Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Calculate an interval's average velocity.
- Simplify a difference quotient.
- Interpret a derivative with units.
Before you start
Evaluate a quadratic and expand a squared binomial.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A toy cart follows s(t) = t² + 2t metres for t ≥ 0 seconds. How do its average velocity from 2 to 3 seconds and instantaneous velocity at 2 seconds compare?
Why this math matters
A journey average summarizes an interval; a speed display aims to describe motion near one instant. Calculus connects these ideas without pretending that distance divided by a zero time interval makes sense. Begin with a nonzero interval, simplify its average rate, and then investigate what happens as the interval shrinks.
Set up the model
A useful answer starts with clear assumptions:
- The formula gives exact one-dimensional position in this classroom model.
- Time is measured in seconds and position in metres.
- Use h > 0 to approach t = 2 from later times; the polynomial also has a matching two-sided derivative.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
What does a speed reading mean at one instant?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A toy cart follows s(t) = t² + 2t metres for t ≥ 0 seconds. How do its average velocity from 2 to 3 seconds and instantaneous velocity at 2 seconds compare?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Find the interval average
s(2) = 8; s(3) = 15; (15 − 8)/(3 − 2) = 7 m/s
The cart changes position by seven metres during this one-second interval. This calculation does not isolate the rate at the starting instant.
Replace one second with h seconds
[s(2 + h) − s(2)]/h = (6h + h²)/h = 6 + h
Expansion gives s(2 + h) = 8 + 6h + h². Cancellation is valid because the interval width h is nonzero.
Take the limiting rate
lim as h → 0 of (6 + h) = 6 m/s
For h = 0.1 and 0.01, the averages are 6.1 and 6.01 m/s. Their limiting value defines the instantaneous velocity.
The result
Average velocity on [2, 3] is 7 m/s; instantaneous velocity at 2 seconds is 6 m/s.
The later interval includes faster motion, so its average exceeds the starting rate. The derivative function is s′(t) = 2t + 2; its units are metres per second.
Common mistakes to catch
- s(2)/2 = 4 m/s averages from the origin, not around t = 2.
- Substituting h = 0 before cancellation creates an undefined quotient.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
For the same cart, find average velocity from t = 0 to t = 2.
Show a hint
Compare the positions at the two endpoints.
Reveal answer and explanation
4 m/s
[s(2) − s(0)]/(2 − 0) = (8 − 0)/2 = 4 m/s.
Practice 2
What is its instantaneous velocity at t = 4?
Show a hint
Evaluate the derivative 2t + 2.
Reveal answer and explanation
10 m/s
s′(4) = 2(4) + 2 = 10; this describes local motion at four seconds.
Take the idea with you
On any position graph, compare a secant across an interval with the limiting tangent at one endpoint.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Can a formula have a hole but still approach one value?
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.