<text><span class="style10">unctions, Graphs and Change (5 of 6)</span><span class="style7"></span><span class="style10">Derivatives and differentiation</span><span class="style7">The process of finding the derivative of a function is called </span><span class="style26">differentiation</span><span class="style7">, and this branch of mathematics is known as the </span><span class="style26">differential calculus</span><span class="style7">. This process can also be interpreted geometrically: as </span><span class="style26">d</span><span class="style7"> be comes smaller, the points P and Q come closer together until finally they coincide, at which point the slope has the value we have calculated, and represents the tangent to the curve.However, it is not necessary always to work out a derivative either as we have done, or by drawing the graph. Instead, certain general principles apply; for example, as we have seen, the derivative of </span><span class="style26">x</span><span class="style7"> to the power of 2 with respect to </span><span class="style26">x</span><span class="style7"> is 2</span><span class="style26">x</span><span class="style7">, and we can generalize this to state that the derivative of </span><span class="style26">axn </span><span class="style7">with respect to </span><span class="style26">x</span><span class="style7"> is </span><span class="style26">anxn</span><span class="style7">-1.The derivative of a function can itself be differentiated; for example, acceleration is the rate of change of velocity, and we can repeat the line of reasoning in the previous example to find the derivative of the velocity function with respect to time. This is the </span><span class="style26">second derivative</span><span class="style7"> of the distance function.In the previous example, the average acceleration over the interval (</span><span class="style26">t</span><span class="style7"> - </span><span class="style26">d</span><span class="style7">, </span><span class="style26">t</span><span class="style7"> + </span><span class="style26">d</span><span class="style7">) is [20-10(t+d)]-[20-10(t-d)] --------------------------- (t+d)-(t-d) which simplifies to the constant -10. It is because acceleration is a second derivative, that is the rate of change of a rate of change, that it is measured in units of distance per second per second. In this case the value -10 approximates to the downward force of gravity, which slows the ball down and returns it to the ground.</span><span class="style10">Totals and integrals</span><span class="style7">So far we have been considering how much we can find out about speed if we are given functional information about distance in terms of time. Now consider the converse problem: if we know a train's velocity as a function of time, how can we calculate the total distance it has traveled? Clearly if we knew the average speed, the total distance would be the result of multiplying the overall average speed by the total duration of the journey; and if the journey was undertaken in stages for which we knew separate average speeds and durations, the total distance traveled would be the sum of the distances calculated by this method for each separate stage. However, this formula requires the journey to be broken down into separable stages whose average speeds are known; where we only know the instantaneous speed at any moment expressed as a function of time, it is of no help at all. On the other hand, we can conjecture that we can approximate to the right answer by breaking the whole journey down into more and more, shorter and shorter stages. Let us now go back to the previous example: the above figure shows the function derived for the velocity of the ball. The complete duration is divided into equal intervals, and in each we take the speed at the midpoint as an approximation to the average. But this really represents the stepped graph (shaded), rather than the true continuous function. However, if we double the number of intervals and halve the duration of each, we get a better approximation, and carrying on in the same way gives a sequence of better and better approximations whose limit can be thought of as the sum of the areas of infinitely many infinitely thin slices of the area under the graph. The </span><span class="style26">integral calculus</span><span class="style7"> is the study of such processes, and it enables us to calculate infinite sums that can be expressed in terms of continuous functions.</span></text>
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<text>ΓÇó MOTION AND FORCEΓÇó MATHEMATICS AND ITS APPLICATIONSΓÇó SETS AND PARADOXESΓÇó CORRESPONDENCE, COUNTING AND INFINITY</text>
