Consider the function f ( x ) = 3x^3 − 3x on the interval [ − 4 , 4 ] . Find the average or mean slope of the function on this interval.
By the Mean Value Theorem, we know there exists at least one c in the open interval ( − 4 , 4 ) such that f ' ( c ) is equal to this mean slope. For this problem, there are two values of c that work.
The smaller one is
the larger one is
i just need the answer with an explanation please and thanks.
the mean slops is just (f(4)-f(-4))/(4+4)
Since f(x) is odd, we know that f(-4) = -f(4), so
the mean slope is just 2f(4)/8 = 2*180/8 = 45
Now, f'(x) = 9x^2-3, so you want c such that
9c^2 - 3 = 45
solve for c.
see the graphs at
https://www.wolframalpha.com/input/?i=plot+y%3D3x%5E3+%E2%88%92+3x%2C+y%3D45x+for+-4%3C%3Dx%3C%3D4
to check your answers.
First:
Consider the function f ( x ) = 3x^3 − 3x on the interval [ − 4 , 4 ] . Find the average or mean slope of the function on this interval.
f(4) = 3*64 - 12 = 180
f(-4) = 3*-64 +12 = -180
distance between = 4 - -4 = 8
so average slope = (180 - -180) / 8 = 360/8 = 45
Then with derivative
f'(x) = 9 x^2 - 3
where does 9 x^2-3 = 45 ?
9 x^2 = 48
x^2 = 5.33
x = + 2.31 or x = -2.31
To find the average or mean slope of a function on an interval, we can use the formula:
Mean slope = (f(b) - f(a))/(b - a)
where a and b are the endpoints of the interval. In this case, a = -4 and b = 4.
Step 1: Calculate f(-4):
f(-4) = 3*(-4)^3 - 3*(-4)
= -192
Step 2: Calculate f(4):
f(4) = 3*(4)^3 - 3*(4)
= 192
Step 3: Calculate the mean slope:
Mean slope = (f(4) - f(-4))/(4 - (-4))
= (192 - (-192))/(4 + 4)
= (192 + 192)/8
= 384/8
= 48
Therefore, the average or mean slope of the function on the interval [-4, 4] is 48.
To find the average or mean slope of the function on the interval [−4, 4], we need to calculate the difference in values of the function divided by the difference in x-values.
The formula for average slope is:
Average slope = (f(b) - f(a)) / (b - a)
where a and b are the endpoints of the interval.
In this case, a = -4 and b = 4. Let's calculate it step by step:
Step 1: Find f(a) and f(b)
Substitute x = -4 into the function to find f(-4):
f(-4) = 3(-4)^3 - 3(-4) = -192 + 12 = -180
Similarly, substitute x = 4 into the function to find f(4):
f(4) = 3(4)^3 - 3(4) = 192 - 12 = 180
Step 2: Calculate the difference in values of the function
f(b) - f(a) = 180 - (-180) = 360
Step 3: Calculate the difference in x-values
b - a = 4 - (-4) = 8
Step 4: Calculate the average slope
Average slope = (f(b) - f(a)) / (b - a) = 360/8 = 45
So, the average or mean slope of the function on the interval [-4, 4] is 45.
However, since you mentioned the Mean Value Theorem, it implies that there exists at least one c in the open interval (-4, 4) such that f'(c) is equal to this mean slope. To find the two values of c, we need to find the derivative of the function and set it equal to the mean slope.
The derivative of f(x) = 3x^3 - 3x can be found using the power rule:
f'(x) = 9x^2 - 3
Setting f'(c) = 45 and solving for c:
9c^2 - 3 = 45
9c^2 = 48
c^2 = 48/9
c^2 = 16/3
c = ±√(16/3)
So, the two values of c that work are c = √(16/3) and c = -√(16/3).