<text><span class="style10">unctions, Graphs and Change (3 of 6)</span><span class="style7"></span><span class="style10">Average rates of change</span><span class="style7">Let us now consider a different example: suppose I throw a ball straight up in the air; it is slowed down by gravity until it stops and falls back to the ground, falling faster all the time. This is an example of a functional relationship between time (</span><span class="style26">t) </span><span class="style7"> and height (</span><span class="style26">h)</span><span class="style7">. If this relationship is described, say, by the equation </span><span class="style26">h</span><span class="style7"> = 20</span><span class="style26">t</span><span class="style7"> - 5</span><span class="style26">t</span><span class="style7"> to the power of 2 then its graph looks like this: Velocity is defined to be the rate of change of displacement (here height), that is, displacement per unit time. Thus in the form increase of heightvelocity = --------------------- time takenthe well known formula for velocity permits the </span><span class="style26">average</span><span class="style7"> velocity over any time interval to be calculated. In the figure, the average velocity between A and B is therefore the change of height, CB, divided by the change of time, AC. Where the dependent variable (in this case height) is represented by the vertical axis, and the independent variable (time) by the horizontal axis, this average is equivalent to the </span><span class="style26">gradient</span><span class="style7"> or </span><span class="style26">slope</span><span class="style7"> of the line joining the points on the curve that correspond to the ends of the interval, as indicated by the bold line above. However, in this case the average clearly conceals more than it reveals, since we know that between </span><span class="style26">t</span><span class="style7"> = A and </span><span class="style26">t</span><span class="style7"> = C, the ball actually changed direction, so that its upward velocity changed from positive to negative.</span></text>
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<text>ΓÇó MOTION AND FORCEΓÇó MATHEMATICS AND ITS APPLICATIONSΓÇó SETS AND PARADOXESΓÇó CORRESPONDENCE, COUNTING AND INFINITY</text>
