Calculus
Calculus · Lesson Review · August 14, 2026

Rates of Change and Instantaneous Velocity

Connect familiar slope to average and instantaneous rates of change, then use shrinking time intervals to estimate a tangent slope.

35-minute edited lesson10 chaptersCaptions + transcript

01 · Watch

Lesson video

VTT captionsSRT captions
0:0001Rates of Change — Secant and Tangent Lines2:1402Secant and Tangent Lines on a Function5:4403Rates and Rates of Change13:1304Velocity Versus Speed16:2405Position Functions18:3606Average Velocity as a Secant Slope21:4307Instantaneous Velocity as a Tangent Slope24:0808Worked Example: Quadratic Position28:5009Shrinking the Time Interval31:5910Estimating the Limiting Velocity

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02 · Understand

Learning targets

03 · Vocabulary

Key definitions

Keep the geometry, the meaning, and the units connected.

01Rate
A comparison of two quantities with different units, such as dollars per hour or frames per second.
02Rate of change
How much the output changes for each unit of input change; on a graph, this is slope.
03Secant line
A line through two points on a curve. Its slope gives an average rate of change over an interval.
04Tangent line
A line that matches a curve's direction at one point. Its slope gives the instantaneous rate of change.
05Position function
A function that gives an object's position at each time.
06Average velocity
Change in position divided by elapsed time over an interval.
07Instantaneous velocity
The rate of change of position at one instant; the slope of the tangent line.
08Velocity
Speed together with direction. Its sign indicates direction along the chosen axis.
09Speed
The magnitude of velocity: speed=∣velocity∣\text{speed}=|\text{velocity}|.

04 · Apply

Worked examples

Each quotient is a secant slope until the interval shrinks to an instant.

Average velocity

Position f(t)=2t2f(t)=2t^2 on [0,3][0,3]

  1. f(0)=0f(0)=0 and f(3)=18f(3)=18
  2. f(3)−f(0)3−0=18−03\frac{f(3)-f(0)}{3-0}=\frac{18-0}{3}
  3. average velocity=6 ft/s\text{average velocity}=6\ \mathrm{ft/s}

This is the slope of the secant line through (0,0)(0,0) and (3,18)(3,18).

Closer secants

Move the left endpoint toward t=3t=3

Average velocity as the interval approaches three seconds
IntervalAverage velocity
[2, 3]10 ft/s
[2.9, 3]11.8 ft/s
[2.999, 3]11.998 ft/s

The values approach 12 ft/s12\ \mathrm{ft/s}. That limiting value is the instantaneous velocity at t=3t=3.

Velocity vs. speed

Velocity

Includes direction. A velocity of −7 m/s-7\ \mathrm{m/s} means motion at 7 meters per second in the negative direction.

Speed

Has no direction and is never negative. For velocity −7 m/s-7\ \mathrm{m/s}, speed is ∣−7∣=7 m/s|-7|=7\ \mathrm{m/s}.

Visual connection

Secants approach the tangent

Two curve points determine a secant. As the second point moves closer to the first, the secant line rotates toward the tangent line.

05 · Avoid

Common mistakes

Leaving off units.

A rate compares two measurements, so report both units: feet per second, dollars per hour, and so on.

Calling a secant instantaneous.

Two separated points give an average rate. The instantaneous rate is the limiting tangent slope.

Mixing up position and velocity.

Position tells where an object is; velocity tells how quickly and in which direction its position is changing.

Using negative speed.

Velocity may be negative. Speed is its absolute value and cannot be negative.

06 · Check yourself

Quick check

Decide first, then open the answer.

1 What geometric object represents average rate of change?

A secant line. Its slope uses two points on the curve.

2 What geometric object represents instantaneous rate of change?

A tangent line. Its slope describes the curve at one instant.

3 If velocity is −12 ft/s-12\ \mathrm{ft/s}, what is speed?

12 ft/s. Speed is ∣velocity∣|\text{velocity}|.

4 For f(t)=2t2f(t)=2t^2, what is average velocity on [0,3][0,3]?

6 ft/s. Compute 18−03−0\frac{18-0}{3-0}.

5 What do the averages 10, 11.8, and 11.998 ft/s suggest?

The instantaneous velocity at t = 3 is 12 ft/s. The secant slopes approach that limiting value.

07 · Revisit

Transcript and captions

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