Calculus for dummies

Jeanious

Hi all, just in need of a bit (yeah right!) of help.

First of all ill just explain my situation, im a good few years out of school at this stage and im goin back to college in sept. to do a fairly mathsy degree. Was actually quite good at maths for LC (a2 in honours) but i never really understood the actually practicalities of some stuff, despite having full knowledge of the method.

Can someone please try to explain, in basic terms, how integration and differentiation actually work? I know HOW to do it, but not WHY to do it, if ya get me, and i always wondered what bearing they both had on reality.

Ive been lookin around the net for a couple of days, and i just cannot get my head around it, like i know differentiation gives ya the slope of a tangent to a function, but what does that tell ya? And why do ya need to know it?

Absolutely bewildered with it all to be honest, and would seriously appreciate any insight at all, thanks!

gerry87

When you differentiate you get dy/dx. dy is the change in y, and dx is the change in x. So dy/dx gives you the change in y for a one unit change in x, which is the same as the slope.

So you can say that if I change x by 10, y will change by 20.

Take f(x) = 2x^2 + 3x, differentiates to dy/dx = 4x+3. So if we look at the function at x=7, dy/dx is 4(7)+3 = 31

f(6) = 2(6)^2 + 3(6) = 90
f(7) = 2(7)^2 + 3(7) = 119
f(8) = 2(8)^2 + 3(8) = 152

Moving x from 6 to 7 causes a change in f(x) from 90 to 119, a change of 29.
Moving x from 7 to 8 causes a change in f(x) from 119 to 152, a change of 33.

These are close to the 31 that was given by dy/dx, this is because differentiation is an approximation. In this example we moved x by 1, if we moved x by .5 then dy/dx would be closer to the real answer.

A real world use from economics would be if you had a function to show how much profit you'd make from selling something at €x, you could work out where the slope of the line is zero (dy/dx = 0), that gives you the price you should sell your thing at to maxamise your profit.

When you integrate [LATEX] $ \int 3x dx $ [/LATEX] you can integrate from one number to another like [LATEX] $ \int^6_2 3x dx $ [/LATEX]. This gives you the area under the curve between the points x=6 and x=2.

An example where you could use it is in probability if you have a probability function like


The area under the curve relates to probability of an event.

Fremen

If you have access to a library, maybe try some first year undergrad physics books (haliday, resnick and walker for example). Getting some physical intuition for how things work might help.

It'll show you why if f(t) represents the position of a body at time t, then df/dt represents the speed, and so on.

Don't sweat it too much though. It'll come to you with practice. One of the strange things about maths is that often you can feel like have no idea what you're doing or why, and everything can seem really difficult. Then, you look back on it a year or two later and it seems really straightforward.
(think back to learning Trigonometry for JC )

Shane_C

Start with first principles. Differentiation as we know it was invented (<insert better term) as an easier way of doing first principles.

Come to think of it, can anyone explain why our fast differentiation tool does the same as FP or is it just a coincidence?

Fremen

It's certainly not coincedence!
Everything follows from a few rules, like the product rule or quotient rule, which are deduced through first principles.