Chapter 14.5, Problem 10AYU

To determine

To solve: The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$,

a. Graph $f$.

In (b)-(e), approximate the area $A$ under $f$ from 0 to 6 as follows:

Answer to Problem 10AYU

a.

Explanation of Solution

Given:

The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$.

Calculation:

$f\left(0\right)=4\left(0\right)=0$; $f\left(1\right)=4\left(1\right)=4$; $f\left(2\right)=4\left(2\right)=8$; $f\left(3\right)=4\left(3\right)=12$; $f\left(4\right)=4\left(4\right)=16$; $f\left(5\right)=4\left(5\right)=20$; $f\left(6\right)=4\left(6\right)=24$

a. Graph $f\left(x\right)=4x$.

To determine

To solve: The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$,

b. Partition $\left[0,6\right]$ into three subintervals of equal length 2 and choose $u$ as the left endpoint of each subinterval.

Answer to Problem 10AYU

b. 48

Explanation of Solution

Given:

The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$.

Calculation:

$f\left(0\right)=4\left(0\right)=0$; $f\left(1\right)=4\left(1\right)=4$; $f\left(2\right)=4\left(2\right)=8$; $f\left(3\right)=4\left(3\right)=12$; $f\left(4\right)=4\left(4\right)=16$; $f\left(5\right)=4\left(5\right)=20$; $f\left(6\right)=4\left(6\right)=24$

b. Partition $\left[0,6\right]$ into three subintervals of equal length 2 and choose $u$ as the left endpoint of each subinterval.

The area $A$ is approximated as

$A=f\left(0\right)2+f\left(2\right)2+f\left(4\right)2$

$=0\left(2\right)+8\left(2\right)+16\left(2\right)$

$=0+16+32$

$=48$

To determine

To solve: The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$,

c. Partition $\left[0,6\right]$ into three subintervals of equal length and choose $u$ as the right endpoint of each subinterval.

Answer to Problem 10AYU

c. 96

Explanation of Solution

Given:

The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$.

Calculation:

$f\left(0\right)=4\left(0\right)=0$; $f\left(1\right)=4\left(1\right)=4$; $f\left(2\right)=4\left(2\right)=8$; $f\left(3\right)=4\left(3\right)=12$; $f\left(4\right)=4\left(4\right)=16$; $f\left(5\right)=4\left(5\right)=20$; $f\left(6\right)=4\left(6\right)=24$

c. Partition $\left[0,6\right]$ into three subintervals of equal length 2 and choose $u$ as the right endpoint of each subinterval.

The area $A$ is approximated as

$A=f\left(2\right)2+f\left(4\right)2+f\left(6\right)2$

$=8\left(2\right)+16\left(2\right)+24\left(2\right)$

$=16+32+48$

$=96$

To determine

To solve: The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$,

d. Partition $\left[0,6\right]$ into six subintervals of equal length and choose $u$ as the left endpoint of each subinterval.

Answer to Problem 10AYU

d. 60

Explanation of Solution

Given:

The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$.

Calculation:

$f\left(0\right)=4\left(0\right)=0$; $f\left(1\right)=4\left(1\right)=4$; $f\left(2\right)=4\left(2\right)=8$; $f\left(3\right)=4\left(3\right)=12$; $f\left(4\right)=4\left(4\right)=16$; $f\left(5\right)=4\left(5\right)=20$; $f\left(6\right)=4\left(6\right)=24$

d. Partition $\left[0,6\right]$ into six subintervals of equal length 1 and choose $u$ as the left endpoint of each subinterval.

The area $A$ is approximated as

$A=f\left(0\right)1+f\left(1\right)1+f\left(2\right)1+f\left(3\right)1+f\left(4\right)1+f\left(5\right)1$

$=0\left(1\right)+4\left(1\right)+8\left(1\right)+12\left(1\right)+16\left(1\right)+20\left(1\right)$

$=0+4+8+12+16+20$

$=60$

To determine

To solve: The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$,

e. Partition $\left[0,6\right]$ into six subintervals of equal length and choose $u$ as the right endpoint of each subinterval.

Answer to Problem 10AYU

e. 84

Explanation of Solution

Given:

The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$.

Calculation:

$f\left(0\right)=4\left(0\right)=0$; $f\left(1\right)=4\left(1\right)=4$; $f\left(2\right)=4\left(2\right)=8$; $f\left(3\right)=4\left(3\right)=12$; $f\left(4\right)=4\left(4\right)=16$; $f\left(5\right)=4\left(5\right)=20$; $f\left(6\right)=4\left(6\right)=24$

e. Partition $\left[0,6\right]$ into six subintervals of equal length 1 and choose $u$ as the right endpoint of each subinterval.

The area $A$ is approximated as

$A=f\left(1\right)1+f\left(2\right)1+f\left(3\right)1+f\left(4\right)1+f\left(5\right)1+f\left(6\right)1$

$=4\left(1\right)+8\left(1\right)+12\left(1\right)+16\left(1\right)+20\left(1\right)+24\left(1\right)$

$=4+8+12+16+20+24$

$=84$

To determine

To solve: The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$,

f. What is the actual area $A$?

Answer to Problem 10AYU

f. 72

Explanation of Solution

Given:

The function $f\left(x\right)=4x$ is defined on the interval $\left[0,6\right]$.

Calculation:

$f\left(0\right)=4\left(0\right)=0$; $f\left(1\right)=4\left(1\right)=4$; $f\left(2\right)=4\left(2\right)=8$; $f\left(3\right)=4\left(3\right)=12$; $f\left(4\right)=4\left(4\right)=16$; $f\left(5\right)=4\left(5\right)=20$; $f\left(6\right)=4\left(6\right)=24$

f. The actual area under the graph of $f\left(x\right)=4x$ from 0 to 6 is the area of a right triangle whose base is of length 6 and whose height is 24. The actual area $A$ is

Therefore

$A= \frac{1}{2}\text{base }\times \text{ height}$

$A= \frac{1}{2}\left(6\right)\left(24\right)=72$