Chapter 4.5, Problem 27E

a.

To determine

To find: To find the true change in the given function.

Answer to Problem 27E

The true change in the given function $f(x)=x^2+2x$ , $a=0$ and $\Delta x=0.1$ is $\Delta f=0.21$ .

Explanation of Solution

Given: The function $f(x)=x^2+2x$ , $a=0$ , $\Delta x=0.1$

Formula used: $\Delta f=f(a+\Delta x)-f(a)$

Calculation:

The given function is $f(x)=x^2+2x$ .

The true change can be found using the formula $\Delta f=f(a+\Delta x)-f(a)$

The given conditions are $a=0$ , $\Delta x=0.1$ .

By substituting the above values in the formula,

  $\begin{array}{c}\Delta f=f(0+0.1)-f(0)\\ \Delta f=f(0.1)-f(0)\\ f(0.1)={(}^{0.1}+2(0.1)\\ f(0.1)=0.01+0.2\\ f(0.1)=0.21\\ f(0)={(}^0+2(0)\\ f(0)=0\\ \Delta f=f(0.1)-f(0)\\ \Delta f=0.21-0\\ \Delta f=0.21\end{array}$

Conclusion:

The true change in the given function $f(x)=x^2+2x$ , $a=0$ and $\Delta x=0.1$ is $\Delta f=0.21$ .

b.

To determine

To find: To find the estimated change in the given function.

Answer to Problem 27E

The estimated change in the given function $f(x)=x^2+2x$ , $a=0$ and $\Delta x=0.1$ is $\Delta f=0.2$ .

Explanation of Solution

Given: The function $f(x)=x^2+2x$ , $a=0$ , $\Delta x=0.1$

Formula used: $\Delta f=f\text{'}(a)\Delta x$

Calculation:

The given function is $f(x)=x^2+2x$ .

The estimated change can be found using the formula $\Delta f=f\text{'}(a)\Delta x$.

The given conditions are $a=0$ , $\Delta x=0.1$ .

By substituting the above values in the formula,

  $\Delta f=f\text{'}(0)(0.1)$

Differentiating the function with respect to x,

  $f\text{'}(x)=2x+2$

Substituting $x=0$ in the above equation,

  $\begin{array}{l}f\text{'}(0)=2(0)+2\\ f\text{'}(0)=2\end{array}$

Substituting in the formula,

  $\begin{array}{l}\Delta f=(2)(0.1)\\ \Delta f=0.2\end{array}$

Conclusion:

The estimated change in the given function $f(x)=x^2+2x$ , $a=0$ and $\Delta x=0.1$ is $\Delta f=0.2$ .

c.

To determine

To find: To find the approximate error in the given function.

Answer to Problem 27E

The approximate error in the given function $f(x)=x^2+2x$ , $a=0$ and $\Delta x=0.1$ is $\left|\Delta f-f\text{'}(a)\Delta x\right|=0.01$ .

Explanation of Solution

Given: The function $f(x)=x^2+2x$ , $a=0$ , $\Delta x=0.1$ .

Formula used: $\left|\Delta f-f\text{'}(a)\Delta x\right|$

Calculation:

The given function is $f(x)=x^2+2x$ .

The true change can be found using the formula $\left|\Delta f-f\text{'}(a)\Delta x\right|$.

The given conditions are $a=0$ , $\Delta x=0.1$ .

By substituting the above values in the formula,

  $\begin{array}{c}\Delta f=f(0+0.1)-f(0)\\ \Delta f=f(0.1)-f(0)\\ $ f(0.1)={(0.1)}^2+2(0.1)$f(0.1)=0.01+0.2\\ f(0.1)=0.21\\ f(0)={(}^0+2(0)\\ f(0)=0\\ \Delta f=f(0.1)-f(0)\\ \Delta f=0.21-0\\ \Delta f=0.21\end{array}$

  $\begin{array}{l}f\text{'}(a)\Delta x\\ f\text{'}(a)\Delta x=f\text{'}(0)(0.1)\\ f\text{'}(a)\Delta x=(2)(0.1)\\ f\text{'}(a)\Delta x=0.2\end{array}$

Approximate error will be

  $\begin{array}{l}\left|\Delta f-f\text{'}(a)\Delta x\right|=0.21-0.2\\ \left|\Delta f-f\text{'}(a)\Delta x\right|=0.01\end{array}$

Conclusion:

The approximate error in the given function $f(x)=x^2+2x$ , $a=0$ and $\Delta x=0.1$ is $\left|\Delta f-f\text{'}(a)\Delta x\right|=0.01$ .