Let f be defined as follows.
y = f(x) = x^2 - 4 x
(a) Find the average rate of change of y with respect to x in the following intervals.
from x = 5 to x = 6
from x = 5 to x = 5.5
from x = 5 to x = 5.1
(b) Find the (instantaneous) rate of change of y at x = 5.
4
To find the average rate of change of y with respect to x in different intervals, we need to find the difference in y-values and divide by the difference in x-values.
(a)
1. From x=5 to x=6:
- Calculate the y-values at x=5 and x=6:
y(5) = (5)^2 - 4(5) = 25 - 20 = 5
y(6) = (6)^2 - 4(6) = 36 - 24 = 12
- Calculate the difference in y-values: 12 - 5 = 7
- Calculate the difference in x-values: 6 - 5 = 1
- Average rate of change = (change in y) / (change in x) = 7 / 1 = 7
2. From x=5 to x=5.5:
- Calculate the y-values at x=5 and x=5.5:
y(5) = (5)^2 - 4(5) = 25 - 20 = 5
y(5.5) = (5.5)^2 - 4(5.5) = 30.25 - 22 = 8.25
- Calculate the difference in y-values: 8.25 - 5 = 3.25
- Calculate the difference in x-values: 5.5 - 5 = 0.5
- Average rate of change = (change in y) / (change in x) = 3.25 / 0.5 = 6.5
3. From x=5 to x=5.1:
- Calculate the y-values at x=5 and x=5.1:
y(5) = (5)^2 - 4(5) = 25 - 20 = 5
y(5.1) = (5.1)^2 - 4(5.1) = 26.01 - 20.4 = 5.61
- Calculate the difference in y-values: 5.61 - 5 = 0.61
- Calculate the difference in x-values: 5.1 - 5 = 0.1
- Average rate of change = (change in y) / (change in x) = 0.61 / 0.1 = 6.1
(b) To find the instantaneous rate of change of y at x=5, we need to find the derivative of f(x) with respect to x and substitute x=5 into the derivative.
- First, find the derivative of f(x) using the power rule for differentiation:
f'(x) = d/dx (x^2 - 4x)
= 2x - 4
- Substitute x=5 into the derivative:
f'(5) = 2(5) - 4
= 10 - 4
= 6
Therefore, the instantaneous rate of change of y at x=5 is 6.
To find the average rate of change of y with respect to x in a given interval, we can use the formula:
Average rate of change = (change in y) / (change in x)
(a) From x = 5 to x = 6:
To find the change in y and change in x, substitute the given x values into the function and calculate the corresponding y values.
Let's calculate the change in y first:
f(x = 6) = (6)^2 - 4(6) = 36 - 24 = 12
f(x = 5) = (5)^2 - 4(5) = 25 - 20 = 5
Change in y = 12 - 5 = 7
Now, let's calculate the change in x:
x = 6 - x = 5 = 1
Change in x = 1
Average rate of change = (change in y) / (change in x) = 7 / 1 = 7
From x = 5 to x = 6, the average rate of change of y with respect to x is 7.
Now, let's calculate the average rate of change for the other intervals.
From x = 5 to x = 5.5:
Change in y:
f(x = 5.5) = (5.5)^2 - 4(5.5) = 30.25 - 22 = 8.25
f(x = 5) = (5)^2 - 4(5) = 25 - 20 = 5
Change in y = 8.25 - 5 = 3.25
Change in x: x = 5.5 - 5 = 0.5
Average rate of change = (change in y) / (change in x) = 3.25 / 0.5 = 6.5
From x = 5 to x = 5.5, the average rate of change of y with respect to x is 6.5.
From x = 5 to x = 5.1:
Change in y:
f(x = 5.1) = (5.1)^2 - 4(5.1) = 26.01 - 20.4 = 5.61
f(x = 5) = (5)^2 - 4(5) = 25 - 20 = 5
Change in y = 5.61 - 5 = 0.61
Change in x: x = 5.1 - 5 = 0.1
Average rate of change = (change in y) / (change in x) = 0.61 / 0.1 = 6.1
From x = 5 to x = 5.1, the average rate of change of y with respect to x is 6.1.
(b) To find the instantaneous rate of change at x = 5, we can compute the derivative of the function f(x) with respect to x.
(a)
Average rate of change of f(x) on an interval I = [x1,x2] equals (f(x2)-f(x1))/(x2-x1) where x2≠x1.
So if f(x)=x^2 - 4 x,
average rate of change between 5 and 6
= (f(6)-f(5))/(6-2)
= (12-5)/(6-5)
= 7
(b)
Instantaneous rate of change at x=5
=f'(5)
where
f'(x)=2x-4
so
f'(5)=10-4=6