08Daveragerate

= Average Rate of Change =


 * If there is a variable rate of change,
 * sometimes it is useful to talk about the ** average rate of change ** over a particular interval


 * For example, a tree grew from 6.2m to 6.75 m between 1st January one year to 1st January the following year
 * The rate of change will not have been constant during the year because trees grow at different rates depending on the season, the temperature, the amount of rainfall, etc
 * Nevertheless, we can talk about the average rate of growth during that time

math . \qquad \qquad \text{Average rate of growth in cm/month} \\.\\ . \qquad \qquad = \dfrac{ \text{growth in cm} }{ \text{time elapsed in months}} \\.\\ . \qquad \qquad = \dfrac{55 \; cm}{12 \; months} \\.\\ . \qquad \qquad = 4.58 \; cm / month math


 * The ** average rate of change ** between two points is simply the **__gradient__** of a straight line joining those two points


 * For example, the average rate of change between points A and B on the graph shown here, is given by:




 * Don't forget that a rate of change is often expressed with units
 * Calculate the gradient, then write it with the units as your answer
 * eg in the tree growth example shown above
 * the gradient of a graph would have been 4.58
 * but the rate of growth was 4.58 cm/month

... ... Find the average rate of change between the points A and B
 * Example 1 **


 * Solution:**

math . \qquad m = \dfrac{y_2 - y_1}{x_2 - x_1} \\.\\ . \qquad m = \dfrac{4-1}{6-2} \\.\\ . \qquad m = \dfrac{3}{4} math

math . \qquad \text{Hence, average rate of change is:} \\.\\ . \qquad m = \dfrac{3}{4} math


 * Example 2 **

... ... An ice cube was allowed to melt and the remaining size measured every minute. ... ... The results are shown in the table below



... ... ** (a) ** .. Find the average rate of change (mm 3 /minute) in the first 3 minutes (from t = 0 to t = 3)

... ... ** (b) ** .. Find the average rate of change (mm 3 /minute) in the last 4 four minutes (from t = 3 to t = 7)

... ... ** (c) ** .. Comment on the rates of change you calculated


 * Solution:**

... ... ** (a) ** ..

math . \qquad \qquad m_1 = \dfrac{605 - 1000}{3 - 0} \\.\\ . \qquad \qquad m_1 = \dfrac{-395}{3} \\.\\ . \qquad \qquad m_1 = -131.67 \; mm^3/minute math

... ... ** (b) **

math . \qquad \qquad m_2 = \dfrac{0 - 605}{7 - 3} \\.\\ . \qquad \qquad m_2 = \dfrac{-605}{4} \\.\\ . \qquad \qquad m_2 = -151.25 \; mm^3/minute math

... ... ** (c) **


 * Both rates were negative (the ice was melting so the size was decreasing)
 * There was not a constant rate of change
 * The ice melted more quickly in the 2nd half of the experiment than in the first

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