Chapter 8.9, Problem 3OE

a.

To determine

To find: the value by using linear interpolation.

Answer to Problem 3OE

The values are $f\left(0\right)=0\text{ and }f\left(2\right)=4$ .

Explanation of Solution

Given information:

The function values, $f\left(x\right)=x^2,x_1=0,x_2=2$ .

Calculation: to find the value of $f\left(0\right)\text{ and }f\left(2\right)$ , substitute in the function and simplify,

  $\begin{array}{l}f\left(0\right)={\left(0\right)}^2\text{ [substitute]}\\ f\left(0\right)=0\end{array}$

  $\begin{array}{l}f\left(2\right)={\left(2\right)}^2\text{ [substitute]}\\ f\left(2\right)=4\end{array}$

Thus, The values are $f\left(0\right)=0\text{ and }f\left(2\right)=4$ .

b.

To determine

To find: the function that have the same value as function $f$ .

Answer to Problem 3OE

The function is $g\left(x\right)=2x$ .

Explanation of Solution

Given information:

The function values, $f\left(x\right)=x^2,x_1=0,x_2=2$ .

Calculation: from the function and values it can be observed that the function can be $g\left(x\right)=2x$ .

c.

To determine

To find: the value of the linear function at $x=1$ .

Answer to Problem 3OE

The value is $f\left(1\right)=1$ .

Explanation of Solution

Given information:

The function values, $f\left(x\right)=x^2,x_1=0,x_2=2$ .

Calculation: to find the value of $f\left(1\right)$ , substitute in the function and simplify,

  $\begin{array}{l}f\left(1\right)={\left(1\right)}^2\text{ [substitute]}\\ f\left(1\right)=1\end{array}$

Thus, The value is $f\left(1\right)=1$ .

d.

To determine

To state: the difference between function values.

Answer to Problem 3OE

Both values are same.

Explanation of Solution

Given information:

The function values, $f\left(x\right)=x^2,x_1=0,x_2=2$ .

by the part (c) it can be observed that the both values are same.

e.

To determine

To state: that how could the values choose for the approximation for $f\left(1\right)$ .

Answer to Problem 3OE

To find the approximation of $f\left(1\right)$ , the $x_1\text{ and }{x}_2$ can be choose where the $1$ lie between the two points.

Explanation of Solution

Given information:

The function values, $f\left(x\right)=x^2,x_1=0,x_2=2$ .

to find the approximation of $f\left(1\right)$ , the $x_1\text{ and }{x}_2$ can be choose where the $1$ lie between the two points.