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Fofino [41]
3 years ago
15

Make a table, write an equation and graph: (2,1) (3,2) 4,3)

Mathematics
2 answers:
Sav [38]3 years ago
4 0
We can’t graph for you here, sorry
grigory [225]3 years ago
4 0
We can't graph here for you, sorry
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Need help with AP CAL
anzhelika [568]

Answer: Choice C

\displaystyle \frac{1}{2}\left(1 - \frac{1}{e^2}\right)

============================================================

Explanation:

The graph is shown below. The base of the 3D solid is the blue region. It spans from x = 0 to x = 1. It's also above the x axis, and below the curve y = e^{-x}

Think of the blue region as the floor of this weirdly shaped 3D room.

We're told that the cross sections are perpendicular to the x axis and each cross section is a square. The side length of each square is e^{-x} where 0 < x < 1

Let's compute the area of each general cross section.

\text{area} = (\text{side})^2\\\\\text{area} = (e^{-x})^2\\\\\text{area} = e^{-2x}\\\\

We'll be integrating infinitely many of these infinitely thin square slabs to find the volume of the 3D shape. Think of it like stacking concrete blocks together, except the blocks are side by side (instead of on top of each other). Or you can think of it like a row of square books of varying sizes. The books are very very thin.

This is what we want to compute

\displaystyle \int_{0}^{1}e^{-2x}dx\\\\

Apply a u-substitution

u = -2x

du/dx = -2

du = -2dx

dx = du/(-2)

dx = -0.5du

Also, don't forget to change the limits of integration

  • If x = 0, then u = -2x = -2(0) = 0
  • If x = 1, then u = -2x = -2(1) = -2

This means,

\displaystyle \int_{0}^{1}e^{-2x}dx = \int_{0}^{-2}e^{u}(-0.5du) = 0.5\int_{-2}^{0}e^{u}du\\\\\\

I used the rule that \displaystyle \int_{a}^{b}f(x)dx = -\int_{b}^{a}f(x)dx which says swapping the limits of integration will have us swap the sign out front.

--------

Furthermore,

\displaystyle 0.5\int_{-2}^{0}e^{u}du = \frac{1}{2}\left[e^u+C\right]_{-2}^{0}\\\\\\= \frac{1}{2}\left[(e^0+C)-(e^{-2}+C)\right]\\\\\\= \frac{1}{2}\left[1 - \frac{1}{e^2}\right]

In short,

\displaystyle \int_{0}^{1}e^{-2x}dx = \frac{1}{2}\left[1 - \frac{1}{e^2}\right]

This points us to choice C as the final answer.

5 0
2 years ago
Evaluate -6(x-2y) when x= -3 and y= -5<br> a. -42<br> b. 80<br> c. -80<br> d. -16
Vinil7 [7]
Simply substitute x for -3 and y for -5, creating the equation shown below:
     
     -6 (-3 - 2(-5) )
Now that you have your values set up, now you can do the math by starting off with Multiplying -2 by -5, which will get you 10. 

Now that you have the parenthesis simplified in the parenthesis, you can use the distributive property by multiplying all the numbers with -6, like so:
-6(-3+10)= 18 - 60, which simplifies to -42, in which (A) is your answer.
 
 
5 0
3 years ago
Read 2 more answers
Use the slider to rotate AABC about point P through different angles of rotation (a). As you move the slider, you'll see the ima
Hitman42 [59]

Answer:

Step-by-step explanation:

7 0
3 years ago
Find the absolute value of 2 and 2 3rd and negative 9 over 4
gayaneshka [121]

we are given

absolute value of 2 and 2 3rd and negative 9 over 4

Firstly , we need write it in terms of expression

2 and 2 3rd is

2+\frac{2}{3}

so, we get as

|2+\frac{2}{3} -\frac{9}{4} |

now, we can simplify it

Firstly , we will find common denominator

|\frac{2*12}{1*12} +\frac{2*4}{3*4} -\frac{9*3}{4*3} |

|\frac{24}{12} +\frac{8}{12} -\frac{27}{12} |

we can see that

all denominators are same

so, we can combine numerators

=|\frac{24+8-27}{12}  |

=|\frac{5}{12}  |

=\frac{5}{12}..............Answer

6 0
3 years ago
Which relation is a function?
zhuklara [117]

Answer: c

Step-by-step explanation:

4 0
3 years ago
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