TN 12th Maths — Learn, Visualize, Practice

1. The derivative as slope and rate of change

The derivative $f'(x)$ measures how fast a function changes. Geometrically it is the slope of the tangent line to the curve $y = f(x)$ at each point. Physically, if $s(t)$ is position, then $s'(t)$ is velocity — an instantaneous rate of change.

DefinitionDerivative at a point

$f'(a) = \displaystyle\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$ , the limit of average rates of change over shorter and shorter intervals.

Key idea: The sign of $f'(a)$ tells you the direction of motion of the curve: rising if positive, falling if negative, flat if zero. Explore this live in the Visualize tab.

2. Tangent and normal lines

At the point $(a, f(a))$ the tangent has slope $m = f'(a)$ . The normal is perpendicular to the tangent, so its slope is $-\dfrac{1}{f'(a)}$ (when $f'(a) \neq 0$ ).

Tangent at (a, f(a))

y - f(a) = f'(a)\,(x - a)

Normal at (a, f(a))

y - f(a) = -\dfrac{1}{f'(a)}\,(x - a)

ExampleTangent to a curve

Find the equation of the tangent to

y = x^2

at the point

(2, 4)

.

ExampleNormal line

Find the normal to

y = x^2

at

(2, 4)

.

3. Mean Value Theorems

These theorems connect a function's overall behaviour on an interval to its derivative at some point inside. They underpin much of calculus.

DefinitionRolle's Theorem

If $f$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $f(a) = f(b)$ , then there exists $c \in (a, b)$ with $f'(c) = 0$ .

DefinitionLagrange's Mean Value Theorem

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then there exists $c \in (a, b)$ with $f'(c) = \dfrac{f(b) - f(a)}{b - a}$ .

Note: Lagrange's MVT says: somewhere inside the interval, the instantaneous slope equals the average slope of the chord from $(a, f(a))$ to $(b, f(b))$. Rolle's is the special case where the chord is horizontal.

ExampleApplying Rolle's theorem

Verify Rolle's theorem for

f(x) = x^2 - 4x + 3

on

[1, 3]

and find

c

.

4. Taylor and Maclaurin series

A smooth function can be approximated near a point by a polynomial built from its derivatives. The Taylor series expands about $x = a$ ; the Maclaurin series is the special case $a = 0$ .

Taylor series about x = a

f(x) = f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2!}(x-a)^2 + \dfrac{f'''(a)}{3!}(x-a)^3 + \cdots

Maclaurin series (a = 0)

f(x) = f(0) + f'(0)\,x + \dfrac{f''(0)}{2!}x^2 + \dfrac{f'''(0)}{3!}x^3 + \cdots

Standard Maclaurin expansions

$e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots$
$\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots$
$\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots$
$\log(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \cdots$ (for $|x| < 1$ )

ExampleDeriving a series

Find the first three non-zero terms of the Maclaurin series of

\cos x

.

5. Indeterminate forms and L'Hôpital's Rule

When a limit gives a meaningless form such as $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ , you cannot read off the answer directly. L'Hôpital's rule replaces the ratio by the ratio of derivatives.

DefinitionL'Hôpital's Rule

If $\displaystyle\lim_{x \to a} \dfrac{f(x)}{g(x)}$ has the form $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ , then $\displaystyle\lim_{x \to a} \dfrac{f(x)}{g(x)} = \lim_{x \to a} \dfrac{f'(x)}{g'(x)}$ , provided the second limit exists.

Watch out: Check the indeterminate form first. Applying L'Hôpital to a limit that is not $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ gives wrong answers. Forms like $0 \cdot \infty$ or $\infty - \infty$ must first be rewritten as a quotient.

ExampleA 0/0 limit

Evaluate

\displaystyle\lim_{x \to 0} \dfrac{\sin x}{x}

.

6. Increasing, decreasing and stationary points

The first derivative reveals where a function rises or falls. Where $f'(x) = 0$ (or is undefined) the curve momentarily levels off — a stationary point.

$f'(x) > 0$ on an interval $\Rightarrow$ $f$ is increasing there.
$f'(x) < 0$ on an interval $\Rightarrow$ $f$ is decreasing there.
$f'(x) = 0$ $\Rightarrow$ a stationary point (candidate max, min, or inflection).

ExampleIntervals of increase

Find where

f(x) = x^3 - 3x

is increasing.

7. Maxima and minima

At a stationary point you decide whether it is a peak, a valley, or neither using one of two tests.

First-derivative test at a stationary point x = c

$f'$ changes $+ \to -$ across $c$ : local maximum.
$f'$ changes $- \to +$ across $c$ : local minimum.
$f'$ does not change sign: neither (a point of inflection).

Second-derivative test at a stationary point

f'(c) = 0,\ f''(c) < 0 \Rightarrow \text{local max}; \qquad f'(c) = 0,\ f''(c) > 0 \Rightarrow \text{local min}

Note: If $f''(c) = 0$ the second-derivative test is inconclusive; fall back on the first-derivative test.

ExampleLocal max and min of a cubic

Find the local maximum and minimum values of

f(x) = x^3 - 3x

.

ExampleOptimization word problem

A rectangular field is fenced with

100

m of fencing. What dimensions give the maximum area?

8. Concavity, inflection, asymptotes and curve sketching

The second derivative controls how the curve bends. Where it changes sign, the concavity flips — a point of inflection.

$f''(x) > 0$ : curve is concave up (holds water).
$f''(x) < 0$ : curve is concave down.
$f''(x) = 0$ with a sign change: point of inflection.

DefinitionAsymptotes

A vertical asymptote occurs where $f \to \pm\infty$ (e.g. $x = 2$ for $\dfrac{1}{x-2}$ ). A horizontal asymptote $y = L$ occurs when $\displaystyle\lim_{x \to \pm\infty} f(x) = L$ .

Curve-sketching checklist

Domain, intercepts, and symmetry.
Stationary points and their nature (first/second derivative test).
Intervals of increase/decrease and concavity; points of inflection.
Asymptotes and end behaviour as $x \to \pm\infty$ .