Chapter 10, Problem 56RE

(a)

To determine

The value of f(4) and $f"\text{'}(4)$

Answer to Problem 56RE

The value is

  $\begin{array}{l}f\left(4\right)=7\\ f"\text{'}\left(4\right)=-12\end{array}$

Explanation of Solution

Given information:

The given polynomial of order four of the function

  $\begin{array}{l}f\left(x\right)\text{at}x=4\text{is}\\ P_4\left(x\right)=7-3\left(x-4\right)+5{\left(x-4\right)}^2-2{\left(x-4\right)}^3+6{\left(x-4\right)}^4\end{array}$

Formula used:

  $f\left(x\right)=f\left(a\right)+\left[f\text{'}\left(a\right)\right]\left(x-a\right)+\frac{\left[f"\left(a\right)\right]{\left(x-a\right)}^2}{2!}+\frac{\left[f"\text{'}\left(a\right)\right]{\left(x-a\right)}^3}{3!}+\mathrm{...}$

Calculation:

Polynomial of order four of a function f (x) is given by at x = a

  $f\left(x\right)=f\left(a\right)+\left[f\text{'}\left(a\right)\right]\left(x-a\right)+\frac{\left[f"\left(a\right)\right]{\left(x-a\right)}^2}{2!}+\frac{\left[f"\text{'}\left(a\right)\right]{\left(x-a\right)}^3}{3!}+\mathrm{...}$

Now, comparing the expression with the given polynomial, where a = 4

  $\begin{array}{l}f\left(4\right)=7f\text{'}\left(4\right)=-3\\ \frac{f"\left(4\right)}{2!}=5\\ \Rightarrow f"\left(4\right)=10\\ \frac{f"\text{'}\left(4\right)}{3!}=-2\\ \Rightarrow f"\text{'}\left(4\right)=-12\end{array}$

  $\begin{array}{l}\frac{f""\left(4\right)}{4!}=6\\ f""\left(4\right)=144\end{array}$

Therefore,

  $\begin{array}{l}f\left(4\right)=7\\ f"\text{'}\left(4\right)=-12\end{array}$

Conclusion:

The value is

  $\begin{array}{l}f\left(4\right)=7\\ f"\text{'}\left(4\right)=-12\end{array}$

(b)

To determine

The second Taylor polynomial for $f\text{'}$ at x=4.

Answer to Problem 56RE

The value is $f\text{'}$ (4.3) = -0.54

Explanation of Solution

Given information:

The given polynomial of order four of the function

  $\begin{array}{l}f\left(x\right)\text{at}x=4\text{is}\\ P_4\left(x\right)=7-3\left(x-4\right)+5{\left(x-4\right)}^2-2{\left(x-4\right)}^3+6{\left(x-4\right)}^4\end{array}$

Formula used:

Substitution method is used.

Calculation:

Since,

  $f\text{'}\left(4\right)=-3f"\left(4\right)=10f"\text{'}\left(4\right)=-12$

Series of $f\text{'}\left(x\right)$ at x = 4

  $\begin{array}{l}{P}_2\left(x\right)=-3+10\left(x-4\right)+\frac{\left(-12\right)}{2!}{\left(x-4\right)}^2\\ P_2\left(x\right)=-3+10\left(x-4\right)-6{\left(x-4\right)}^2\end{array}$

Therefore,

The value of

  $\begin{array}{l}f\text{'}\left(x\right)\text{at}x=4.3\\ f\text{'}\left(4.3\right)=-3+10\left(4.3-4\right)-6{\left(4.3-4\right)}^2\\ f\text{'}\left(4.3\right)=-0.54\end{array}$

Conclusion:

The value is $f\text{'}$ (4.3) = -0.54

(c)

To determine

The fourth Taylor polynomial for $g\left(x\right)={\displaystyle \underset{4}{\overset{x}{\int }}f\left(t\right)}dt$

Answer to Problem 56RE

The value is $g\left(x\right)=7\left(x-4\right)-\frac{3}{2}{\left(x-4\right)}^2+\frac{5}{3}{\left(x-4\right)}^3-\frac{1}{2}{\left(x-4\right)}^4$

Explanation of Solution

Given information:

  $g\left(x\right)={\displaystyle \underset{4}{\overset{x}{\int }}f\left(t\right)}dt$

Formula used:

Substitute the value of x=4

Calculation:

The fourth order polynomial for $g\left(x\right)\text{at }x=4\text{is},$

  $\begin{array}{l}g\left(x\right)={\displaystyle \underset{4}{\overset{x}{\int }}\left[7-3t+5t^2-2t^3\right]dt}\\ g\left(x\right)=7\left(x-4\right)-\frac{3}{2}{\left(x-4\right)}^2+\frac{5}{3}{\left(x-4\right)}^3-\frac{1}{2}{\left(x-4\right)}^4\end{array}$

Therefore,

  $g\left(x\right)=7\left(x-4\right)-\frac{3}{2}{\left(x-4\right)}^2+\frac{5}{3}{\left(x-4\right)}^3-\frac{1}{2}{\left(x-4\right)}^4$

Conclusion:

The value is $g\left(x\right)=7\left(x-4\right)-\frac{3}{2}{\left(x-4\right)}^2+\frac{5}{3}{\left(x-4\right)}^3-\frac{1}{2}{\left(x-4\right)}^4$

(d)

To determine

The value of f(3)

Answer to Problem 56RE

The exact value of f (3) if the function converge to f (x) at x = 3

Explanation of Solution

Given information:

  $g\left(x\right)={\displaystyle \underset{4}{\overset{x}{\int }}f\left(t\right)}dt$

Formula used:

Substitution method is used.

Calculation:

No, it is not possible to determine the exact value of f (3), to determine the exact value the entire series. The exact value of f (3) if the function converge to f (x) at x = 3

Conclusion:

The exact value of f (3) if the function converge to f (x) at x = 3