Search topics

Search

Calculate and interpret an average rate of change

Start a set of 8
Step 11 of 15

What the calculation does not establish

An average rate need not equal the instantaneous rate at either endpoint. A negative average does not prove that the function decreases throughout the interval. Later work estimates tangent gradients using nearby points

  1. 1Interpret the interval A positive average rate means the final value is higher; a negative rate means it is lower. Zero means equal endpoint values, not necessarily a constant function
  2. 2Keep the units Rate units are output units per input unit. An average rate describes the whole interval, not the instantaneous rate at a particular point
Year 11, VCE Units 1 & 2, Chapter 12, 4 difficulty levels, No calculatorBuild a worksheet

Average rate of change

Choose the interval endpoints, calculate matched changes and divide. The same method works for a rule, a table or labelled points

\[\overline r=\frac{f(b)-f(a)}{b-a},\quad a<b\]
  1. Compare two inputs Read or calculate the function value at each end of the interval. A rate compares changes; a single function value is not a rate \(\displaystyle \Delta y=f(b)-f(a),\quad\Delta x=b-a\)
  2. Divide by the interval width Use matched subtraction orders. Reversing both differences preserves the rate; reversing only one changes its sign \(\displaystyle \overline r=\frac{f(b)-f(a)}{b-a}=\frac{f(a)-f(b)}{a-b},\quad a\ne b\)
  3. Interpret the interval A positive average rate means the final value is higher; a negative rate means it is lower. Zero means equal endpoint values, not necessarily a constant function
  4. Keep the units Rate units are output units per input unit. An average rate describes the whole interval, not the instantaneous rate at a particular point

Worked example

Find the exact average rate over the stated interval

\[f(x)=2x^2+x,\quad\frac12\le x\le2\]
  1. \(\displaystyle f(\tfrac12)=2(\tfrac12)^2+\tfrac12=1\)Evaluate the earlier endpoint
  2. \(\displaystyle f(2)=2(2)^2+2=10\)Evaluate the later endpoint
  3. \(\displaystyle \Delta y=10-1=9\)Subtract output values in later-minus-earlier order
  4. \(\displaystyle \Delta x=2-\tfrac12=\tfrac32\)Use the same order for the inputs
  5. \(\displaystyle \overline r=\frac{9}{3/2}=6\)Divide. The exact average rate is 6 output units per input unit

Common mistakes

Stopping at the change
✕\(\displaystyle \overline r=f(b)-f(a)\)✓\(\displaystyle \overline r=\frac{f(b)-f(a)}{b-a}\)
Output change alone is not change per unit of input
Mismatched subtraction order
✕\(\displaystyle \frac{f(a)-f(b)}{b-a}\)✓\(\displaystyle \frac{f(a)-f(b)}{a-b}\)
Reverse both differences together to preserve the sign
Overinterpreting zero
✕Zero average rate proves the function is constant✓Zero average rate means the endpoint values are equal
Values inside the interval may differ from the endpoint values

Difficulty levels

introRead endpoint values from points or a table, including reversed display order
standardEvaluate linear and quadratic rules, then calculate the signed average
challengeUse cubic rules or fractional endpoints and retain exact values
examCalculate a contextual average and recognise what its sign and units establish

Questions about this topic

Must I differentiate?+

No. Average rates use endpoint differences. A derivative concerns an instantaneous rate

Does a negative average mean the function always decreases?+

No. It means the final value is lower than the initial value. The function may rise and fall within the interval

Can I average the rates for two smaller intervals?+

Equal-width intervals permit an ordinary arithmetic mean. With unequal widths, use a width-weighted mean: add the output changes and divide by the total input width

What if the endpoints have the same input?+

Then the interval width is zero and this average-rate quotient is undefined. Use distinct inputs

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 12: Rates and derivatives

In this chapter

Calculate and interpret an average rate of change12A Average and instantaneous rates: Relate graph direction and instantaneous rateEstimate instantaneous rates with secants and tangentsDerive polynomial gradients from first principlesDifferentiate polynomialsApply polynomial derivatives12C Polynomial differentiation: Relate graph direction and instantaneous rate