Trying to understand basic calculus.

Hello, I'm in year 9 and I'm trying to grasp basic calculus but I just don't understand it. Can someone please explain the basics of calculus as simply as possible so I can understand it. I've looked on sources like Wikipedia but they just explain everything in a really advanced manner. I know calculus isn't part of the GCSE syllabus but I just want to be one step ahead so when it comes down to A-levels; I can comprehend what the teacher is saying. Any help would be most appreciated.

Reply 1

goku24

Original post by Nerdcubed

Calculus is essentially a limiting procedure - we go down to infinitely small values and see what happens around them. These have no meaning in the usual sense, so we have to use calculus to get somewhere with them.

The typical place to start is differentiation, so I'll start there as well. Say you want to find the gradient of the curve x^2 at the point (1,1). We can accomplish this by drawing a triangle from x=1 (1,1) to x=2 (2,4). This gives a gradient of 3/1 = 3. By moving our point closer and closer, say from x=1 to x=1.5, we can get more and more accurate results. With calculus, we use algebra to represent an infinitely small quantity,

\delta x

and see what happens when this approaches 0. With some clever algebraic manipulation, we notice that the gradient at x=1 is 2. In general, we can show that the gradient at x=a is 2a using the process of differentiation.

Don't feel compelled to look at this, but if you want a bit more detail...

Let's say we have a function, y=f(x). We can approximate the gradient of this function at x=a by using our earlier limiting procedure, but with

\delta x

instead of a numerical change. By using the formula for gradient, we get the instantaneous gradient at any point to be

\dfrac{\delta y}{\delta x}

and we can find delta y by doing

f ( x + \delta x ) - f(x)

. By doing this, and letting all our values approach zero so they become infinetisimal, we get the definition of the derivative to be

\displaystyle\lim_{\delta x \to 0} \dfrac{f(x + \delta x) - f(x)}{\delta x}

. By subbing in f(x) = x^2, we get the derivative to be

\displaystyle\lim_{\delta x \to 0} \dfrac{ x^2 + 2 x \delta x + ( \delta x ) ^2 - x^2}{\delta x}

which simplifies down to

\displaystyle\lim_{\delta x \to 0} 2x + \delta x

which approaches 2x as dx becomes 0. This is the gradient function for the curve y=x^2.

If you want any more help, feel free to ask, but I suggest getting your GCSE maths 100% down before you start looking at any further topics, as a good grasp of algebraic manipulation is essential for calculus.

HTH.

Posted from TSR Mobile

(edited 9 years ago)

Reply 2