Let f(x)=5x2+5x−12. Answer the following questions.Find the average slope of the function f on the interval [−1,1].Verify the Mean Value Theorem by finding a number c in (−1,1) such that f′(c)=m¯¯¯.
To find the average slope of the function f on the interval [-1,1], we need to calculate the slope between the two endpoints of the interval.
Step 1: Calculate the value of f(x) at the endpoints of the interval.
- Calculate f(-1):
f(-1) = 5(-1)^2 + 5(-1) - 12
= 5 + (-5) - 12
= -12
- Calculate f(1):
f(1) = 5(1)^2 + 5(1) - 12
= 5 + 5 - 12
= -2
Step 2: Calculate the difference in the y-values and the difference in the x-values:
- Difference in y-values: f(1) - f(-1) = -2 - (-12) = 10
- Difference in x-values: 1 - (-1) = 2
Step 3: Calculate the average slope:
- Average slope = (difference in y-values) / (difference in x-values) = 10 / 2 = 5
Therefore, the average slope of the function f on the interval [-1,1] is 5.
To verify the Mean Value Theorem, we need to find a number c in the interval (-1,1) such that f'(c) is equal to the average slope we calculated above, which is 5.
Step 1: Calculate the derivative of f(x):
- f'(x) = d/dx (5x^2 + 5x - 12)
= 10x + 5
Step 2: Set f'(x) equal to the average slope and solve for x:
- 10x + 5 = 5
- 10x = 0
- x = 0
Therefore, the number c in the interval (-1,1) such that f'(c) is equal to the average slope of 5 is c = 0.
To find the average slope of the function f on the interval [-1,1], we need to calculate the difference in the function values divided by the difference in the x-values.
Step 1: Finding f(x) at the endpoints of the interval:
Let's calculate f(-1) and f(1):
f(-1) = 5(-1)^2 + 5(-1) - 12
= 5 + (-5) - 12
= -12
f(1) = 5(1)^2 + 5(1) - 12
= 5 + 5 - 12
= -2
Step 2: Finding the difference in function values and x-values:
The difference in function values is: f(1) - f(-1) = -2 - (-12) = -2 + 12 = 10
The difference in x-values is: 1 - (-1) = 1 + 1 = 2
Step 3: Calculating the average slope:
Average slope = (difference in function values) / (difference in x-values)
= 10 / 2
= 5
Therefore, the average slope of the function f on the interval [-1,1] is 5.
To verify the Mean Value Theorem, we need to find a number c in the interval (-1,1) such that f'(c) = m¯¯¯.
Step 1: Finding f'(x):
To find f'(x), we need to differentiate f(x) with respect to x.
f(x) = 5x^2 + 5x - 12
Taking the derivative:
f'(x) = 10x + 5
Step 2: Substituting the values from the previous step:
We need to find a number c such that f'(c) = 5.
So, f'(c) = 10c + 5 = 5
Step 3: Solving for c:
10c + 5 = 5
10c = 0
c = 0
Therefore, the number c in the interval (-1,1) such that f'(c) = 5 is c = 0.
Hence, the Mean Value Theorem is verified for the function f on the interval [-1,1] with c = 0.
the average slope is the slope of the line joining the two endpoints on the graph. Once you find that, the MVT says there is a c in the interval such that
f'(c) = that slope.
So, find that slope (call it m) and then just solve
f'(c) = 10c+5 = m