Chapter 2.3, Problem 80AYU

To determine

To find: The following values using the given function $f(x)=x^2$ :

a) Average rate of change for the function $f$ from 1 to 2 .

b) Average rate of change for the function $f$ from 1 to 1.5.

c) Average rate of change for the function $f$ from 1 to 1.1 .

d) Average rate of change for the function $f$ from 1 to 1.01 .

e) Average rate of change for the function $f$ from 1 to 1.001 .

f) Graph each of the secant lines along with $f$ .

g) Effects of secant lines.

h) Effects of the slopes of the secant lines.

Answer to Problem 80AYU

Solution:

a) The average rate of change for the function $f$ from 1 to 2 is 3

b) The average rate of change for the function $f$ from 1 to 1.5 is 2.5

c) The average rate of change for the function from 1 to 1.1 is 2.1.

d) The average rate of change for the function $f$

  $f$ from 1 to 1.01 is 2.01.

e) The average rate of change for the function $f$ from 1 to 1.001 is 2.001.

f.  

g.They are getting closer to the straight line y=2x .

h.They are getting closer to 2.

Explanation of Solution

Function is defined as $f(x)=x^2$

a) The average rate of change for the function $f$ from 1 to 2 is

  $\frac{\Delta y}{\Delta x}=\frac{f(2)-f(1)}{2-1}=\frac{4-1}{1}=3$

b) The average rate of change for the function $f$ from 1 to 1.5 is

  $\frac{\Delta y}{\Delta x}=\frac{f(1.5)-f(1)}{1.5-1}=\frac{2.25-1}{0.5}=2.5$

Function is defined as $f(x)=x^2$

a) The average rate of change for the function $f$ from 1 to 2 is

  $\frac{\Delta y}{\Delta x}=\frac{f(2)-f(1)}{2-1}=\frac{4-1}{1}=3$

b) The average rate of change for the function $f$ from 1 to 1.5 is

  $\frac{\Delta y}{\Delta x}=\frac{f(1.5)-f(1)}{1.5-1}=\frac{2.25-1}{0.5}=2.5$

c) The average rate of change for the function $f$ from 1 to 1.1 is

  $\frac{\Delta y}{\Delta x}=\frac{f(1.1)-f(1)}{1.1-1}=\frac{1.21-1}{0.1}=2.1$

The average rate of change for the function $f$ from 1 to 1.01 is

  $\frac{\Delta y}{\Delta x}=\frac{f(1.01)-f(1)}{1.01-1}=\frac{1.0201-1}{0.01}=2.01$

e) The average rate change for the function $f$ from 1 to 1.001 is

  $\frac{\Delta y}{\Delta x}=\frac{f(1.001)-f(0)}{1.001-1}=\frac{1.002001-1}{0.001}=2.001$

f.  

g) They are getting closer to the2 straight line $y=2x$

h) They are getting closer to 2 .