The Laplace transform for the function f ( t ) = 1 − t by using the definition of the Laplace transform.

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Answer 1

Question

Chapter 8, Problem 1RE

To determine

The Laplace transform for the function f(t)=1−t by using the definition of the Laplace transform.

Expert Solution & Answer

Answer to Problem 1RE

The Laplace transform for the function f(t)=1−t is s−1s2_.

Explanation of Solution

Procedure used:

If f(t) is a function defined in the interval (0,∞). Then, the Laplace transform of the function f(t) is:

F(s)=ℒ{f(t)}=∫0∞e−stf(t)dt

The formula for evaluating improper integrals:

∫0∞f(t)dt=limM→∞∫0Mf(t)dt

Calculation:

Consider the function,

f(t)=1−t

Apply the Laplace transform.

F(s)=∫0∞e−st(1−t)dt=∫0∞e−stdt−∫0∞e−sttdt

Apply the formula for evaluation improper integrals:

F(s)=limM→∞∫0Me−stdt−limM→∞∫0Me−sttdt=limM→∞∫0Me−stdt−{limM→∞[−1ste−st]0M+limM→∞1s∫0Me−stdt}=limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M−limM→∞1s∫0Me−stdt=(1−1s)∫0Me−stdt−limM→∞[−1ste−st]0M

Simplify the above equation.

(1−1s)limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M=(1−1s)limM→∞[−1se−st]0M−0=(1−1s)limM→∞[−1se−sM+1s]

If s>0, then limM→∞e−sM=0.

(1−1s)limM→∞[−1se−sM+1s]=(1−1s)1s=s−1s2

Therefore, the Laplace transform for the function f(t)=1−t is s−1s2_.

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Answer 2

Question

Chapter 8, Problem 1RE

To determine

The Laplace transform for the function f(t)=1−t by using the definition of the Laplace transform.

Expert Solution & Answer

Answer to Problem 1RE

The Laplace transform for the function f(t)=1−t is s−1s2_.

Explanation of Solution

Procedure used:

If f(t) is a function defined in the interval (0,∞). Then, the Laplace transform of the function f(t) is:

F(s)=ℒ{f(t)}=∫0∞e−stf(t)dt

The formula for evaluating improper integrals:

∫0∞f(t)dt=limM→∞∫0Mf(t)dt

Calculation:

Consider the function,

f(t)=1−t

Apply the Laplace transform.

F(s)=∫0∞e−st(1−t)dt=∫0∞e−stdt−∫0∞e−sttdt

Apply the formula for evaluation improper integrals:

F(s)=limM→∞∫0Me−stdt−limM→∞∫0Me−sttdt=limM→∞∫0Me−stdt−{limM→∞[−1ste−st]0M+limM→∞1s∫0Me−stdt}=limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M−limM→∞1s∫0Me−stdt=(1−1s)∫0Me−stdt−limM→∞[−1ste−st]0M

Simplify the above equation.

(1−1s)limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M=(1−1s)limM→∞[−1se−st]0M−0=(1−1s)limM→∞[−1se−sM+1s]

If s>0, then limM→∞e−sM=0.

(1−1s)limM→∞[−1se−sM+1s]=(1−1s)1s=s−1s2

Therefore, the Laplace transform for the function f(t)=1−t is s−1s2_.

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Answer 3

Question

Chapter 8, Problem 1RE

To determine

The Laplace transform for the function f(t)=1−t by using the definition of the Laplace transform.

Expert Solution & Answer

Answer to Problem 1RE

The Laplace transform for the function f(t)=1−t is s−1s2_.

Explanation of Solution

Procedure used:

If f(t) is a function defined in the interval (0,∞). Then, the Laplace transform of the function f(t) is:

F(s)=ℒ{f(t)}=∫0∞e−stf(t)dt

The formula for evaluating improper integrals:

∫0∞f(t)dt=limM→∞∫0Mf(t)dt

Calculation:

Consider the function,

f(t)=1−t

Apply the Laplace transform.

F(s)=∫0∞e−st(1−t)dt=∫0∞e−stdt−∫0∞e−sttdt

Apply the formula for evaluation improper integrals:

F(s)=limM→∞∫0Me−stdt−limM→∞∫0Me−sttdt=limM→∞∫0Me−stdt−{limM→∞[−1ste−st]0M+limM→∞1s∫0Me−stdt}=limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M−limM→∞1s∫0Me−stdt=(1−1s)∫0Me−stdt−limM→∞[−1ste−st]0M

Simplify the above equation.

(1−1s)limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M=(1−1s)limM→∞[−1se−st]0M−0=(1−1s)limM→∞[−1se−sM+1s]

If s>0, then limM→∞e−sM=0.

(1−1s)limM→∞[−1se−sM+1s]=(1−1s)1s=s−1s2

Therefore, the Laplace transform for the function f(t)=1−t is s−1s2_.

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Answer 4

Question

Chapter 8, Problem 1RE

To determine

The Laplace transform for the function f(t)=1−t by using the definition of the Laplace transform.

Expert Solution & Answer

Answer to Problem 1RE

The Laplace transform for the function f(t)=1−t is s−1s2_.

Explanation of Solution

Procedure used:

If f(t) is a function defined in the interval (0,∞). Then, the Laplace transform of the function f(t) is:

F(s)=ℒ{f(t)}=∫0∞e−stf(t)dt

The formula for evaluating improper integrals:

∫0∞f(t)dt=limM→∞∫0Mf(t)dt

Calculation:

Consider the function,

f(t)=1−t

Apply the Laplace transform.

F(s)=∫0∞e−st(1−t)dt=∫0∞e−stdt−∫0∞e−sttdt

Apply the formula for evaluation improper integrals:

F(s)=limM→∞∫0Me−stdt−limM→∞∫0Me−sttdt=limM→∞∫0Me−stdt−{limM→∞[−1ste−st]0M+limM→∞1s∫0Me−stdt}=limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M−limM→∞1s∫0Me−stdt=(1−1s)∫0Me−stdt−limM→∞[−1ste−st]0M

Simplify the above equation.

(1−1s)limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M=(1−1s)limM→∞[−1se−st]0M−0=(1−1s)limM→∞[−1se−sM+1s]

If s>0, then limM→∞e−sM=0.

(1−1s)limM→∞[−1se−sM+1s]=(1−1s)1s=s−1s2

Therefore, the Laplace transform for the function f(t)=1−t is s−1s2_.

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Answer 5

Chapter 8, Problem 1RE

To determine

The Laplace transform for the function f(t)=1−t by using the definition of the Laplace transform.

Expert Solution & Answer

Answer to Problem 1RE

The Laplace transform for the function f(t)=1−t is s−1s2_.

Explanation of Solution

Procedure used:

If f(t) is a function defined in the interval (0,∞). Then, the Laplace transform of the function f(t) is:

F(s)=ℒ{f(t)}=∫0∞e−stf(t)dt

The formula for evaluating improper integrals:

∫0∞f(t)dt=limM→∞∫0Mf(t)dt

Calculation:

Consider the function,

f(t)=1−t

Apply the Laplace transform.

F(s)=∫0∞e−st(1−t)dt=∫0∞e−stdt−∫0∞e−sttdt

Apply the formula for evaluation improper integrals:

F(s)=limM→∞∫0Me−stdt−limM→∞∫0Me−sttdt=limM→∞∫0Me−stdt−{limM→∞[−1ste−st]0M+limM→∞1s∫0Me−stdt}=limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M−limM→∞1s∫0Me−stdt=(1−1s)∫0Me−stdt−limM→∞[−1ste−st]0M

Simplify the above equation.

(1−1s)limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M=(1−1s)limM→∞[−1se−st]0M−0=(1−1s)limM→∞[−1se−sM+1s]

If s>0, then limM→∞e−sM=0.

(1−1s)limM→∞[−1se−sM+1s]=(1−1s)1s=s−1s2

Therefore, the Laplace transform for the function f(t)=1−t is s−1s2_.

Answer 6

Chapter 8, Problem 1RE

To determine

The Laplace transform for the function f(t)=1−t by using the definition of the Laplace transform.

Expert Solution & Answer

Answer to Problem 1RE

The Laplace transform for the function f(t)=1−t is s−1s2_.

Explanation of Solution

Procedure used:

If f(t) is a function defined in the interval (0,∞). Then, the Laplace transform of the function f(t) is:

F(s)=ℒ{f(t)}=∫0∞e−stf(t)dt

The formula for evaluating improper integrals:

∫0∞f(t)dt=limM→∞∫0Mf(t)dt

Calculation:

Consider the function,

f(t)=1−t

Apply the Laplace transform.

F(s)=∫0∞e−st(1−t)dt=∫0∞e−stdt−∫0∞e−sttdt

Apply the formula for evaluation improper integrals:

F(s)=limM→∞∫0Me−stdt−limM→∞∫0Me−sttdt=limM→∞∫0Me−stdt−{limM→∞[−1ste−st]0M+limM→∞1s∫0Me−stdt}=limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M−limM→∞1s∫0Me−stdt=(1−1s)∫0Me−stdt−limM→∞[−1ste−st]0M

Simplify the above equation.

(1−1s)limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M=(1−1s)limM→∞[−1se−st]0M−0=(1−1s)limM→∞[−1se−sM+1s]

If s>0, then limM→∞e−sM=0.

(1−1s)limM→∞[−1se−sM+1s]=(1−1s)1s=s−1s2

Therefore, the Laplace transform for the function f(t)=1−t is s−1s2_.

Answer 7

Chapter 8, Problem 1RE

To determine

The Laplace transform for the function f(t)=1−t by using the definition of the Laplace transform.

Answer 8

Expert Solution & Answer

Answer to Problem 1RE

The Laplace transform for the function f(t)=1−t is s−1s2_.

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

Procedure used:

If f(t) is a function defined in the interval (0,∞). Then, the Laplace transform of the function f(t) is:

F(s)=ℒ{f(t)}=∫0∞e−stf(t)dt

The formula for evaluating improper integrals:

∫0∞f(t)dt=limM→∞∫0Mf(t)dt

Calculation:

Consider the function,

f(t)=1−t

Apply the Laplace transform.

F(s)=∫0∞e−st(1−t)dt=∫0∞e−stdt−∫0∞e−sttdt

Apply the formula for evaluation improper integrals:

F(s)=limM→∞∫0Me−stdt−limM→∞∫0Me−sttdt=limM→∞∫0Me−stdt−{limM→∞[−1ste−st]0M+limM→∞1s∫0Me−stdt}=limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M−limM→∞1s∫0Me−stdt=(1−1s)∫0Me−stdt−limM→∞[−1ste−st]0M

Simplify the above equation.

(1−1s)limM→∞∫0Me−stdt−limM→∞[−1ste−st]0M=(1−1s)limM→∞[−1se−st]0M−0=(1−1s)limM→∞[−1se−sM+1s]

If s>0, then limM→∞e−sM=0.

(1−1s)limM→∞[−1se−sM+1s]=(1−1s)1s=s−1s2

Therefore, the Laplace transform for the function f(t)=1−t is s−1s2_.

Answer 12

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

The Laplace transform for the function f ( t ) = 1 − t by using the definition of the Laplace transform.

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The Laplace transform for the function f(t)=1−t by using the definition of the Laplace transform.

Answer to Problem 1RE

Explanation of Solution

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Chapter 8 Solutions