Chapter 6.5, Problem 21E

(a)

To determine

Average temperature when endpoint values are doubled.

Answer to Problem 21E

The average temperature is 70.08 when endpoint values are doubled.

Explanation of Solution

Given information:

In the Trapezoidal rule,

Except endpoint values, every function value is doubled.

According to Trapezoidal rule,

  $T_n=\frac{b-a}{2n}\left(y_0+2y_1+2y_2+\mathrm{...}+2y_{n-1}+y_n\right)$

Substitute the values,

  $\begin{array}{l}{\displaystyle {\int }_0^{12}f\left(x\right)dx}\approx \frac{1}{2}\left[\begin{array}{l}63+2\left(65\right)+2\left(66\right)+2\left(68\right)+2\left(70\right)+2\left(69\right)+2\left(68\right)\\ +2\left(68\right)+2\left(65\right)+2\left(64\right)+2\left(62\right)+2\left(58\right)+55\end{array}\right]\\ =\frac{1}{2}\left[1564\right]\\ =782\end{array}$

Now,

When the endpoint values of 63 and 55 are doubled,

  $\begin{array}{l}{\displaystyle {\int }_0^{12}f\left(x\right)dx}\approx \frac{1}{2}\left[\begin{array}{l}2\left(63\right)+2\left(65\right)+2\left(66\right)+2\left(68\right)+2\left(70\right)+2\left(69\right)+2\left(68\right)\\ +2\left(68\right)+2\left(65\right)+2\left(64\right)+2\left(62\right)+2\left(58\right)+2\left(55\right)\end{array}\right]\\ =\frac{1}{2}\left[63+1564+55\right]\\ =\frac{1}{2}\left[1682\right]\\ =841\end{array}$

Thus,

The average temperature:

  $\begin{array}{l}av\left(f\right)=\frac{1}{b-a}{\displaystyle {\int }_a^{b}f\left(x\right)dx}\\ =\frac{1}{12-0}{\displaystyle {\int }_0^{12}f\left(x\right)dx}\\ \approx \frac{1}{12}\left(841\right)\\ \approx 70.08\end{array}$

(b)

To determine

Significance for not to double the endpoint values if the average temperature is considered over the entire 12 − hour period.

Answer to Problem 21E

It does not make any sense to double the endpoint values.

Explanation of Solution

Given information:

Average temperature is considered over 12 − hour period.

From Part (a) result,

We came to know that

The average temperature changes when the endpoints are doubled.

After doubling the endpoint values,

The error in first and last trapezoids increases.

Therefore,

It does not make any sense to double the endpoint values.