Find the average rate of change of f(x)=-3x^2+1:
a) From 1 to 3____
b) From 3 to 5____
c) From 1 to 4______
(a) [f(3) - f(1)]/(3-1)
= (28-4)/2 = 12
(b) [f(5) -f(3)]/(5-3)
= (76-28)/2 = 24
(c) [f(4) - f(1)]/(4-1)
= (your turn)
To find the average rate of change of a function, we need to calculate the difference in the function values between two given points and divide it by the difference in the x-values of those points. Let's calculate the average rate of change for each interval:
a) From 1 to 3:
To find the average rate of change from x = 1 to x = 3, we need to find the difference in the function values at these points and divide it by the difference in x-values.
- Calculate the function values at x = 1 and x = 3:
f(1) = -3(1)^2 + 1 = -2
f(3) = -3(3)^2 + 1 = -26
- Calculate the difference in the function values:
-26 - (-2) = -26 + 2 = -24
- Calculate the difference in the x-values:
3 - 1 = 2
- Calculate the average rate of change:
Average rate of change = (-24)/(2) = -12
Therefore, the average rate of change of f(x) from 1 to 3 is -12.
b) From 3 to 5:
Using the same steps as above, let's find the average rate of change from x = 3 to x = 5.
- Calculate the function values at x = 3 and x = 5:
f(3) = -3(3)^2 + 1 = -26
f(5) = -3(5)^2 + 1 = -74
- Calculate the difference in the function values:
-74 - (-26) = -74 + 26 = -48
- Calculate the difference in the x-values:
5 - 3 = 2
- Calculate the average rate of change:
Average rate of change = (-48)/(2) = -24
Therefore, the average rate of change of f(x) from 3 to 5 is -24.
c) From 1 to 4:
Let's calculate the average rate of change from x = 1 to x = 4.
- Calculate the function values at x = 1 and x = 4:
f(1) = -3(1)^2 + 1 = -2
f(4) = -3(4)^2 + 1 = -47
- Calculate the difference in the function values:
-47 - (-2) = -47 + 2 = -45
- Calculate the difference in the x-values:
4 - 1 = 3
- Calculate the average rate of change:
Average rate of change = (-45)/(3) = -15
Therefore, the average rate of change of f(x) from 1 to 4 is -15.