Chapter 11, Problem 37RE

Solution

a.

To prove: The given function is nonnegative.

The function, nonnegative, is proved.

Given Information:

The function is $f\left(x\right)={\left(x-5\right)}^2;\left[0,4\right]$ .

Calculation:

Consider the given polynomial,

  $f\left(x\right)={\left(x-5\right)}^2;\left[0,4\right]$

Apply point-plotting to graph $f\left(x\right)$ in the interval $\left[0,4\right]$ . As the graph of the function lies on and above the x -axis for this interval, then it is nonnegative.

Make the table for the function.

  

And the required is show as following,

  

Therefore, the function is positive.

b.

To determine: The value of LRAM, RRAM, and the average approximation.

The obtained values are $\text{LRAM}=42.2976$ , $\text{RRAM}=40.3776$ , and $\text{average}=41.3376$ .

The function is $f\left(x\right)={\left(x-5\right)}^2;\left[0,4\right]$ .

Calculation:

Consider the given polynomial,

  $f\left(x\right)={\left(x-5\right)}^2;\left[0,4\right]$

Since we will be approximating using 50 rectangles, we will use the program for LRAM and RRAM using a graphing calculator from the exercise in Section 11.4.

  

And,

  

First, enter the function in Y1. For LRAM, enter $A=0, B=4, N=50$. This will give an area of:

  $\text{LRAM}=42.2976$

For LRAM, enter $A=0, B=4, N=50.$. This will give an area of:

  $\text{RRAM}=40.3776$

The average approximation is defined as,

  $\begin{array}{c}\text{average}=\frac{42.2976+40.3776}{2}\\ =41.3376\end{array}$

Therefore, the obtained values are $\text{LRAM}=42.2976$ , $\text{RRAM}=40.3776$ , and $\text{average}=41.3376$ .