y = f(x) = x^2 - 6 x
(a) Find the average rate of change of y with respect to x in the following intervals.
from x = 2 to x = 3
from x = 2 to x = 2.5
from x = 2 to x = 2.1
(b) Find the (instantaneous) rate of change of y at x = 2.
to find the average slope between x = a and x = b
average slope = [Y(b)-Y(a)] / [b-a]
so for example from x = 2 to x = 3
average slope = [ -9 - (-8) ] / [3-2]
= -1
for part b you want dy/dx at x = 2
dy/dx = 2 x - 6
= -2
To find the average rate of change of y with respect to x, we can use the formula:
Average rate of change = (change in y) / (change in x)
(a)
For the interval from x = 2 to x = 3:
- Plug in x = 2 and x = 3 into the given equation to find the corresponding values of y:
y(2) = (2^2) - 6(2) = 4 - 12 = -8
y(3) = (3^2) - 6(3) = 9 - 18 = -9
- Calculate the change in y: -9 - (-8) = -9 + 8 = -1
- Calculate the change in x: 3 - 2 = 1
- Plug the values of change in y and change in x into the formula to find the average rate of change:
Average rate of change = (-1) / 1 = -1
For the interval from x = 2 to x = 2.5:
- Plug in x = 2 and x = 2.5 into the equation to find the corresponding values of y:
y(2) = (2^2) - 6(2) = 4 - 12 = -8
y(2.5) = (2.5^2) - 6(2.5) = 6.25 - 15 = -8.75
- Calculate the change in y: -8.75 - (-8) = -8.75 + 8 = -0.75
- Calculate the change in x: 2.5 - 2 = 0.5
- Plug the values of change in y and change in x into the formula to find the average rate of change:
Average rate of change = (-0.75) / 0.5 = -1.5
For the interval from x = 2 to x = 2.1:
- Plug in x = 2 and x = 2.1 into the equation to find the corresponding values of y:
y(2) = (2^2) - 6(2) = 4 - 12 = -8
y(2.1) = (2.1^2) - 6(2.1) = 4.41 - 12.6 = -8.19
- Calculate the change in y: -8.19 - (-8) = -8.19 + 8 = -0.19
- Calculate the change in x: 2.1 - 2 = 0.1
- Plug the values of change in y and change in x into the formula to find the average rate of change:
Average rate of change = (-0.19) / 0.1 = -1.9
(b)
To find the instantaneous rate of change of y at x = 2, we can use the concept of the derivative. In this case, we can calculate the derivative of the given equation f(x) = x^2 - 6x.
Taking the derivative of f(x) with respect to x, we get:
f'(x) = 2x - 6
Then, evaluate f'(2) to find the instantaneous rate of change at x = 2:
f'(2) = 2(2) - 6
= 4 - 6
= -2
Therefore, the instantaneous rate of change of y at x = 2 is -2.