Question

T/F: Let \(f\) be a position function. The average rate of change on \([a, b]\) is the slope of the line through the points \((a, f(a))\) and \((b, f(b))\).

Short answer

True.

Step-by-step solution

Understand the Average Rate of Change

The average rate of change of a function between two points is calculated as the change in the function's output divided by the change in input. It is similar to finding the slope of a secant line that passes through two given points on the graph of the function.

Calculate the Slope of the Line

Given two points on the graph of the function, \((a, f(a))\) and \((b, f(b))\), the slope (m) of the line through these points can be expressed as: \[ m = \frac{f(b) - f(a)}{b - a} \] This formula matches the formula for the average rate of change on \([a, b]\).

Relate the Concepts

Since the formula for average rate of change is identical to the formula for the slope of the line through these two points, the statement that the average rate of change on \([a, b]\) is the slope of the line through the points \((a, f(a))\) and \((b, f(b))\) is true.

Key concepts

Slope of a Secant Line

The slope of a secant line is a foundational concept when dealing with graphs of functions. Essentially, the secant line cuts through a curve at two distinct points. The slope is a measure of how steep the line is between these two points. It is calculated as the change in the vertical direction (known as the "rise") over the change in the horizontal direction (known as the "run").

Mathematically, if you have two points \( (a, f(a)), (b, f(b)) \), the slope \( m \) is calculated using the formula:
\[ m = \frac{f(b) - f(a)}{b - a} \]
Where:

• \( f(b) \) is the function value at point b
• \( f(a) \) is the function value at point a
• \( b - a \) is the difference between the x-values

This slope effectively represents the average rate of change of the function over the interval \([a, b] \). By understanding this, we can analyze how a function behaves over different intervals, bridging connections to calculus concepts like derivatives, which involve limits as intervals approach zero.

Position Function

A position function is a great way to understand movement over time. It's a type of function that gives an output for the position of an object at a time \( t \).

These functions can describe:

• Where an object is located in space at a particular time
• The path along which an object moves
• How far an object has traveled from a starting point

For instance, suppose you are analyzing the position of a car along a straight road. With a position function \( f(t) \), you can represent where the car is at any time \( t \)

By knowing how to read and calculate values from a position function, students can predict and describe motion in physics and applications requiring dynamic models.

Calculus Concepts

Calculus is an integral part of understanding dynamic changes in systems. It delves into how quantities change and provides tools to compute these changes. Some core calculus concepts include:

• Limits: Critical for defining derivatives and integrals, limits help understand behavior as inputs approach a specific value.
• Derivatives: Often seen as the instant rate of change, derivatives are foundational for calculating velocities and other real-world "instantaneous" phenomena.
• Integrals: These provide tools for calculating areas under curves, total accumulated quantities, and have applications in areas like physics and engineering.

The average rate of change offers a simplistic introduction to these deeper concepts. For instance, by calculating the slope of a secant line, we inch closer to understanding instantaneous rates of change, a principal calculus topic. Mastery of this paves the way to unraveling more intricate calculus principles and their applications across science and engineering.