Chapter 14.5, Problem 19AYU

To determine

To solve: The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$,

a. Graph $f$,

indicating the area $A$ under $f$ from $-1$ to 3.

Answer to Problem 19AYU

a.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$.

Calculation:

$ƒ\left(-1\right)=0.368;ƒ(\frac{-1}{2})=0.607;ƒ\left(0\right)=1;ƒ(\frac{1}{2})=1.649;ƒ\left(1\right)=2.718;ƒ(\frac{3}{2})=4.482;ƒ\left(2\right)=7.389;ƒ(\frac{5}{2})=12.182$

a. Graph $ƒ\left(x\right)=e^x$

To determine

To solve: The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$,

b. Approximate the area $A$ by Partition $\left[-1,3\right]$ into four subintervals of equal length and choose $u$ as the left endpoint of each subinterval.

Answer to Problem 19AYU

b. $11.475$

Explanation of Solution

Given:

The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$.

Calculation:

$ƒ\left(-1\right)=0.368;ƒ(\frac{-1}{2})=0.607;ƒ\left(0\right)=1;ƒ(\frac{1}{2})=1.649;ƒ\left(1\right)=2.718;ƒ(\frac{3}{2})=4.482;ƒ\left(2\right)=7.389;ƒ(\frac{5}{2})=12.182$

b. Partition $\left[-1,3\right]$ into four subintervals of equal length 1 and choose $u$ as the left endpoint of each subinterval.

The area $A$ is approximated as

$A=f\left(-1\right)1+f\left(0\right)1+f\left(1\right)1+f\left(2\right)1$

$=0.368\left(1\right)+1\left(1\right)+2.718\left(1\right)+7.389\left(1\right)$

$=0.368+1+2.718+7.389$

$=11.475$

To determine

To solve: The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$,

c. Approximate the area $A$ by Partition $\left[-1,3\right]$ into eight subintervals of equal length and choose $u$ as the left endpoint of each subinterval.

Answer to Problem 19AYU

c. $15.1975$

Explanation of Solution

Given:

The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$.

Calculation:

$ƒ\left(-1\right)=0.368;ƒ(\frac{-1}{2})=0.607;ƒ\left(0\right)=1;ƒ(\frac{1}{2})=1.649;ƒ\left(1\right)=2.718;ƒ(\frac{3}{2})=4.482;ƒ\left(2\right)=7.389;ƒ(\frac{5}{2})=12.182$

c. Partition $\left[-1,3\right]$ into eight subintervals of equal length $\frac{1}{2}$ and choose $i$ as the left endpoint of each subinterval.

The area $A$ is approximated as

$A=f\left(-1\right) \frac{1}{2}+f(- \frac{1}{2})\frac{1}{2}+f\left(0\right)(\frac{1}{2})+f(\frac{1}{2})(\frac{1}{2})+f\left(1\right)(\frac{1}{2})+f(\frac{3}{2})(\frac{1}{2})+f\left(2\right)(\frac{1}{2})+f(\frac{5}{2})(\frac{1}{2})$

$=0.368(\frac{1}{2})+\left(0.607\right)\text{(}\frac{1}{2})+1(\frac{1}{2})+\left(1.649\right)(\frac{1}{2})+\left(2.718\right)(\frac{1}{2})+\left(4.482\right)(\frac{1}{2})+\left(7.389\right)(\frac{1}{2})+\left(12.182\right)(\frac{1}{2})$

$=0.184+0.3035+0.5+0.8245+1.359+2.241+3.6945+6.091$

$=15.1975$

To determine

To solve: The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$,

d. Express the area $A$ as an integral.

Answer to Problem 19AYU

d. $\underset{-1}{\overset{3}{{\displaystyle \int }}}{e}^{x}dx$

Explanation of Solution

Given:

The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$.

Calculation:

$ƒ\left(-1\right)=0.368;ƒ(\frac{-1}{2})=0.607;ƒ\left(0\right)=1;ƒ(\frac{1}{2})=1.649;ƒ\left(1\right)=2.718;ƒ(\frac{3}{2})=4.482;ƒ\left(2\right)=7.389;ƒ(\frac{5}{2})=12.182$

d. Express the area $A$ as an integral.

The area $A$ as an integral is $\underset{-1}{\overset{3}{{\displaystyle \int }}}{e}^{x}dx$.

To determine

To solve: The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$,

e. Use a graphing utility to approximate the integral.

Answer to Problem 19AYU

e. $19.718$

Explanation of Solution

Given:

The function $ƒ\left(x\right)=e^x$ is defined on the interval $\left[-1,3\right]$.

Calculation:

$ƒ\left(-1\right)=0.368;ƒ(\frac{-1}{2})=0.607;ƒ\left(0\right)=1;ƒ(\frac{1}{2})=1.649;ƒ\left(1\right)=2.718;ƒ(\frac{3}{2})=4.482;ƒ\left(2\right)=7.389;ƒ(\frac{5}{2})=12.182$

e. Use a graphing utility to approximate the integral.

That is evaluate the integral

The value of the integral $\underset{-1}{\overset{3}{{\displaystyle \int }}}{e}^{x}dx$ is $19.718$.

so the area under the graph of $ƒ$ from $-1$ to 3 is $19.718$.