Rate of Change Math Example 1

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Example 1

easy
The position of a particle at time tt seconds is s(t)=3t2−2t+1s(t) = 3t^2 - 2t + 1 metres. Find the average rate of change of position from t=1t = 1 to t=4t = 4, and the instantaneous rate of change at t=2t = 2.

Solution

  1. 1
    Average rate of change: s(4)−s(1)4−1\frac{s(4) - s(1)}{4 - 1}.
  2. 2
    s(4)=3(16)−2(4)+1=48−8+1=41s(4) = 3(16) - 2(4) + 1 = 48 - 8 + 1 = 41. s(1)=3−2+1=2s(1) = 3 - 2 + 1 = 2.
  3. 3
    Average rate: 41−23=393=13\frac{41 - 2}{3} = \frac{39}{3} = 13 m/s.
  4. 4
    Instantaneous rate: s′(t)=6t−2s'(t) = 6t - 2. At t=2t = 2: s′(2)=12−2=10s'(2) = 12 - 2 = 10 m/s.

Answer

Average rate of change: 1313 m/s; instantaneous rate at t=2t=2: 1010 m/s
The average rate of change is the slope of the secant line between two points. The instantaneous rate of change is the derivative evaluated at the specific time. They are equal only when the function is linear.

About Rate of Change

A measure of how quickly one quantity changes with respect to another; the ratio of the change in output to the change in input.

Learn more about Rate of Change →

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Example 3 medium

Find the average rate of change of [formula] on [formula].

Example 4 easy

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Example 5 medium

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