The picture is below of how to separate this into 2 different regions, which you have to because it's not continuous over the whole function. It "breaks" at x = 2. So the way to separate this is to take the integral from x = 0 to x = 2 and then add it to the integral for x = 2 to x = 3. In order to integrate each one of those "parts" of that absolute value function we have to determine the equation for each line that makes up that part.

For the integral from [0, 2], the equation of the line is -3x + 6;

For the integral from [2, 3], the equation of the line is 3x - 6.

We integrate then:

$\int\limits^2_0 {-3x+6} \, dx+\int\limits^3_2 {3x-6} \, dx$ and

$-\frac{3x^2}{2}+6x\left \} {{2} \atop {0}} \right. +\frac{3x^2}{2}-6x\left \} {{3} \atop {2}} \right.$ sorry for the odd representation; that's as good as it gets here!

Using the First Fundamental Theorem of Calculus, we get:

(6 - 0) + (-4.5 - (-6)) = 6 + 1.5 = 7.5

5 0

2 years ago

Manuela is having mugs made for a fundraiser. The Cup Company will make them for $4 each plus a $30 set-up charge. Mugs Are Us w

amm1812

Answer:

The price will be the same for both companies at 60 mugs

Step-by-step explanation:

The form of the linear equation is y = m x + b, where

m is the rate of change

b is the fixed amount

Assume that the number of mugs is x and the total cost of them is $y

∵ The Cup Company will make them for $4 each plus a $30

set-up charge

∴ The cost per mug = $4

∴ m = 4

∴ The fixed amount = $30

∴ b = 30

∵ x is the number of mugs

∵ y is the cost of them

→ Substitute the values of m and b in the form of the equation

∴ y = 4x + 30 ⇒ (1)

∵ Mugs Are Us will make them for $4.50 each with no set-up charge

∴ The cost per mug = $4.50

∴ m = 4.5

∴ The fixed amount = 0

∴ b = 0

∵ x is the number of mugs

∵ y is the cost of them

→ Substitute the values of m and b in the form of the equation

∴ y = 4.5x + 0

∴ y= 4.5x ⇒ (2)

∵ The price be the same for both companies

→ Equate the right sides of (1) and (2)

∴ 4.5x = 4x + 30

→ Subtract 4x from both sides

∵ 4.5x - 4x = 4x - 4x + 30

∴ 0.5x = 30

→ Divide both sides by 0.5

∴ x = 60

∴ The price will be the same for both companies at 60 mugs

4 0

2 years ago

Please help, I am not able to find the answer

Anton [14]

Answer:

Assuming you are asked for the equation of the graph, the answer would be y=|x+2|-1.

Step-by-step explanation:

The parent function for the "V-shape" is y=|x|. This particular function is translated to the left two units and down one. This is shown by adding two to x within the absolute value and subtracting one from the absolute value of x+2, leaving you with y=|x+2|-1. If you plug this equation into desmos online graphing calculator, you will see that the graph is identical to the one given.

6 0

2 years ago

Factor the GCF: 21x3y4 + 15x2y2 − 12xy3.

-Dominant- [34]

Ugh, these questions.

21x^3y^4 + 15x^2y^2 - 12xy^3

3xy^2 (7x^2y^2 + 5x - 4y)

Clearing up clutter...

3xy² (7x²y² + 5x - 4y)

That's your answer. Thanks for working my brain. ;)

6 0

3 years ago

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