Differential Calculus

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Introduction

Differential Calculus, or differentiation, is a branch of mathematics that studies rates of change. It appears in Questions 6, 7 and 8 of the Ordinary Level Maths Paper 1 and Questions 6 and 7 of the Higher Level Maths Paper 1. <math>Insert non-formatted text here</math><math>Insert formula here</math>

Limits

The idea of limits is fundamental to all of calculus.

$\displaystyle\lim_{x \to a} f(x) = B$ basically means that as x gets closer to a, f(x) gets closer to B.

Evaluating limits is fairly straightforward. Take for example

$\displaystyle\lim_{x \to 4} \displaystyle\frac{5x+9}{4x+4}$

We simply write 4 where ever there is an x. So we now have

$\displaystyle\frac{5(4)+9}{4(4)+4} = \displaystyle\frac{20+9}{16+4} = \displaystyle\frac{29}{20}$

Sometimes, when we evaluate a limit we may get $\frac{0}{0}$ , $\frac{\infty}{\infty}$ or $\infty - \infty$

We cannot determine the value of the limit in this form but there are two things we can do. We can find and cancel a common factor or use the special limit, that is to say $\lim_{\theta\to0} \frac{sin\theta}{\theta} = 1$

Definition of the Derivative

The slope of a line with points $(x_1, y_1)$ and $(x_2, y_2)$ is $\frac{y_2 - y_1}{x_2 - x_1}$ .

Let's take a curve, y = f(x).

Two points of this curve are $(x, f(x))$ and $(x+h, f(x+h))$ .

The slope however, is not $\frac{f(x+h) - f(x)}{x + h - x} = \frac{f(x+h) - f(x)}{h}$ .

However, as the value of h gets smaller, the portion of the curve between x and x+h more closely resembles a line and this formulae becomes valid. So it's fair to say that the slope is equal to

$lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ .