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faust18 [17]
2 years ago
15

Section 8.1 Introduction to the Laplace Transforms

Mathematics
1 answer:
sukhopar [10]2 years ago
7 0

Presumably you've proven exercise 6, that the Laplace transform of t^k f(t) is (-1)^k F^{(k)}(s).

Let F(s) = 1/s, whose inverse Laplace transform is f(t) = 1. Differentiate F with respect to s :

F'(s) = -\dfrac1{s^2}

By the claim from ex.6, this is the Laplace transform of t • f(t) = t.

Differentiate F again with respect to s :

F''(s) = \dfrac2{s^3}

and this is the Laplace transform of t² • f(t) = t². And so on.

We can prove the general claim by induction. Assume it's true for n = k, that t^k \leftrightarrow \frac{k!}{s^{k+1}}. Then using the result of ex.6, we have

F(s) = \dfrac{k!}{s^{k+1}} \implies F'(s) = -\dfrac{(k+1)!}{s^{k+2}} \leftrightarrow t^{k+1}

QED

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Every weekday after tennis practice, Will buys either a carton of milk or a bottle
Gemiola [76]

Answer:

The correct options are;

The cost of milk powder is $3 per bottle

An equivalent expression for the total cost of Will's beverages during this month is 60 - m dollars

Step-by-step explanation:

The given information are;

The items Will buys every weekday after tennis practice = A carton of milk or a bottle of powder drink

The number of milk purchased during the month = m

Will's total cost for beverages in dollars last month = 2·m + 3·(20 - m)

1) Therefore, given that the total amount is factored in dollars, we have;

2·m and 3·(20 - m) are dollar amounts, while, m is a quantity;

Therefore;

Where;

2·m = The total cost of the cartons of milk, then the factor 2 represents $2.00 which is the cost per carton of milk

Similarly, 3·(20 - m) can be said to represent the total cost of the powder drink then the factor 3 represents $3.00 which is the cost per powder drink and the factor (20 - m), represents the number of powder drink Will buys for the month

From which we found the cost of milk powder = $3.00 per bottle

2) The expression, 2·m + 3·(20 - m) simplifies to 2·m + 60 - 3·m, which gives;

2·m + 60 - 3·m = 60 - m which is an equivalent expression for the total cost of Will's beverages for the month.

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3 years ago
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The answer and work is in the SS below

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Mitra created a mosaic design using square 2x2 cm white tiles and 2x3 cm rectangular centimeter black tiles. She used twice as m
zvonat [6]

Mitra used 156 white tiles of 2x2 cm and 76 black tiles of 2x3 cm to form a pattern with an area of 1092 cm²

<h3>What is an equation?</h3>

An equation is an expression that shows the relationship between two or more numbers and variables.

Let x represent the number of white tiles and y represent the number of black tiles.

Area of white tiles = 2 cm * 2 cm = 4 cm²

Area of black tiles = 2 cm * 3 cm = 6 cm²

She used twice as many white tiles as black tiles. Hence:

x = 2y   (1)

Also, the finished pattern was 1092 square centimeters in area. Hence:

4x + 6y = 1092   (2)

From both equations:

x = 156, y = 78

Mitra used 156 white tiles of 2x2 cm and 76 black tiles of 2x3 cm to form a pattern with an area of 1092 cm²

Find out more on equation at: brainly.com/question/2972832

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3 0
2 years ago
Lynn is making custom bricks. She combines 39 pounds of water and 48 pounds of brick dust
vova2212 [387]

Answer:

10, and 4.

Step-by-step explanation:

39+48=87

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olganol [36]

Answer:

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Step 2: Divide both sides by -5.

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3 years ago
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