Chapter 4.1, Problem 113AYU

(a)

To determine

To create: a table by using a graphing utility

Answer to Problem 113AYU

X	y
-0.9	0.181
-0.8	0.328
-0.7	0.447
-0.6	0.544
-0.5	0.625
-0.4	0.696
-0.3	0.763
-0.2	0.832
-0.1	0.909
0	1.000
0.1	1.111
0.2	1.248
0.3	1.417
0.4	1.624
0.5	1.88
0.6	2.176
0.7	2.533
0.8	2.952
0.9	3.439

Explanation of Solution

Given:

  $Y_1=f\left(x\right)=\frac{1}{1-x}$

  $Y_2=g_2\left(x\right)=1+x+x^2+x^3$

For $-1<x<1$ with $\Delta \text{Tbl}=0.1$ .

Calculation:

In the first part, make a table for two functions given as

  $Y_1=f\left(x\right)=\frac{1}{1-x}$ and $Y_2=1+x+x^2+x^3$ between $-1<x<1$ .

So, let us make a table with the values of x and $f\left(x\right)$ with $\Delta x=0.1$ .

For $f\left(x\right)=\frac{1}{1-x}$

X	y
-0.9	0.522
-0.8	0.556
-0.7	0.588
-0.6	0.625
-0.5	0.667
-0.4	0.714
-0.3	0.76
-0.2	0.834
-0.1	0.909
0	1.000
0.1	1.111
0.2	1.25
0.3	1.42
0.4	1.666
0.5	2.000
0.6	2.500
0.7	3.332
0.8	5.000
0.9	10.00

For $f\left(x\right)=1+x+x^2+x^3$

X	y
-0.9	0.181
-0.8	0.328
-0.7	0.447
-0.6	0.544
-0.5	0.625
-0.4	0.696
-0.3	0.763
-0.2	0.832
-0.1	0.909
0	1.000
0.1	1.111
0.2	1.248
0.3	1.417
0.4	1.624
0.5	1.88
0.6	2.176
0.7	2.533
0.8	2.952
0.9	3.439

Conclusion:

Therefore, the table for the given values is completed.

(b)

To determine

To create: a table by using a graphing utility

Answer to Problem 113AYU

x	y
-0.9	0.8371
-0.8	0.7376
-0.7	0.6871
-0.6	0.6736
-0.5	0.6875
-0.4	0.7216
-0.3	0.7711
-0.2	0.8336
-0.1	0.9091
0	1.0000
0.1	1.1111
0.2	1.2496
0.3	1.4251
0.4	1.6496
0.5	1.9375
0.6	2.3056
0.7	2.7732
0.8	3.3616
0.9	4.0951

Explanation of Solution

Given:

  $Y_1=f\left(x\right)=\frac{1}{1-x}$ and $Y_3=g_3\left(x\right)=1+x+x^2+x^3+x^4$ for $-1<x<1$ with $\Delta \text{Tbl}=0.1$ .

Calculation:

Here evaluate the same way but now the functions will be

  $Y_1=f\left(x\right)=\frac{1}{1-x}$ and $Y_2=f\left(x\right)=1+x+x^2+x^3$ between $-1<x<1$ . So let us make a table with the values of x and $f\left(x\right)$ with $\Delta x=0.1$ .

For $f\left(x\right)=\frac{1}{1-x}$

x	y
-0.9	0.522
-0.8	0.556
-0.7	0.588
-0.6	0.625
-0.5	0.667
-0.4	0.714
-0.3	0.76
-0.2	0.834
-0.1	0.909
0	1.000
0.1	1.111
0.2	1.25
0.3	1.42
0.4	1.666
0.5	2.000
0.6	2.500
0.7	3.332
0.8	5.000
0.9	10.00

For $f\left(x\right)=1+x+x^2+x^3+x^4$

x	y
-0.9	0.8371
-0.8	0.7376
-0.7	0.6871
-0.6	0.6736
-0.5	0.6875
-0.4	0.7216
-0.3	0.7711
-0.2	0.8336
-0.1	0.9091
0	1.0000
0.1	1.1111
0.2	1.2496
0.3	1.4251
0.4	1.6496
0.5	1.9375
0.6	2.3056
0.7	2.7732
0.8	3.3616
0.9	4.0951

Conclusion:

Therefore, the table for the given values is completed.

(c)

To determine

To create: a table by using a graphing utility

Answer to Problem 113AYU

x	y
-0.9	0.4094
-0.8	0.0003
-0.7	0.2789
-0.6	0.4662
-0.5	0.5937
-0.4	0.6857
-0.3	0.7601
-0.2	0.8316
-0.1	0.9090
0	1.0000
0.1	1.1111
0.2	1.2483
0.3	1.4194
0.4	1.6342
0.5	1.9062
0.6	2.6342
0.7	2.7010
0.8	3.2796
0.9	4.0290

Explanation of Solution

Given:

  $Y_1=f\left(x\right)=\frac{1}{1-x}$ and $Y_4=g_3\left(x\right)=1+x+x^2+x^3+x^4+x^5$ for $-1<x<1$ with $\Delta \text{Tbl}=0.1$ .

Calculation:

Here evaluate the same way but now the functions will be

  $Y_1=f\left(x\right)=\frac{1}{1-x}$ and $Y_2=f\left(x\right)=1+x+x^2+x^3$ between $-1<x<1$ .So, let us make a table with the values of x and f ( x ) with $\Delta \text{Tbl}=0.1$

For $f\left(x\right)=\frac{1}{1-x}$

x	y
-0.9	0.522
-0.8	0.556
-0.7	0.588
-0.6	0.625
-0.5	0.667
-0.4	0.714
-0.3	0.76
-0.2	0.834
-0.1	0.909
0	1.000
0.1	1.111
0.2	1.25
0.3	1.42
0.4	1.666
0.5	2.000
0.6	2.500
0.7	3.332
0.8	5.000
0.9	10.00

For $f\left(x\right)=1+x+x^2+x^3+x^4$

x	y
-0.9	0.4094
-0.8	0.0003
-0.7	0.2789
-0.6	0.4662
-0.5	0.5937
-0.4	0.6857
-0.3	0.7601
-0.2	0.8316
-0.1	0.9090
0	1.0000
0.1	1.1111
0.2	1.2483
0.3	1.4194
0.4	1.6342
0.5	1.9062
0.6	2.6342
0.7	2.7010
0.8	3.2796
0.9	4.0290

Conclusion:

Therefore, the table for the given values is completed.

(d)

To determine

To explain: the values of the function when more terms are added to the polynomial

Answer to Problem 113AYU

There is a very less difference between the respective values as compared to values obtained by polynomial of a lesser degree.

Explanation of Solution

Calculation:

As more terms are added to the polynomial, there is a very less difference between the respective values as compared to values obtained by polynomial of a lesser degree. This implies that there are certain values of x which provide a better approximation for calculating the value of the series.

Conclusion:

Therefore, there is a very less difference between the respective values as compared to values obtained by polynomial of a lesser degree.