Consider the function
f(x) = 2 x^3 - 12 x^2 - 30 x+ 10
on the interval [ -6 , 8 ]. Find the average or mean slope of the function on this interval.
Average slope =
By the Mean Value Theorem, we know there exists at least one c in the open interval ( -6 , 8 ) such that f'( c) is equal to this mean slope. Find all values of c that work and list them (separated by commas) in the box below.
List of numbers:
To find the average or mean slope of the function on the interval [-6, 8], we need to find the difference in the values of the function at the endpoints and divide it by the difference in the x-values.
Step 1: Find the value of f(x) at the endpoints -6 and 8.
f(-6) = 2(-6)^3 - 12(-6)^2 - 30(-6) + 10 = 334
f(8) = 2(8)^3 - 12(8)^2 - 30(8) + 10 = -478
Step 2: Find the difference in the values of the function.
Difference in y-values = f(8) - f(-6) = -478 - 334 = -812
Step 3: Find the difference in the x-values.
Difference in x-values = 8 - (-6) = 14
Step 4: Calculate the average slope.
Average slope = Difference in y-values / Difference in x-values = -812 / 14 = -58
Therefore, the average slope of the function on the interval [-6, 8] is -58.
According to the Mean Value Theorem, there exists at least one c in the open interval (-6, 8) such that f'(c) is equal to this mean slope. To find the values of c that work, we can solve the equation f'(c) = -58.
Step 5: Find the derivative of the function f(x).
f'(x) = d/dx(2x^3 - 12x^2 - 30x + 10) = 6x^2 - 24x - 30
Step 6: Set f'(c) equal to -58 and solve for c.
6c^2 - 24c - 30 = -58
6c^2 - 24c + 28 = 0
Step 7: Solve the quadratic equation.
Using the quadratic formula, we have:
c = (-(-24) ± √((-24)^2 - 4(6)(28))) / (2(6))
c = (24 ± √(576 - 672)) / 12
c = (24 ± √(-96)) / 12
Since the discriminant is negative, there are no real solutions. Therefore, there are no values of c in the open interval (-6, 8) such that f'(c) is equal to the mean slope of -58.
List of numbers: N/A
To find the average slope of the function f(x) = 2x^3 - 12x^2 - 30x + 10 on the interval [ -6 , 8 ], we need to calculate the difference in the function values at the endpoints of the interval and divide it by the difference in x-values.
Using the formula for average slope:
Average slope = (f(8) - f(-6)) / (8 - (-6))
First, let's calculate f(8):
f(8) = 2(8)^3 - 12(8)^2 - 30(8) + 10
= 1024 - 768 - 240 + 10
= 26
Next, let's calculate f(-6):
f(-6) = 2(-6)^3 - 12(-6)^2 - 30(-6) + 10
= -432 - 432 + 180 + 10
= -674
Now we can calculate the average slope:
Average slope = (26 - (-674)) / (8 - (-6))
= 700 / 14
= 50
So the average slope of the function on the interval [ -6 , 8 ] is 50.
Now, let's use the Mean Value Theorem to find the values of c for which f'(c) is equal to the mean slope. The Mean Value Theorem states that there exists at least one c in the open interval ( -6 , 8 ) such that f'(c) is equal to the mean slope (50 in this case).
To find these values of c, we need to find the derivative of the function f(x) and solve the resulting equation f'(x) = 50.
Taking the derivative of f(x):
f'(x) = 6x^2 - 24x - 30
Setting f'(x) equal to 50:
6x^2 - 24x - 30 = 50
Simplifying the equation:
6x^2 - 24x - 80 = 0
Now we need to solve this quadratic equation to find the values of x that satisfy it. We can use factoring, completing the square, or the quadratic formula to solve this equation.
Factoring the quadratic equation:
6(x^2 - 4x - 20/6) = 0
(x - 5)(x + 2/3) = 0
From this, we get two solutions:
x - 5 = 0 => x = 5
x + 2/3 = 0 => x = -2/3
Therefore, the values of c that satisfy f'(c) = 50 are c = 5 and c = -2/3.
Thus, the list of values of c is 5, -2/3.