Write the expression for the function given in Figure P12.1 (a) using step functions.

Question

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Answer 1

Question

Chapter 12, Problem 1P

(a)

To determine

Write the expression for the function given in Figure P12.1 (a) using step functions.

(a)

Expert Solution

Answer to Problem 1P

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10).

Explanation of Solution

Given data:

Refer to Figure P12.1 (a) in the textbook for the given function.

Calculation:

The function in Figure P12.1 (a) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−10s;10+t, −10s≥t≥0s10−t, 0s≤t≤10s0, t≥10s

Form a single continuous definition for the function using step function as follows:

10+t, on at t=−10, off at t=0;10−t, on at t=0, off at t=10.

Use the values and write the expression for f(t) as follows:

f(t)=(10+t)[u(t+10)−u(t)]+(10−t)[u(t)−u(t−10)]

Rearrange the expression as follows:

f(t)={10u(t+10)+tu(t+10)−10u(t)−tu(t)+10u(t)−tu(t)−10u(t−10)+tu(t−10)}=(t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

Conclusion:

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

(b)

To determine

Write the expression for the function given in Figure P12.1 (b) using step functions.

(b)

Expert Solution

Answer to Problem 1P

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Explanation of Solution

Given data:

Refer to Figure P12.1 (b).in the textbook for the given function.

Calculation:

The function in Figure P12.1 (b) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−3s;−24−8t, −3s≤t≤−2s−8, −2s≤t≤−1s8t, −1s≤t≤1s8, 1s≤t≤2s−8t+24, 2s≤t≤3s0, t≥3s

Form a single continuous definition for the function using step function as follows:

−24−8t, on at t=−3, off at t=−2;−8, on at t=−2, off at t=−1;8t, on at t=−1, off at t=1;8, on at t=1, off at t=2;−8t+24,on at t=2, off at t=3;

Use the values and write the expression for f(t) as follows:

f(t)={(−24−8t)[u(t+3)−u(t+2)]−8[u(t+2)−u(t+1)]+8t[u(t+1)−u(t−1)]+8[u(t−1)−u(t−2)](24−8t)[u(t−2)−u(t−3)]}

Rearrange the expression as follows:

f(t)={−24u(t+3)−8tu(t+3)+24u(t+2)+8tu(t+2)−8u(t+2)+8u(t+1)+8tu(t+1)−8tu(t−1)+8u(t−1)−8u(t−2)+24u(t−2)−8tu(t−2)−24u(t−3)+8tu(t−3)}={−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}

Conclusion:

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

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Answer 2

Question

Chapter 12, Problem 1P

(a)

To determine

Write the expression for the function given in Figure P12.1 (a) using step functions.

(a)

Expert Solution

Answer to Problem 1P

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10).

Explanation of Solution

Given data:

Refer to Figure P12.1 (a) in the textbook for the given function.

Calculation:

The function in Figure P12.1 (a) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−10s;10+t, −10s≥t≥0s10−t, 0s≤t≤10s0, t≥10s

Form a single continuous definition for the function using step function as follows:

10+t, on at t=−10, off at t=0;10−t, on at t=0, off at t=10.

Use the values and write the expression for f(t) as follows:

f(t)=(10+t)[u(t+10)−u(t)]+(10−t)[u(t)−u(t−10)]

Rearrange the expression as follows:

f(t)={10u(t+10)+tu(t+10)−10u(t)−tu(t)+10u(t)−tu(t)−10u(t−10)+tu(t−10)}=(t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

Conclusion:

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

(b)

To determine

Write the expression for the function given in Figure P12.1 (b) using step functions.

(b)

Expert Solution

Answer to Problem 1P

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Explanation of Solution

Given data:

Refer to Figure P12.1 (b).in the textbook for the given function.

Calculation:

The function in Figure P12.1 (b) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−3s;−24−8t, −3s≤t≤−2s−8, −2s≤t≤−1s8t, −1s≤t≤1s8, 1s≤t≤2s−8t+24, 2s≤t≤3s0, t≥3s

Form a single continuous definition for the function using step function as follows:

−24−8t, on at t=−3, off at t=−2;−8, on at t=−2, off at t=−1;8t, on at t=−1, off at t=1;8, on at t=1, off at t=2;−8t+24,on at t=2, off at t=3;

Use the values and write the expression for f(t) as follows:

f(t)={(−24−8t)[u(t+3)−u(t+2)]−8[u(t+2)−u(t+1)]+8t[u(t+1)−u(t−1)]+8[u(t−1)−u(t−2)](24−8t)[u(t−2)−u(t−3)]}

Rearrange the expression as follows:

f(t)={−24u(t+3)−8tu(t+3)+24u(t+2)+8tu(t+2)−8u(t+2)+8u(t+1)+8tu(t+1)−8tu(t−1)+8u(t−1)−8u(t−2)+24u(t−2)−8tu(t−2)−24u(t−3)+8tu(t−3)}={−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}

Conclusion:

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

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Answer 3

Question

Chapter 12, Problem 1P

(a)

To determine

Write the expression for the function given in Figure P12.1 (a) using step functions.

(a)

Expert Solution

Answer to Problem 1P

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10).

Explanation of Solution

Given data:

Refer to Figure P12.1 (a) in the textbook for the given function.

Calculation:

The function in Figure P12.1 (a) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−10s;10+t, −10s≥t≥0s10−t, 0s≤t≤10s0, t≥10s

Form a single continuous definition for the function using step function as follows:

10+t, on at t=−10, off at t=0;10−t, on at t=0, off at t=10.

Use the values and write the expression for f(t) as follows:

f(t)=(10+t)[u(t+10)−u(t)]+(10−t)[u(t)−u(t−10)]

Rearrange the expression as follows:

f(t)={10u(t+10)+tu(t+10)−10u(t)−tu(t)+10u(t)−tu(t)−10u(t−10)+tu(t−10)}=(t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

Conclusion:

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

(b)

To determine

Write the expression for the function given in Figure P12.1 (b) using step functions.

(b)

Expert Solution

Answer to Problem 1P

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Explanation of Solution

Given data:

Refer to Figure P12.1 (b).in the textbook for the given function.

Calculation:

The function in Figure P12.1 (b) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−3s;−24−8t, −3s≤t≤−2s−8, −2s≤t≤−1s8t, −1s≤t≤1s8, 1s≤t≤2s−8t+24, 2s≤t≤3s0, t≥3s

Form a single continuous definition for the function using step function as follows:

−24−8t, on at t=−3, off at t=−2;−8, on at t=−2, off at t=−1;8t, on at t=−1, off at t=1;8, on at t=1, off at t=2;−8t+24,on at t=2, off at t=3;

Use the values and write the expression for f(t) as follows:

f(t)={(−24−8t)[u(t+3)−u(t+2)]−8[u(t+2)−u(t+1)]+8t[u(t+1)−u(t−1)]+8[u(t−1)−u(t−2)](24−8t)[u(t−2)−u(t−3)]}

Rearrange the expression as follows:

f(t)={−24u(t+3)−8tu(t+3)+24u(t+2)+8tu(t+2)−8u(t+2)+8u(t+1)+8tu(t+1)−8tu(t−1)+8u(t−1)−8u(t−2)+24u(t−2)−8tu(t−2)−24u(t−3)+8tu(t−3)}={−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}

Conclusion:

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

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Answer 4

Question

Chapter 12, Problem 1P

(a)

To determine

Write the expression for the function given in Figure P12.1 (a) using step functions.

(a)

Expert Solution

Answer to Problem 1P

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10).

Explanation of Solution

Given data:

Refer to Figure P12.1 (a) in the textbook for the given function.

Calculation:

The function in Figure P12.1 (a) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−10s;10+t, −10s≥t≥0s10−t, 0s≤t≤10s0, t≥10s

Form a single continuous definition for the function using step function as follows:

10+t, on at t=−10, off at t=0;10−t, on at t=0, off at t=10.

Use the values and write the expression for f(t) as follows:

f(t)=(10+t)[u(t+10)−u(t)]+(10−t)[u(t)−u(t−10)]

Rearrange the expression as follows:

f(t)={10u(t+10)+tu(t+10)−10u(t)−tu(t)+10u(t)−tu(t)−10u(t−10)+tu(t−10)}=(t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

Conclusion:

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

(b)

To determine

Write the expression for the function given in Figure P12.1 (b) using step functions.

(b)

Expert Solution

Answer to Problem 1P

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Explanation of Solution

Given data:

Refer to Figure P12.1 (b).in the textbook for the given function.

Calculation:

The function in Figure P12.1 (b) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−3s;−24−8t, −3s≤t≤−2s−8, −2s≤t≤−1s8t, −1s≤t≤1s8, 1s≤t≤2s−8t+24, 2s≤t≤3s0, t≥3s

Form a single continuous definition for the function using step function as follows:

−24−8t, on at t=−3, off at t=−2;−8, on at t=−2, off at t=−1;8t, on at t=−1, off at t=1;8, on at t=1, off at t=2;−8t+24,on at t=2, off at t=3;

Use the values and write the expression for f(t) as follows:

f(t)={(−24−8t)[u(t+3)−u(t+2)]−8[u(t+2)−u(t+1)]+8t[u(t+1)−u(t−1)]+8[u(t−1)−u(t−2)](24−8t)[u(t−2)−u(t−3)]}

Rearrange the expression as follows:

f(t)={−24u(t+3)−8tu(t+3)+24u(t+2)+8tu(t+2)−8u(t+2)+8u(t+1)+8tu(t+1)−8tu(t−1)+8u(t−1)−8u(t−2)+24u(t−2)−8tu(t−2)−24u(t−3)+8tu(t−3)}={−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}

Conclusion:

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

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Answer 5

Chapter 12, Problem 1P

(a)

To determine

Write the expression for the function given in Figure P12.1 (a) using step functions.

(a)

Expert Solution

Answer to Problem 1P

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10).

Explanation of Solution

Given data:

Refer to Figure P12.1 (a) in the textbook for the given function.

Calculation:

The function in Figure P12.1 (a) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−10s;10+t, −10s≥t≥0s10−t, 0s≤t≤10s0, t≥10s

Form a single continuous definition for the function using step function as follows:

10+t, on at t=−10, off at t=0;10−t, on at t=0, off at t=10.

Use the values and write the expression for f(t) as follows:

f(t)=(10+t)[u(t+10)−u(t)]+(10−t)[u(t)−u(t−10)]

Rearrange the expression as follows:

f(t)={10u(t+10)+tu(t+10)−10u(t)−tu(t)+10u(t)−tu(t)−10u(t−10)+tu(t−10)}=(t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

Conclusion:

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

(b)

To determine

Write the expression for the function given in Figure P12.1 (b) using step functions.

(b)

Expert Solution

Answer to Problem 1P

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Explanation of Solution

Given data:

Refer to Figure P12.1 (b).in the textbook for the given function.

Calculation:

The function in Figure P12.1 (b) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−3s;−24−8t, −3s≤t≤−2s−8, −2s≤t≤−1s8t, −1s≤t≤1s8, 1s≤t≤2s−8t+24, 2s≤t≤3s0, t≥3s

Form a single continuous definition for the function using step function as follows:

−24−8t, on at t=−3, off at t=−2;−8, on at t=−2, off at t=−1;8t, on at t=−1, off at t=1;8, on at t=1, off at t=2;−8t+24,on at t=2, off at t=3;

Use the values and write the expression for f(t) as follows:

f(t)={(−24−8t)[u(t+3)−u(t+2)]−8[u(t+2)−u(t+1)]+8t[u(t+1)−u(t−1)]+8[u(t−1)−u(t−2)](24−8t)[u(t−2)−u(t−3)]}

Rearrange the expression as follows:

f(t)={−24u(t+3)−8tu(t+3)+24u(t+2)+8tu(t+2)−8u(t+2)+8u(t+1)+8tu(t+1)−8tu(t−1)+8u(t−1)−8u(t−2)+24u(t−2)−8tu(t−2)−24u(t−3)+8tu(t−3)}={−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}

Conclusion:

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Answer 6

Chapter 12, Problem 1P

(a)

To determine

Write the expression for the function given in Figure P12.1 (a) using step functions.

(a)

Expert Solution

Answer to Problem 1P

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10).

Explanation of Solution

Given data:

Refer to Figure P12.1 (a) in the textbook for the given function.

Calculation:

The function in Figure P12.1 (a) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−10s;10+t, −10s≥t≥0s10−t, 0s≤t≤10s0, t≥10s

Form a single continuous definition for the function using step function as follows:

10+t, on at t=−10, off at t=0;10−t, on at t=0, off at t=10.

Use the values and write the expression for f(t) as follows:

f(t)=(10+t)[u(t+10)−u(t)]+(10−t)[u(t)−u(t−10)]

Rearrange the expression as follows:

f(t)={10u(t+10)+tu(t+10)−10u(t)−tu(t)+10u(t)−tu(t)−10u(t−10)+tu(t−10)}=(t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

Conclusion:

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

Answer 7

Chapter 12, Problem 1P

(a)

To determine

Write the expression for the function given in Figure P12.1 (a) using step functions.

Answer 8

(a)

Expert Solution

Answer to Problem 1P

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10).

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Given data:

Refer to Figure P12.1 (a) in the textbook for the given function.

Calculation:

The function in Figure P12.1 (a) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−10s;10+t, −10s≥t≥0s10−t, 0s≤t≤10s0, t≥10s

Form a single continuous definition for the function using step function as follows:

10+t, on at t=−10, off at t=0;10−t, on at t=0, off at t=10.

Use the values and write the expression for f(t) as follows:

f(t)=(10+t)[u(t+10)−u(t)]+(10−t)[u(t)−u(t−10)]

Rearrange the expression as follows:

f(t)={10u(t+10)+tu(t+10)−10u(t)−tu(t)+10u(t)−tu(t)−10u(t−10)+tu(t−10)}=(t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

Conclusion:

The expression for the given function is (t+10)u(t+10)−2tu(t)+(t−10)u(t−10)

Answer 13

(b)

To determine

Write the expression for the function given in Figure P12.1 (b) using step functions.

(b)

Expert Solution

Answer to Problem 1P

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Explanation of Solution

Given data:

Refer to Figure P12.1 (b).in the textbook for the given function.

Calculation:

The function in Figure P12.1 (b) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−3s;−24−8t, −3s≤t≤−2s−8, −2s≤t≤−1s8t, −1s≤t≤1s8, 1s≤t≤2s−8t+24, 2s≤t≤3s0, t≥3s

Form a single continuous definition for the function using step function as follows:

−24−8t, on at t=−3, off at t=−2;−8, on at t=−2, off at t=−1;8t, on at t=−1, off at t=1;8, on at t=1, off at t=2;−8t+24,on at t=2, off at t=3;

Use the values and write the expression for f(t) as follows:

f(t)={(−24−8t)[u(t+3)−u(t+2)]−8[u(t+2)−u(t+1)]+8t[u(t+1)−u(t−1)]+8[u(t−1)−u(t−2)](24−8t)[u(t−2)−u(t−3)]}

Rearrange the expression as follows:

f(t)={−24u(t+3)−8tu(t+3)+24u(t+2)+8tu(t+2)−8u(t+2)+8u(t+1)+8tu(t+1)−8tu(t−1)+8u(t−1)−8u(t−2)+24u(t−2)−8tu(t−2)−24u(t−3)+8tu(t−3)}={−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}

Conclusion:

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Answer 14

(b)

To determine

Write the expression for the function given in Figure P12.1 (b) using step functions.

Answer 15

(b)

Expert Solution

Answer to Problem 1P

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Given data:

Refer to Figure P12.1 (b).in the textbook for the given function.

Calculation:

The function in Figure P12.1 (b) is a piecewise linear function

Write the piecewise-linear definition for the function as follows:

f(t)={0, t≤−3s;−24−8t, −3s≤t≤−2s−8, −2s≤t≤−1s8t, −1s≤t≤1s8, 1s≤t≤2s−8t+24, 2s≤t≤3s0, t≥3s

Form a single continuous definition for the function using step function as follows:

−24−8t, on at t=−3, off at t=−2;−8, on at t=−2, off at t=−1;8t, on at t=−1, off at t=1;8, on at t=1, off at t=2;−8t+24,on at t=2, off at t=3;

Use the values and write the expression for f(t) as follows:

f(t)={(−24−8t)[u(t+3)−u(t+2)]−8[u(t+2)−u(t+1)]+8t[u(t+1)−u(t−1)]+8[u(t−1)−u(t−2)](24−8t)[u(t−2)−u(t−3)]}

Rearrange the expression as follows:

f(t)={−24u(t+3)−8tu(t+3)+24u(t+2)+8tu(t+2)−8u(t+2)+8u(t+1)+8tu(t+1)−8tu(t−1)+8u(t−1)−8u(t−2)+24u(t−2)−8tu(t−2)−24u(t−3)+8tu(t−3)}={−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}

Conclusion:

The expression for the given function is {−8(t+3)u(t+3)+8(t+2)u(t+2)+8(t+1)u(t+1)−8(t−1)u(t−1)−8(t−2)u(t−2)+8(t−3)u(t−3)}.

Answer 20

Ch. 12.4 - Prob. 1AP Ch. 12.5 - Prob. 2AP Ch. 12.7 - Prob. 3AP Ch. 12.7 - Prob. 4AP Ch. 12.7 - Prob. 5AP Ch. 12.7 - Prob. 6AP Ch. 12.7 - Prob. 7AP Ch. 12.7 - Prob. 8AP Ch. 12.7 - Prob. 9AP Ch. 12.7 - Prob. 10AP

Write the expression for the function given in Figure P12.1 (a) using step functions.

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Write the expression for the function given in Figure P12.1 (a) using step functions.

Answer to Problem 1P

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Write the expression for the function given in Figure P12.1 (b) using step functions.

Answer to Problem 1P

Explanation of Solution

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