Average velocity
Position f(t) = 2t² on [0, 3]
- f(0) = 0 and f(3) = 18
- (f(3) − f(0)) / (3 − 0) = (18 − 0) / 3
- average velocity = 6 ft/s
This is the slope of the secant line through (0, 0) and (3, 18).
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.
01 · Watch
Lesson video
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: f(t) = 2t²28:5009Shrinking the Time Interval31:5910Estimating the Limiting Velocity
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02 · Understand
Learning targets
- Interpret a rate and include the correct compound units.
- Connect average rate of change to the slope of a secant line.
- Distinguish velocity from speed and explain the sign of velocity.
- Read position from a position function or position-versus-time graph.
- Estimate instantaneous velocity with shrinking time intervals.
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|.
04 · Apply
Worked examples
Each quotient is a secant slope until the interval shrinks to an instant.
Closer secants
Move the left endpoint toward t = 3
| Interval | Average 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/s. That limiting value is the instantaneous velocity at t = 3.
Velocity vs. speed
Velocity
Includes direction. A velocity of −7 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, speed is |−7| = 7 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.
secanttangent
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, what is speed?
12 ft/s. Speed is |velocity|.
4 For f(t)=2t², what is average velocity on [0,3]?
6 ft/s. Compute (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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