Problem 36
Question
What is the average rate of change of a function?
Step-by-Step Solution
Verified Answer
The average rate of change of a function is the ratio of the change in the function value to the change in the independent variable. It is calculated as \[ \frac{f(b) - f(a)}{b - a} \] over any interval [a, b] on the function's domain.
1Step 1: Understanding the Concept
The average rate of change is calculated as follows: \[ \text{Average Rate of Change} = \frac{\text{change in } y}{\text{change in } x} \] This refers to the change in the value of the function (y) over the change in the independent variable (x).
2Step 2: Choosing an Interval
The interval chosen must be clear. In most cases this will be given in the problem, for example from \(x = a\) to \(x = b\). If no specific interval is given we are referring to the average rate of change of the function over its entire domain.
3Step 3: Applying the Formula
If we have the function values at \(x = a\) and \(x = b\) as \(f(a)\) and \(f(b)\) respectively, we substitute these into our formula, the average rate of change becomes \[ \frac{f(b) - f(a)}{b - a} \].
Other exercises in this chapter
Problem 36
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View solution Problem 36
Find \(f+g, f-g,\) fg, and \(\frac{f}{g} .\) Determine the domain for each function. $$f(x)=6 x^{2}-x-1, g(x)=x-1$$
View solution Problem 36
Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((1,-3)\) with \(x\) -intercept \(=-1
View solution Problem 37
write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(-3,-1), r=\sqrt{3} $$
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