Chapter 4.5, Problem 30E

a.

To determine

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

Answer to Problem 30E

The true change in the given function $f(x)=x^4$ , $a=1$ and $\Delta x=0.01$ is $\Delta f=0.0406$ .

Explanation of Solution

Given: The function $f(x)=x^4$ , $a=1$ , $\Delta x=0.01$ .

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

Calculation:

The given function is $f(x)=x^4$ .

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

The given conditions are $a=1$ , $\Delta x=0.01$ ,

By substituting the above values in the formula,

  $\begin{array}{c}\Delta f=f(1+0.01)-f(1)\\ \Delta f=f(1.01)-f(1)\\ $ f(1.01)={(1.01)}^4$f(1.01)=1.0406\\ $ f(1)={(1)}^4$f(1)=1\\ \Delta f=f(1.01)-f(0)\\ \Delta f=1.0406-1\\ \Delta f=0.0406\end{array}$

Conclusion:

The true change in the given function $f(x)=x^4$ , $a=1$ and $\Delta x=0.01$ is $\Delta f=0.0406$ .

b.

To determine

To find the estimated change in the given function.

Answer to Problem 30E

The estimated change in the given function $f(x)=x^4$ , $a=1$ and $\Delta x=0.01$ is $\Delta f=0.2$ .

Explanation of Solution

Given: The function $f(x)=x^4$ , $a=1$ , $\Delta x=0.01$

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

Calculation:

The given function is $f(x)=x^4$

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

The given conditions are $a=1$ , $\Delta x=0.01$

By substituting the above values in the formula

  $\Delta f=f\text{'}(1)(0.01)$

Differentiating the function with respect to x

  $f\text{'}(x)=4x^3$

Substituting $x=0$ in the above equation

  $\begin{array}{l}f\text{'}(1)=4{(}^1\\ f\text{'}(1)=4\end{array}$

Substituting in the formula

  $\begin{array}{l}\Delta f=(4)(0.01)\\ \Delta f=0.04\end{array}$

Conclusion:

The estimated change in the given function $f(x)=x^4$ , $a=1$ and $\Delta x=0.01$ is $\Delta f=0.04$ .

c.

To determine

To find the approximate error in the given function.

Answer to Problem 30E

The approximate error in the given function $f(x)=x^4$ , $a=1$ and $\Delta x=0.01$ is $\left|\Delta f-f\text{'}(a)\Delta x\right|=0.0006$ .

Explanation of Solution

Given: The function $f(x)=x^4$ , $a=1$ , $\Delta x=0.01$

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

Calculation:

The given function is $f(x)=x^4$

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

The given conditions are $a=1$ , $\Delta x=0.01$

By substituting the above values in the formula

  $\begin{array}{l}\Delta f=f(1+0.01)-f(1)\\ \Delta f=f(1.01)-f(1)\\ $ f(1.01)={(1.01)}^4$f(1.01)=1.0406\\ $ f(1)={(1)}^4$f(1)=1\\ \Delta f=f(1.01)-f(0)\\ \Delta f=1.0406-1\\ \Delta f=0.0406\end{array}$

  $\begin{array}{l}f\text{'}(a)\Delta x\\ f\text{'}(a)\Delta x=f\text{'}(1)(0.01)\\ \Delta f=(4)(0.01)\\ \Delta f=0.04\end{array}$

Approximate error

  $\begin{array}{l}\left|\Delta f-f\text{'}(a)\Delta x\right|=0.0406-0.04\\ \left|\Delta f-f\text{'}(a)\Delta x\right|=0.0006\end{array}$

Conclusion:

The approximate error in the given function $f(x)=x^4$ , $a=1$ and $\Delta x=0.01$ is $\left|\Delta f-f\text{'}(a)\Delta x\right|=0.0006$ .