Question

In Exercises 93–96, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. $$ f(t)=t^{2}-7, \quad[3,3.1] $$

Name: Problem 94, Chapter 2: Calculus of a Single Variable
Uploaded: 2020-10-09T16:43:04-08:00
Duration: 1 min 19 s
Channel: Audrey Fong
Description: In Exercises 93–96, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(t)=t^2-7, [3,3.1]

   In Exercises 93–96, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints
of the interval.
$$
f(t)=t^{2}-7, \quad[3,3.1]
$$

Calculus of a Single Variable

Ron Larson, Bruce… 9th Edition

Chapter 2, Problem 94

Instant Answer

Solved by Expert Audrey Fong

10/09/2020

Step 1

The average rate of change is given by the formula: $$ \frac{f(b) - f(a)}{b - a} $$ where $a$ and $b$ are the endpoints of the interval. In this case, $a = 3$ and $b = 3.1$. Show more…

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In Exercises 93–96, find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. $$ f(t)=t^{2}-7, \quad[3,3.1] $$

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Key Concepts

Average Rate of Change

The average rate of change of a function over an interval is the ratio of the change in the function's value to the change in the independent variable over that interval. It provides a measure of the overall, or 'average', change between two points and is often interpreted as the slope of the secant line connecting them.

Instantaneous Rate of Change

The instantaneous rate of change at a specific point is the limit of the average rate of change as the interval shrinks to zero. It gives the slope of the tangent line to the function at that point and reflects how rapidly the function is changing at that exact moment.

Derivative

The derivative is a central concept in calculus that quantifies the instantaneous rate of change of a function with respect to its variable. It is computed as the limit of the average rate of change as the interval tends to zero, and it plays a key role in understanding and analyzing the behavior of functions.

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