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Svetlanka [38]
3 years ago
12

Evaluate the integral (3 - 2x)dx for x = -1 to x = 3 byinterpreting it in terms of areas.

Mathematics
1 answer:
suter [353]3 years ago
5 0

Answer:

4 unit^2

explanation is given at the end.

Step-by-step explanation:

What this integral represents is the net area between the function f(x) = 3 - 2x

and the x-axis, between the range of x between -1 and 3.

\int^{3}_{-1} {3-2x} \, dx \\\left|3x - 2\dfrac{x^2}{2}\right|^{3}_{-1}\\\left|3x - x^2\right|^{3}_{-1}

we have integrated the equation, and now we're going to put the limits find the area under the function f(x)

\left|3x - x^2\right|^{3}_{-1}\\(3(3)-(3)^2)-(3(-1)-(-1)^2)

-------------

if these seems like a big jump, try to understand it through this:

\left|3x - x^2\right|^{b}_a\\(3(b)-(b)^2)-(3(a)-(a)^2)\\

we've only distributed the limits each time to the same integrated expression.

--------------

coming back to our solution:

(3(3)-(3)^2)-(3(-1)-(-1)^2)

(0)-(-4)

4

so our area is 4

the area above the x-axis is positive, and the area below the x-axis is negative. Since, our answer is +4. <u>We now know that if within this range [-1,3] the area above and below the x-axis exist, there is more area above the x-axis than below the x-axis</u><u>. In other words, the net area is above the x-axis and that is equal to 4 unit^2 </u>

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Answer:

<u>Question 1</u>

The function f(x) = x + 6 is one-to-one, so it does have an inverse.

The inverse of +6 is -6, so f^{-1}(x)=x-6

Therefore, g(x) is the inverse of f(x).

<u>Question 2</u>

The function f(x) = -3x -9 is one-to-one, so it does have an inverse.

To find the inverse, replace f(x) with y:

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Rearrange the equation to make x the subject:

\implies y+9= -3x

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Replace x with f^{-1}(x) and y with x:

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3 years ago
Solve for x. 7 + 11x + 60 = -13 + 3x
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Answer:

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Step-by-step explanation:

Step 1: Write equation

7 + 11x + 60 = -13 + 3x

Step 2: Solve for <em>x</em>

  1. Combine like terms: 11x + 67 = -13 + 3x
  2. Subtract 3x on both sides: 8x + 67 = -13
  3. Subtract 67 on both sides: 8x = -80
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Step 3: Check

<em>Plug in x to verify it's a solution.</em>

7 + 11(-10) + 60 = -13 + 3(-10)

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