Answer a. Explanation Given: The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] . Calculation: ƒ ( − 1 ) = 0.368 ; ƒ ( − 1 2 ) = 0.607 ; ƒ ( 0 ) = 1 ; ƒ ( 1 2 ) = 1.649 ; ƒ ( 1 ) = 2.718 ; ƒ ( 3 2 ) = 4.482 ; ƒ ( 2 ) = 7.389 ; ƒ ( 5 2 ) = 12.182 a. Graph ƒ ( x ) = e x To determine To solve: The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] , b. Approximate the area A by Partition [ − 1 , 3 ] into four subintervals of equal length and choose u as the left endpoint of each subinterval. Answer b. 11.475 Explanation Given: The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] . Calculation: ƒ ( − 1 ) = 0.368 ; ƒ ( − 1 2 ) = 0.607 ; ƒ ( 0 ) = 1 ; ƒ ( 1 2 ) = 1.649 ; ƒ ( 1 ) = 2.718 ; ƒ ( 3 2 ) = 4.482 ; ƒ ( 2 ) = 7.389 ; ƒ ( 5 2 ) = 12.182 b. Partition [ − 1 , 3 ] 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 ( − 1 ) 1 + f ( 0 ) 1 + f ( 1 ) 1 + f ( 2 ) 1 = 0.368 ( 1 ) + 1 ( 1 ) + 2.718 ( 1 ) + 7.389 ( 1 ) = 0.368 + 1 + 2.718 + 7.389 = 11.475 To determine To solve: The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] , c. Approximate the area A by Partition [ − 1 , 3 ] into eight subintervals of equal length and choose u as the left endpoint of each subinterval. Answer c. 15.1975 Explanation Given: The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] . Calculation: ƒ ( − 1 ) = 0.368 ; ƒ ( − 1 2 ) = 0.607 ; ƒ ( 0 ) = 1 ; ƒ ( 1 2 ) = 1.649 ; ƒ ( 1 ) = 2.718 ; ƒ ( 3 2 ) = 4.482 ; ƒ ( 2 ) = 7.389 ; ƒ ( 5 2 ) = 12.182 c. Partition [ − 1 , 3 ] into eight subintervals of equal length 1 2 and choose i as the left endpoint of each subinterval. The area A is approximated as A = f ( − 1 ) 1 2 + f ( − 1 2 ) 1 2 + f ( 0 ) ( 1 2 ) + f ( 1 2 ) ( 1 2 ) + f ( 1 ) ( 1 2 ) + f ( 3 2 ) ( 1 2 ) + f ( 2 ) ( 1 2 ) + f ( 5 2 ) ( 1 2 ) = 0.368 ( 1 2 ) + ( 0.607 ) ( 1 2 ) + 1 ( 1 2 ) + ( 1.649 ) ( 1 2 ) + ( 2.718 ) ( 1 2 ) + ( 4.482 ) ( 1 2 ) + ( 7.389 ) ( 1 2 ) + ( 12.182 ) ( 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 ( x ) = e x is defined on the interval [ − 1 , 3 ] , d. Express the area A as an integral. Answer d. ∫ − 1 3 e x d x Explanation Given: The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] . Calculation: ƒ ( − 1 ) = 0.368 ; ƒ ( − 1 2 ) = 0.607 ; ƒ ( 0 ) = 1 ; ƒ ( 1 2 ) = 1.649 ; ƒ ( 1 ) = 2.718 ; ƒ ( 3 2 ) = 4.482 ; ƒ ( 2 ) = 7.389 ; ƒ ( 5 2 ) = 12.182 d. Express the area A as an integral. The area A as an integral is ∫ − 1 3 e x d x . To determine To solve: The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] , e. Use a graphing utility to approximate the integral. Answer e. 19.718 Explanation Given: The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] . Calculation: ƒ ( − 1 ) = 0.368 ; ƒ ( − 1 2 ) = 0.607 ; ƒ ( 0 ) = 1 ; ƒ ( 1 2 ) = 1.649 ; ƒ ( 1 ) = 2.718 ; ƒ ( 3 2 ) = 4.482 ; ƒ ( 2 ) = 7.389 ; ƒ ( 5 2 ) = 12.182 e. Use a graphing utility to approximate the integral. That is evaluate the integral The value of the integral ∫ − 1 3 e x d x is 19.718 . so the area under the graph of ƒ from − 1 to 3 is 19.718 .

9th Edition

ISBN: 9780321716835

Author: Michael Sullivan

Publisher: Addison Wesley

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Question

Chapter 14.5, Problem 19AYU

To determine

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

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

Expert Solution

Answer to Problem 19AYU

Precalculus, Chapter 14.5, Problem 19AYU , additional homework tip 1

Explanation of Solution

Given:

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

Calculation:

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

a. Graph $ƒ (x) = e^{x}$

Precalculus, Chapter 14.5, Problem 19AYU , additional homework tip 2

To determine

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

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

Expert Solution

Answer to Problem 19AYU

b. $11.475$

Explanation of Solution

Given:

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

Calculation:

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

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

The area $A$ is approximated as

$A = f (- 1) 1 + f (0) 1 + f (1) 1 + f (2) 1$

$= 0.368 (1) + 1 (1) + 2.718 (1) + 7.389 (1)$

$= 0.368 + 1 + 2.718 + 7.389$

$= 11.475$

To determine

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

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

Expert Solution

Answer to Problem 19AYU

c. $15.1975$

Explanation of Solution

Given:

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

Calculation:

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

c. Partition $[- 1, 3]$ 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

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

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

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

$= 15.1975$

To determine

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

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

Expert Solution

Answer to Problem 19AYU

d. $\int_{- 1}^{3} e^{x} d x$

Explanation of Solution

Given:

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

Calculation:

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

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

The area $A$ as an integral is $\int_{- 1}^{3} e^{x} d x$ .

To determine

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

e. Use a graphing utility to approximate the integral.

Expert Solution

Answer to Problem 19AYU

e. $19.718$

Explanation of Solution

Given:

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

Calculation:

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

e. Use a graphing utility to approximate the integral.

That is evaluate the integral

The value of the integral $\int_{- 1}^{3} e^{x} d x$ is $19.718$ .

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

Chapter 14.5, Problem 18AYU

Chapter 14.5, Problem 20AYU

Chapter 14 Solutions

Precalculus

Ch. 14.1 - Graph f( x )={ 3x2ifx2 3ifx=2 (pp.100-102)Ch. 14.1 - Prob. 2AYU Ch. 14.1 - Prob. 3AYU Ch. 14.1 - Prob. 4AYU Ch. 14.1 - True or False lim xc f( x )=N may be described by...Ch. 14.1 - Prob. 6AYU Ch. 14.1 - lim x2 ( 4 x 3 )Ch. 14.1 - lim x3 ( 2 x 2 +1 )Ch. 14.1 - lim x0 x+1 x 2 +1 Ch. 14.1 - Prob. 10AYU

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Knowledge Booster

The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] , a. Graph f , indicating the area A under f from − 1 to 3.

Solution Summary: The author illustrates how the function (x) = e x is defined on the interval.

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

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

Answer to Problem 19AYU

Explanation of Solution

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

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

Answer to Problem 19AYU

Explanation of Solution

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

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

Answer to Problem 19AYU

Explanation of Solution

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

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

Answer to Problem 19AYU

Explanation of Solution

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

e. Use a graphing utility to approximate the integral.

Answer to Problem 19AYU

Explanation of Solution

Chapter 14 Solutions

Additional Math Textbook Solutions

The function ƒ ( x ) = e x is defined on the interval [ − 1 , 3 ] , a. Graph f , indicating the area A under f from − 1 to 3.

Solution Summary: The author illustrates how the function (x) = e x is defined on the interval.

To solve: The function ƒ( x ) = e x is defined on the interval [ −1, 3 ] , a. Graph f , indicating the area A under f from −1 to 3.

Answer to Problem 19AYU

Explanation of Solution

To solve: The function ƒ( x ) = e x is defined on the interval [ −1, 3 ] , b. Approximate the area A by Partition [ −1, 3 ] into four subintervals of equal length and choose u as the left endpoint of each subinterval.

Answer to Problem 19AYU

Explanation of Solution

To solve: The function ƒ( x ) = e x is defined on the interval [ −1, 3 ] , c. Approximate the area A by Partition [ −1, 3 ] into eight subintervals of equal length and choose u as the left endpoint of each subinterval.

Answer to Problem 19AYU

Explanation of Solution

To solve: The function ƒ( x ) = e x is defined on the interval [ −1, 3 ] , d. Express the area A as an integral.

Answer to Problem 19AYU

Explanation of Solution

To solve: The function ƒ( x ) = e x is defined on the interval [ −1, 3 ] , e. Use a graphing utility to approximate the integral.

Answer to Problem 19AYU

Explanation of Solution

Chapter 14 Solutions

Additional Math Textbook Solutions

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

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

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

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

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

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

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

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

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

e. Use a graphing utility to approximate the integral.