The average value of the given function is
.
According to the statement
we have given that the function f(x) and we have to find the average value of that function.
So, For this purpose, we know that the
The given function f(x) is
![f(x) = -x^{2} + 2x +1](https://tex.z-dn.net/?f=f%28x%29%20%3D%20-x%5E%7B2%7D%20%2B%202x%20%2B1)
And now integrate this function with the limit 0 to a then
![f_{avg} = \frac{1}{b - a} \int\limits^a_0 {f(x)} \, dx = -x^{2} + 2x +1](https://tex.z-dn.net/?f=f_%7Bavg%7D%20%20%3D%20%5Cfrac%7B1%7D%7Bb%20-%20a%7D%20%5Cint%5Climits%5Ea_0%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%20%3D%20-x%5E%7B2%7D%20%2B%202x%20%2B1)
Now integrate this then
![f_{avg} = \frac{1}{a} \int\limits^a_0 {-x^{2} + 2x +1} \, dx](https://tex.z-dn.net/?f=f_%7Bavg%7D%20%20%3D%20%5Cfrac%7B1%7D%7Ba%7D%20%5Cint%5Climits%5Ea_0%20%7B-x%5E%7B2%7D%20%2B%202x%20%2B1%7D%20%5C%2C%20dx)
Then the value becomes according to the integration rules is:
![f_{avg} = \frac{1}{a} \int\limits^a_0 {-\frac{x^{3} }{3} + \frac{2x^{2} }{2} +x} \,](https://tex.z-dn.net/?f=f_%7Bavg%7D%20%20%3D%20%5Cfrac%7B1%7D%7Ba%7D%20%5Cint%5Climits%5Ea_0%20%7B-%5Cfrac%7Bx%5E%7B3%7D%20%7D%7B3%7D%20%2B%20%5Cfrac%7B2x%5E%7B2%7D%20%7D%7B2%7D%20%20%2Bx%7D%20%5C%2C)
Now put the limits then answer will become as output is:
![f_{avg} = \frac{1}{a} [ {-\frac{a^{3} }{3} + \frac{2a^{2} }{2} +a} \,]](https://tex.z-dn.net/?f=f_%7Bavg%7D%20%20%3D%20%5Cfrac%7B1%7D%7Ba%7D%20%5B%20%7B-%5Cfrac%7Ba%5E%7B3%7D%20%7D%7B3%7D%20%2B%20%5Cfrac%7B2a%5E%7B2%7D%20%7D%7B2%7D%20%20%2Ba%7D%20%5C%2C%5D)
Now solve this equation then
![f_{avg} = [ {-\frac{a^{2} }{3} + \frac{2a }{2} +1} \,]](https://tex.z-dn.net/?f=f_%7Bavg%7D%20%20%3D%20%20%5B%20%7B-%5Cfrac%7Ba%5E%7B2%7D%20%7D%7B3%7D%20%2B%20%5Cfrac%7B2a%20%7D%7B2%7D%20%20%2B1%7D%20%5C%2C%5D)
Now
![f_{avg} = \frac{-2a^{2} + 6a + 6 }{6}](https://tex.z-dn.net/?f=f_%7Bavg%7D%20%20%3D%20%20%5Cfrac%7B-2a%5E%7B2%7D%20%2B%206a%20%2B%206%20%7D%7B6%7D)
This is the value which represent the average of the given function in the statement.
So, The average value of the given function is
.
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Answer:
(2, -3), (3, -3)
Step-by-step explanation:
y = x^2 - 5x + 3
y = -3
x^2 - 5x + 3 = -3
x^2 - 5x + 6 = 0
(x - 3)(x - 2) = 0
x - 3 = 0 or x - 2 = 0
x = 3 or x = 2
We are given y = -3.
Answer: (2, -3), (3, -3)
Answer:
the dimensions of the most economical shed are height = 10 ft and side 5 ft
Step-by-step explanation:
Given data
volume = 250 cubic feet
base costs = $4 per square foot
material for the roof costs = $6 per square foot
material for the sides costs = $2.50 per square foot
to find out
the dimensions of the most economical shed
solution
let us consider length of side x and height is h
so we can say x²h = 250
and h = 250 / x²
now cost of material = cost of base + cost top + cost 4 side
cost = x²(4) + x²(6) + 4xh (2.5)
cost = 10 x² + 10xh
put here h = 250 / x²
cost = 10 x² + 10x (250/ x² )
cost = 10 x² + (2500/ x )
differentiate and we get
c' = 20 x - 2500 / x²
put c' = 0 solve x
20 x - 2500 / x² = 0
x = 5
so we say one side is 5 ft base
and height is h = 250 / x²
h = 250 / 5²
height = 10 ft
Answer:
a translation moves a thing up and down or left and right.
Step-by-step explanation:
so true?
Answer:
-4
Step-by-step explanation:
Ok first we can factor out 3x and get
![f(x)=3(x^3+4x^2+x+6)](https://tex.z-dn.net/?f=f%28x%29%3D3%28x%5E3%2B4x%5E2%2Bx%2B6%29)
then factor it
![f(x)=3(x^2(x-1)+5x(x-1)+6(x-1))\\f(x) = 3(x-1)(x^2+5x+6)\\f(x) = 3(x-1)(x+3)(x+2)\\x = 1, -3, -2](https://tex.z-dn.net/?f=f%28x%29%3D3%28x%5E2%28x-1%29%2B5x%28x-1%29%2B6%28x-1%29%29%5C%5Cf%28x%29%20%3D%203%28x-1%29%28x%5E2%2B5x%2B6%29%5C%5Cf%28x%29%20%3D%203%28x-1%29%28x%2B3%29%28x%2B2%29%5C%5Cx%20%3D%201%2C%20-3%2C%20-2)
add em together, we get -4