Consider the function f(x)=3x.
a. Determine an expression, in terms of a and h, for the average rate of change between the points (a,f(a)) and ( a + h, f(a+h) ) for the function . Show all steps needed to find a simplified algebraic expression.
b. Using your expression from (a), determine the average rate of change from x = -1 to x = 2. Express your answer as an exact value - no decimals!
This is the same type of question as your other previous ones.
Except it is the easiest of them all since f(x) = 3x is just a straight line
Just follow the same steps
I'm a bit confused with what steps I should take though
a. The average rate of change of a function between two points is defined as the ratio of the change in the output values to the change in the input values. In this case, we need to find the average rate of change between the points (a, f(a)) and (a + h, f(a + h)) for the function f(x) = 3x.
First, let's find the change in the output values. The output value at (a, f(a)) is f(a) = 3a, and the output value at (a + h, f(a + h)) is f(a + h) = 3(a + h). The change in the output values is f(a + h) - f(a) = 3(a + h) - 3a.
Next, let's find the change in the input values. The input value at (a, f(a)) is a, and the input value at (a + h, f(a + h)) is a + h. The change in the input values is (a + h) - a = h.
Finally, we can express the average rate of change between the two points as a simplified algebraic expression:
Average rate of change = (f(a + h) - f(a)) / (a + h - a)
Substituting the values we found:
Average rate of change = (3(a + h) - 3a) / h
Simplifying the expression:
Average rate of change = (3a + 3h - 3a) / h
= 3h / h
= 3
Therefore, the algebraic expression for the average rate of change between the points (a, f(a)) and (a + h, f(a + h)) for the function f(x) = 3x is simply 3.
b. To determine the average rate of change from x = -1 to x = 2, we can use the expression we derived in part (a), which is 3.
Substituting the values into the expression:
Average rate of change = 3
Therefore, the average rate of change from x = -1 to x = 2 is 3.