Chapter 11.2, Problem 71PPS

(a)

To determine

Make a table of the partial sums of the series for $1\le k\le 10$

Answer to Problem 71PPS

Explanation of Solution

Given:

Consider ${\displaystyle \sum _{k=1}^{x}2k+2}$

Make a table of the partial sums of the series for $1\le k\le 10$

Concept Used:

Substitute k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 in the expression ${\displaystyle \sum _{k=1}^{x}2k+2}$

Calculation:

Make a table of the partial sums of the series for $1\le k\le 10$

$\text{n}$	${\text{S}}_{\text{n}}$
1	4
2	10
3	18
4	28
5	40
6	54
7	70
8	88
9	108
10	130

(b)

To determine

Graph ( k , partial sum)

Explanation of Solution

Given:

Consider ${\displaystyle \sum _{k=1}^{x}2k+2}$ ; Graph ( k , partial sum)

Concept Used:

Graph k as x − axis and partial sum as y − axis.

Calculation:

Graph the k (number of terms) and the partial sum.

  

(c)

To determine

Graph $\text{f(x)}\text{=}{\text{x}}^2+3\text{x}$

Explanation of Solution

Given:

Graph $\text{f(x)}\text{=}{\text{x}}^2+3\text{x}$

Concept Used:

Graph the quadratic function $\text{f(x)}\text{=}{\text{x}}^2+3\text{x}$

Calculation:

Table for graphing for $1\le x\le 10$ $\text{x}$ $\text{f(x)}\text{=}{\text{x}}^2+3\text{x}$ 1 4 2 10 3 18 4 28 5 40 6 54 7 70 8 88 9 108 10 130

Graph of $\text{f(x)}\text{=}{\text{x}}^2+3\text{x}$

(d)

To determine

Find the similarity of two graphs.

Answer to Problem 71PPS

Explanation of Solution

Given:

What do you notice about the two graphs?

Consider ${\displaystyle \sum _{k=1}^{x}2k+2}$

Concept Used:

The graphs cover the same range. The domain of the series is the natural numbers, while the domain of the quadratic function is all real numbers, $0\le x\le 10$

(e)

To determine

What conclusions can you make about the relationship between quadratic function and the sum of arithmetic series?

Answer to Problem 71PPS

Explanation of Solution

Given:

What conclusions can you make about the relationship between quadratic function and the sum of arithmetic series?

Concept Used:

For every partial sum of an arithmetic series, there is a corresponding quadratic function that shares the same range,

(f)

To determine

Find the arithmetic series that relates to $\text{g(x)}\text{=}{\text{x}}^2+8\text{x}$

Explanation of Solution

Given:

Find the arithmetic series that relates to $\text{g(x)}\text{=}{\text{x}}^2+8\text{x}$

Concept Used:

The arithmetic series is ${\displaystyle \sum _{k=1}^{x}2k+1}$