Consider the function f(x)=4sqrtx+5 on the interval [4,5]. 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 (4,5) such that f'(c) is equal to this mean slope. For this problem, there is only one c that works. Find it.
1) For the average value of the SLOPE of f(x) in that interval, calculate Integral of f'(x) from 4 to 5. You have to divide by the interval to get the average, but that is 1.
f'(x) = 2/sqrtx
Average value = INTEGRAL 2 x^-1/2 dx
from 4 to 5
= 4 (sqrt 5 - sqrt 4) = 0.9443
2) Now that you have the average value of f'(x), find the value of c such that f'(c) = f'(x)(average)
2/sqrtc = 0.9443
sqrt c = 2.118
c = 4.486
To find the average or mean slope of the function f(x) on the interval [4,5], we need to calculate the difference between the values of f(x) at the endpoints of the interval and divide it by the difference between the x-values.
First, let's find the value of f(x) at the endpoints:
f(4) = 4*sqrt(4) + 5 = 4*2 + 5 = 8 + 5 = 13
f(5) = 4*sqrt(5) + 5 = 4*sqrt(5) + 5 (keep it in this form)
Next, let's calculate the difference between f(5) and f(4):
f(5) - f(4) = (4*sqrt(5) + 5) - 13 = 4*sqrt(5) + 5 - 13 = 4*sqrt(5) - 8
Now, let's calculate the difference between the x-values:
5 - 4 = 1
Finally, let's divide the difference in y-values by the difference in x-values to get the average slope:
Average slope = (4*sqrt(5) - 8) / 1 = 4*sqrt(5) - 8
According to the Mean Value Theorem, there exists a c in the open interval (4,5) such that f'(c) is equal to this mean slope. To find this c, we need to find the derivative of f(x) and then solve for c.
Let's find the derivative of f(x):
f'(x) = d/dx (4*sqrt(x) + 5) = 4*(1/2)*sqrt(x) + 0 = 2*sqrt(x)
Now, let's set f'(c) equal to the mean slope we found:
2*sqrt(c) = 4*sqrt(5) - 8
To solve for c, we isolate the square root term:
sqrt(c) = 2*sqrt(5) - 4
Squaring both sides to eliminate the square root:
c = (2*sqrt(5) - 4)^2
Simplifying the right side:
c = (4*5 - 16*sqrt(5) + 16)
Further simplification:
c = 20 - 16*sqrt(5) + 16
Therefore, the value of c is 20 - 16*sqrt(5) + 16, which can be simplified to 36 - 16*sqrt(5).