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An easy way to understand what differential calculus is about!. Click on the green arrow to return to the previous page
Normally differential calculus is introduced by considering gradients of curves but I found a less abstract approach more helpful as below though mathematicians would severely criticise it I think.
Consider linear motion with a constant acceleration. Call that acceleration "a" and it could be the acceleration due to gravity and we could consider Newton's famous apple falling down vertically from a branch to the ground. When the apple hits the ground its velocity (v) will be the sum of any intial velocity(u) plus the product of the acceleration and the time taken to fall (t).This applies in the general case though for an apple falling from a branch the initial velocity is zero. 24 ft per second.
And the distance moved from start to finish (height the apple drops from) will be the displacement"s". So displacement (s)= average velocity multiplied by the time taken and this is =(u+v)/2 multiplied by the time taken
So s=(u+v)/2 multiplied by t
Substituting for v from above into our equation for s gives s =[u+ (u+at)]/2 times t and this is =( 2ut+at)/2 times t
Multiplying through by t gives s = ( 2ut+at²)/2 or that s = ut + at²/2. Since we know the initial velocity u is zero then the product ut=0 and hence
So s= at²/2
The acceleration due to gravity in feet is very nearly 32 ft per second per second so s = 16t²
To make the arithmetic easy, lets say the apple takes just 1 second to hit the ground when its dropped so the AVERAGE velocity of the apple will be 16 feet per second and the apple will have fallen a distance of 16 feet when it hits the ground but it can't have a velocity of 16 feet per second then because it has started with zero velocity and increased its speed as it was accelerated by gravity so its final velocity when it hits the ground must be greater to balance out its slow speed at the start of its descent.
In the first ½ second, the apple will fall a distance s = 16 (1/2)times(1/2) =16/4 = 4ft and it has a further 12 ft to fall in the last ½ second so its AVERAGE velocity during that final half second will be equal to the distance left to fall divided by the time left to fall =12 divided by ½
Again in the last ¼ second, the distance already travelled in the preceding ¾ second will be S=16.(3/4).(3/4) = 16.(9/16) = 9 ft and the apple has another 16-9 =7 ft to travel so the AVERAGE velocity in the last quarter second will be 7 divided by ¼ = 28 ft per second.
Now in the final 1/8th second, the apple will have already fallen s= 16.(7/8).(7/8) =49/4 ft and will have a further distance to fall of 16-49/4 =64-49/4 =15/4 ft and so its AVERAGE velocity in the last 1/8 th second will be (15/4).(1/8) = 30 ft per second.
Similarly by another calculation, in the last 1/100th second the AVERAGE velocity will be 31.84 ft per second.
So its now obvious that as the interval of time remaining is made smaller and smaller the AVERAGE velocity will reach a limiting value of 32 ft per second and this is the instantaneous rate of change of velocity with respect to time. Newton called this the fluxion. We now call this the derivative or dv/dt
Recall for our apple that s=16t² and dv/dt =32 as t approached the limit t=0