Consider the function f(x)=2sqrtx+4 on the interval [2,8]. Find the average or mean slope of the function on this interval. ?
By the Mean Value Theorem, we know there exists a c in the open interval (2,8) such that f'(c) is equal to this mean slope. For this problem, there is only one c that works. Find it. ?
y = 2√x + 4
mean slope is just
∆y/∆x = ((2√8+4)-(2√2+4))/(8-2) = √2/3
So, we want to find c where y'(c) = √2/3
y'(x) = 1/√x
1/√c = √2/3
c = 9/2, which is in (2,8)
If you meant f(x) = 2√(x+4) then adjust the algebra accordingly.
To find the average or mean slope of the function on the interval [2,8], we need to calculate the difference in the function values at the endpoints of the interval divided by the difference in the input values.
Let's begin by calculating the function values at the endpoints of the interval [2,8]:
f(2) = 2 * sqrt(2) + 4
= 2 * 1.414 + 4
= 2.828 + 4
≈ 6.828
f(8) = 2 * sqrt(8) + 4
= 2 * 2.828 + 4
= 5.656 + 4
= 9.656
Now, let's calculate the difference in the function values:
Δy = f(8) - f(2)
= 9.656 - 6.828
≈ 2.828
Next, we calculate the difference in the input values:
Δx = 8 - 2
= 6
Now, we can calculate the average or mean slope:
m = Δy/Δx
= 2.828/6
≈ 0.4713
According to the Mean Value Theorem, there exists a value c in the open interval (2,8) such that f'(c) is equal to this mean slope. To find this value, we need to find the derivative of the function f(x) and then solve for c.
The derivative of f(x) = 2 * sqrt(x) + 4 is given by:
f'(x) = 1/sqrt(x)
Now, we can set f'(c) = m and solve for c:
1/sqrt(c) = 0.4713
To solve this equation, we can isolate the square root term:
sqrt(c) = 1/0.4713
c = (1/0.4713)^2
Using a calculator, we find:
c ≈ 4.271
Therefore, the value of c that satisfies f'(c) = m is approximately 4.271.
To find the average or mean slope of the function on the interval [2,8], we need to calculate the difference in the function values at the endpoints of the interval and divide it by the difference in the x-coordinates of the endpoints.
First, let's find the value of f(x) at the endpoints of the interval:
- f(2) = 2√2 + 4
- f(8) = 2√8 + 4
Now, let's calculate the difference in the function values: f(8) - f(2)
= (2√8 + 4) - (2√2 + 4)
Next, let's calculate the difference in the x-coordinates: 8 - 2
Finally, divide the difference in the function values by the difference in the x-coordinates to get the average or mean slope:
average slope = (f(8) - f(2)) / (8 - 2)
Evaluate this expression to find the average slope.
To find the value of c that satisfies the Mean Value Theorem, we need to find the derivative of the function f(x) = 2√x + 4.
Start by finding the derivative. The derivative of √x is (1/2) * (√x)^(-1/2). Since we have 2√x, the derivative of 2√x is 2 * (1/2) * (√x)^(-1/2) = (√x)^(-1/2) = 1 / (√x).
So, the derivative of f(x) = 2√x + 4 is f'(x) = 1 / (√x).
Now, we know that by the Mean Value Theorem, there exists a c in the open interval (2,8) such that f'(c) is equal to the mean slope.
Evaluate f'(x) = 1 / (√x) at c to find the exact value of c that satisfies f'(c) = average slope.
Evaluate the derivative at different points within the interval (2,8) until you find a value for c that satisfies f'(c) = average slope.