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 $s'(t) = v(t)$ is velocity and $v'(t) = s''(t)$ is acceleration — each derivative is 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.

ExampleVelocity and direction of a moving particle

A particle moves so that its position is

s(t) = t^3 - 6t^2 + 9t

metres. When does it change direction?

ExampleRate of change of volume

The volume of a cube of side

x

is

V = x^3

. Find the rate of change of

V

with respect to

x

when

x = 5

.

2. Related rates

A related-rates problem involves two or more quantities that change with time. You know some of the rates and want another. The method: write an equation linking the quantities, differentiate both sides with respect to time $t$ (using the chain rule), then substitute the known values.

Key idea: Differentiate the geometric relation with respect to $t$ FIRST, and only substitute the given numbers at the very end — never before differentiating.

ExampleInflating a spherical balloon

Air is blown into a spherical balloon at

1000\ \text{cm}^3/\text{s}

. How fast is the radius changing when

r = 7

cm?

ExampleSpreading ripple

A stone dropped in a pond makes circular ripples whose radius grows at

2

cm/s. How fast is the disturbed area growing when

r = 5

cm?

ExampleTwo cars — a Pythagorean related rate

Car A is

10

km north of a crossing moving north at

80

km/hr; car B is

15

km east moving east at

100

km/hr. How fast is the distance between them changing?

3. 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)

Note: If the tangent is horizontal, $f'(a) = 0$: the tangent is $y = f(a)$ and the normal is the vertical line $x = a$. If the tangent is vertical, swap the roles.

ExampleTangent and normal from the formula

Find the tangent and normal to

y = x^2 + 3x - 2

at

(1, 2)

.

ExampleWhere is the tangent parallel to a line?

Find the points on

y = x^3 - 3x^2 + x - 2

where the tangent is parallel to

y = x

.

ExampleAngle between two curves

Find the acute angle between

y = x^2

and

y = (x-3)^2

at their intersection.

Note: Two curves cut orthogonally when $m_1 m_2 = -1$ at every intersection; they are tangent (touch) when $m_1 = m_2$.

4. Rolle's theorem and Lagrange's Mean Value Theorem

These theorems connect a function's behaviour across an interval to its derivative at some interior point. They are the backbone of the results on monotonicity and optimization.

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 (MVT)

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

ExampleFinding c in Rolle's theorem

Find

c

satisfying Rolle's theorem for

f(x) = x^2(1-x)^2

on

[0, 1]

.

ExampleFinding c in the Mean Value Theorem

Find the value of

c

from the MVT for

f(x) = x - x^2

on

[1, 2]

.

ExampleMVT as an inequality bound

Prove that

|\sin\alpha - \sin\beta| \le |\alpha - \beta|

for all real

\alpha, \beta

.

Watch out: Rolle/MVT need ALL hypotheses. $f(x) = |x|$ on $[-1, 1]$ has $f(-1) = f(1)$ but is not differentiable at $0$, so Rolle's fails there. Always check continuity and differentiability first.

5. Taylor and Maclaurin series

An infinitely differentiable function can be written as a power series 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 (memorise these)

$e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots$
$\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots$
$\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots$
$\log(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots$ (for $-1 < x \le 1$ )
$\tan x = x + \dfrac{x^3}{3} + \dfrac{2x^5}{15} + \cdots$ (for $|x| < \tfrac{\pi}{2}$ )

ExampleDeriving the series for e^x

Find the Maclaurin series of

e^x

.

ExampleSeries for log(1+x) up to 4 terms

Expand

\log(1 + x)

as a Maclaurin series up to the first four non-zero terms.

ExampleA Taylor expansion about a point

Write the Taylor series of

\dfrac{1}{x}

about

x = 2

(first three non-zero terms).

6. 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. L'Hôpital's rule replaces the ratio by the ratio of derivatives.

DefinitionL'Hôpital's Rule

If $\displaystyle\lim_{x \to a} \dfrac{f(x)}{g(x)}$ is of the form $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ and $g'(x) \ne 0$ , then $\displaystyle\lim_{x \to a} \dfrac{f(x)}{g(x)} = \lim_{x \to a} \dfrac{f'(x)}{g'(x)}$ , provided the right-hand limit exists.

Watch out: Check the form FIRST. Applying L'Hôpital when the limit is NOT $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ gives wrong answers. Other forms ($0\cdot\infty$, $\infty - \infty$, $1^\infty$, $0^0$, $\infty^0$) must be rewritten into a quotient first.

ExampleA 0/0 limit, applied twice

Evaluate

\displaystyle\lim_{x \to 0} \dfrac{1 - \cos x}{x^2}

.

ExampleAn ∞/∞ limit

Evaluate

\displaystyle\lim_{x \to \infty} \dfrac{x^2 + 17x + 29}{x^4}

.

ExampleA 0·∞ form

Evaluate

\displaystyle\lim_{x \to 0^+} x\log x

.

ExampleA 1^∞ form via logarithms

Evaluate

\displaystyle\lim_{x \to 0^+} (1 + x)^{1/x}

.

7. Monotonicity: increasing and decreasing functions

The first derivative reveals where a function rises or falls. This is the direct consequence of the Mean Value Theorem.

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

Note: A stationary point has $f'(x_0) = 0$. A critical point has $f'(x_0) = 0$ OR $f'$ undefined. Every stationary point is critical, but not every critical point is stationary — e.g. $f(x) = |x-17|$ has a critical point at $x = 17$ that is not stationary.

ExampleIntervals of increase and decrease

Find where

f(x) = x^3 - 3x

is increasing and where it is decreasing.

ExampleProving strict monotonicity

Show that

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

is strictly increasing on

(2, \infty)

.

8. Maxima and minima (first and second derivative tests)

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. Example: $f(x) = x^4$ has $f''(0) = 0$ yet a clear minimum at $0$.

ExampleLocal max and min of a cubic

Find the local maximum and minimum values of

f(x) = x^3 - 3x

.

ExampleAbsolute extrema on a closed interval

Find the absolute maximum and minimum of

f(x) = 2x^3 + 3x^2 - 12x

on

[-3, 2]

.

9. Applied optimization (word problems)

To optimize a real quantity: introduce variables, write the quantity to be maximized/minimized, use a constraint to reduce it to ONE variable, differentiate, and confirm with the second-derivative test.

ExampleOpen box from a square sheet

A

12 \times 12

square sheet has equal squares of side

x

cut from each corner; the flaps are folded up into an open box. Find

x

that maximizes the volume.

ExampleRectangle of maximum area

A rectangular field is fenced with

100

m of fencing. What dimensions give the maximum area?

ExampleLeast perimeter for a fixed area

Show that among all rectangles of a given area

k

, the square has the least perimeter.

10. Concavity, inflection, asymptotes and curve sketching

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

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

Watch out: $f''(c) = 0$ alone does NOT guarantee an inflection — the sign must actually change. And an inflection need not be a stationary point: $y = \sin x$ has an inflection at $x = \pi$ where the slope is $-1$, not $0$.

ExampleConcavity and point of inflection

Discuss the concavity of

f(x) = x^3 - 6x^2 + 9x + 1

and find its 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 (odd/even).
Stationary points and their nature (first/second derivative test).
Intervals of increase/decrease and of concavity; points of inflection.
Asymptotes and end behaviour as $x \to \pm\infty$ .

Key idea: Read the derivative signs off a number line: mark the critical numbers, test each interval for the sign of $f'$ (rise/fall) and $f''$ (concavity), then the shape draws itself.