Calculus

1. Introduction

Calculus is the largest and most examined unit of Grade 12. It deepens differentiation to the inverse trigonometric, exponential, logarithmic and hyperbolic functions, states the two great mean value theorems (Rolle and Lagrange), and provides L'Hospital's rule for hard limits. On the integral side it develops anti-derivatives, integration by partial fractions, and the solution of first order differential equations. The unifying idea is that differentiation and integration are inverse processes, and the whole apparatus is used to study the local behaviour of curves and the motion of quantities.

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2. Derivatives of standard functions by definition

The derivative from first principles is

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

Applied to the standard functions it produces the results used throughout:

$$ \begin{aligned} \frac{d}{dx}\sin^{-1}x &= \frac{1}{\sqrt{1-x^2}} \ \qquad \frac{d}{dx}\tan^{-1}x &= \frac{1}{1+x^2} \ \frac{d}{dx}e^{x} &= e^{x} \ \qquad \frac{d}{dx}a^{x} &= a^{x}\ln a \ \qquad \frac{d}{dx}\ln x &= \frac{1}{x}. \end{aligned} $$

Worked example. Differentiate $e^x$ from first principles. By definition,

$$ \begin{aligned} \frac{d}{dx}e^{x} &= \lim_{h\to 0}\frac{e^{x+h} - e^{x}}{h} \ &= e^{x}\lim_{h\to 0}\frac{e^{h} - 1}{h} \ &= e^{x}\cdot 1 \ &= e^{x} \end{aligned} $$

using the standard limit $\lim_{h\to 0}\dfrac{e^{h}-1}{h} = 1$.

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3. Continuity and differentiability

A function is continuous at $x=a$ if $\lim_{x\to a} f(x) = f(a)$, and differentiable there if $f'(a)$ exists. The key relationship is one-way: differentiability implies continuity, but not conversely. If $f$ is differentiable at $a$ then

$$ \begin{aligned} \lim_{x\to a}[f(x) - f(a)] &= \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\cdot(x-a) \ &= f'(a)\cdot 0 \ &= 0 \end{aligned} $$

so $f$ is continuous at $a$. The converse fails: $f(x) = |x|$ is continuous at $x=0$ but has no derivative there, because the left and right slopes are $-1$ and $+1$.

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