curve at only one point.

To make the secant line to tangent line at $\left(x,f\left(x\right)\right)$ . Take the secant line through the two points very close to $x$ and decrease the distance between the points as much as possible. Then automatically the secant line will change into tangent line at $\left(x,f\left(x\right)\right)$ .

Therefore, Take the points of the secant line very close to the point $x$ . So, it will change into tangent line.

(b)

To determine

To find: The method of the exact slope of tangent line at $\left(x,f\left(x\right)\right)$ with the help of limit process.

Answer to Problem 75E

The slope of the tangent line with the help of limit process is equal to $\underset{h\to 0}{\lim }\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)$ .

Explanation of Solution

Calculation:

The slope of the tangent line is the average rate of change of the values of function at two points very close to them.

Assume that the secant line that gets transformed into tangent line at $\left(x,f\left(x\right)\right)$ passes through $x$ and $x+h$ . So, the slope of the secant line is the average rate of change of function from $x$ to $x+h$ .

  $\begin{array}{c}m=\frac{f\left(x+h\right)-f\left(x\right)}{x+h-x}\\ =\frac{f\left(x+h\right)-f\left(x\right)}{h}\end{array}$

Now to change the secant line into tangent line, need to decrease the distance between the points i.e. need to decrease the value of $h$ as much as possible or approaches to zero. So, it changes to tangent line and slope of the tangent line is given by:

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

Therefore, the slope of the tangent line with the help of limit process is equal to $\underset{h\to 0}{\lim }\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)$ .