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Glacius
CalculusConcept reference

Average rate of change

The average rate of change is the difference in function outputs divided by the corresponding nonzero difference in inputs.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. From f(2)=4 to f(5)=13, the output rises by 9 over an input increase of 3
  3. Keep both subtractions in the same order
  4. An average rate describes the whole interval
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01From f(2)=4 to f(5)=13, the output rises by 9 over an input increase of 3#

From f(2)=4f(2)=4 to f(5)=13f(5)=13, the output rises by 9 over an input increase of 3. The secant joins these two graph points.

Known graph points are (2,4) and (5,13). The dashed secant joins them. Its horizontal change is 3 and vertical change is 9. The function between these points is unspecified. Axes use 40 pixels per x unit and 8 pixels per output unit.Known graph points are (2,4) and (5,13). The dashed secant joins them. Its horizontal change is 3 and vertical change is 9. The function between these points is unspecified. Axes use 40 pixels per x unit and 8 pixels per output unit.
Figure 1Known graph points are (2,4) and (5,13). The dashed secant joins them. Its horizontal change is 3 and vertical change is 9. The function between these points is unspecified. Axes use 40 pixels per x unit and 8 pixels per output unit.
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For f(2)=4f(2)=4 and f(5)=13f(5)=13, divide 13−413-4 by 5−25-2. The average rate is 3 output units per input unit.

f(b)−f(a)b−a,b≠a\frac{f(b)-f(a)}{b-a},\quad b\ne a
Check your reasoning

f(0)=2f(0)=2, f(3)=14f(3)=14. Average rate on [0,3][0,3]?

Show answer and explanation
4

Divide the output change 12 by the input change 3.

02Keep both subtractions in the same order#

Keep both subtractions in the same order. If f(1)=8f(1)=8 and f(4)=2f(4)=2, the output change is −6-6 and the input change is 3. The rate is −2-2.

Check your reasoning

f(−2)=3f(-2)=3, f(2)=3f(2)=3. Average rate on [−2,2][-2,2]?

  1. A3
  2. B6
  3. C0
Show answer and explanation
0

Equal endpoint outputs give zero net change.

03An average rate describes the whole interval#

An average rate describes the whole interval. It is neither an average of the outputs nor a rate at one instant. Equal endpoint outputs give zero net change, even if values vary between them.

Check your reasoning

A reading is 10 units at minute 1 and 4 units at minute 4. Average change in units per minute?

Show answer and explanation
-2

The change is −6-6 over 3 minutes.

Key takeaway

Output change divided by input change.

  • Compute an average rate of change over an interval

Further questions

Can the average rate be negative?
Yes. Over an increasing input interval, a lower final output gives a negative output change and a negative average rate.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Average rate of change.” Math behind ML. /learn/c-average