What is the simplified average rate of change between x = 2 and x = 2 + h for the function:
(Enter your expression as you would enter an equation in Winplot)
a. ƒ(x) = x2?
ƒ(x) = f(x)=1x ?
1/x**
he average rate of change is the slope between the two points. For f(x) = x^2, that is
∆y/∆x = ((2+h)^2-2^2)/(2+h - 2) = (4+4h+h^2-4)/h = (4h+h^2)/h = 4+h
do likewise for the other one
To calculate the simplified average rate of change between x = 2 and x = 2 + h for the function ƒ(x) = x^2, we can use the formula:
Average rate of change = (ƒ(2 + h) - ƒ(2)) / h
Substituting the function into the formula, we have:
Average rate of change = ( (2 + h)^2 - 2^2 ) / h
Simplifying further:
ƒ(2 + h) = (2 + h)^2 = 4 + 4h + h^2
ƒ(2) = 2^2 = 4
Thus, the simplified average rate of change between x = 2 and x = 2 + h for the function ƒ(x) = x^2 is:
Average rate of change = (4 + 4h + h^2 - 4) / h
= (4h + h^2) / h
= 4 + h
To find the average rate of change of a function, we need to find the difference in the function values between two points divided by the difference in the x-values of those points.
For the function ƒ(x) = x^2, to find the simplified average rate of change between x = 2 and x = 2 + h, we can substitute these values into the function and simplify:
Step 1: Find the function value at x = 2 and x = 2 + h.
ƒ(2) = (2)^2 = 4
ƒ(2 + h) = (2 + h)^2 = 4 + 4h + h^2
Step 2: Find the difference in the function values.
Difference = ƒ(2 + h) - ƒ(2) = (4 + 4h + h^2) - 4 = 4h + h^2
Step 3: Find the difference in the x-values.
Difference in x-values = (2 + h) - 2 = h
Step 4: Calculate the average rate of change.
Average rate of change = Difference in the function values / Difference in x-values
Average rate of change = (4h + h^2) / h
Now that we have the expression for the average rate of change, we can simplify it further:
Average rate of change = (4h + h^2) / h
Average rate of change = 4 + h
The simplified average rate of change between x = 2 and x = 2 + h for the function ƒ(x) = x^2 is 4 + h.
For the second function ƒ(x) = 1/x, we can follow the same steps to find the average rate of change between x = 2 and x = 2 + h:
Step 1: Find the function value at x = 2 and x = 2 + h.
ƒ(2) = 1/2 = 0.5
ƒ(2 + h) = 1/(2 + h)
Step 2: Find the difference in the function values.
Difference = ƒ(2 + h) - ƒ(2) = 1/(2 + h) - 0.5 = 1/(2 + h) - (1/2)(2 + h)/(2 + h) = (2 + h - 2h - h^2) / (2 + h)
Step 3: Find the difference in the x-values.
Difference in x-values = (2 + h) - 2 = h
Step 4: Calculate the average rate of change.
Average rate of change = Difference in the function values / Difference in x-values
Average rate of change = (2 + h - 2h - h^2) / h
The simplified average rate of change between x = 2 and x = 2 + h for the function ƒ(x) = 1/x is (2 + h - 2h - h^2) / h.