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Introduction

We are often interested in computing how fast a quantity is changing. We may be interested in the average rate of change on a given interval, or the instantaneous rate of change at a specified point.

Example: for a moving object with position $s(t)$, the average velocity on the interval from $[t_i, t_f]$ is given by:

$$v_a = \frac{s(t_f) - s(t_i)}{t_f - t_i}$$

Without loss of generality, we can write the average rate of change of a function $f(x)$ in the interval $[a, a+h]$ as:

$$v_a = \frac{f(a+h) - f(a)}{h}$$

How is the average rate of change of a function connected to its graph? Take a look at the graph below:

Construct a right triangle in which the hypotenuse connects the points $a$ and $b$. What are the lengths of the legs of this triangle?

$\textrm{base} = $ $\textrm{height} = $

What is the slope of the line connecting points $a$ and $b$?

$\textrm{slope} = $

Use the expression above for $v_a$ to compute the average rate of change for the function $f(x)$ in the interval between points $a$ and $b$.

$v_a = $

Based on your answers above, how can we relate the average rate of change with a geometric interpretation of the function graph?

Note: this example question includes the introduction of a concept with corresponding short quiz questions. Read more here.

Correct answer

$\textrm{base} = $ 4 $\textrm{height} = $ 8

$\textrm{slope} = $ 2

$v_a = $ 2

In this video, we provide the solution for this example (note that the given function $f(x)$ may not be exactly the same).

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QID:

demo/annotated/LectureVelocity/1-Introduction

Title:

Introduction

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Started at:

2026-08-04 21:52:31 (CDT)

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{
  "va": 2,
  "base": 4,
  "slope": 2,
  "height": 8
}

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