TN 12th Maths — Learn, Visualize, Practice

1. Linear approximation — the tangent line as a calculator

Near a point, a smooth curve looks almost straight — it hugs its tangent line. This is the single most useful idea in differential calculus: over a small step $\Delta x$ , the change in $y = f(x)$ is well approximated by the change along the tangent, whose slope is $f'(x)$ .

Increment approximation

\Delta y \approx f'(x)\,\Delta x

Sliding along the tangent from a point $x_0$ where $f$ is easy to evaluate gives the linear approximation (also called the tangent-line approximation):

DefinitionLinear approximation

The linear approximation of a differentiable $f$ near $x_0$ is $L(x) = f(x_0) + f'(x_0)(x - x_0)$ . For $x$ close to $x_0$ , $f(x) \approx L(x)$ . The trick is to choose $x_0$ as a nearby point where $f(x_0)$ and $f'(x_0)$ are known exactly.

ExampleEstimating a root

Use linear approximation to approximate

\sqrt{36.6}

.

ExampleA cube-root estimate

Using

f(x) = x^{1/3}

at

x_0 = 27

, approximate

27.23^{1/3}

.

Note: Choosing $x_0$ well is the whole game: pick the closest 'nice' value (a perfect square, cube, or multiple of a known angle) so that $f(x_0)$ and $f'(x_0)$ need no calculator.

2. Errors: absolute, relative and percentage

When a measured quantity feeds into a formula, its small error propagates. The differential is the tool that converts a measurement error $\Delta x$ into an estimated error in $f(x)$ .

DefinitionThree kinds of error

$\textbf{Absolute error} = |\text{actual} - \text{approximate}|$ (carries a unit). $\textbf{Relative error} = \dfrac{\text{absolute error}}{\text{actual value}}$ (no unit). $\textbf{Percentage error} = \text{relative error} \times 100\%$ .

If $x$ carries a small error $\Delta x$ , then the induced error in $f(x)$ is about $\Delta f \approx f'(x)\,\Delta x$ . Dividing by $f(x)$ turns this into a relative error.

ExampleSurface area of a soap bubble

A spherical soap bubble's radius grows from

5\text{ cm}

to

5.2\text{ cm}

. Use linear approximation to estimate the increase in surface area

S = 4\pi r^2

, and find the percentage error against the exact change.

ExampleVolume of a cylinder

A right circular cylinder has

r = 10\text{ cm}

,

h = 20\text{ cm}

(fixed). If

r

increases to

10.1\text{ cm}

, estimate the change in volume

V = \pi r^2 h

and the relative error.

Watch out: A clean exam result: the percentage error in the $n$-th root of a number is about $\dfrac{1}{n}$ times the percentage error in the number itself. (From $f = x^{1/n}$, $\dfrac{df}{f} = \dfrac{1}{n}\dfrac{dx}{x}$.)

3. Differentials as a working notation

To turn $\Delta y \approx f'(x)\Delta x$ into a precise object, we introduce the differentials $dx$ and $dy$ . We treat $dx$ as an independent variable (the size of the step), and define $dy$ to be the matching rise along the tangent line.

DefinitionDifferential

For a differentiable $y = f(x)$ , the differential of $y$ is $dy = f'(x)\,dx$ , where $dx = \Delta x$ is the differential of $x$ . The true change $\Delta y$ and the differential $dy$ agree to first order: $\Delta y \approx dy$ for small $dx$ .

Key idea: $\Delta y$ is the exact change measured along the curve; $dy = f'(x)\,dx$ is the change predicted by the tangent line. Their difference shrinks faster than $\Delta x$ itself, which is why the approximation is so good.

Differentials obey the same algebra as derivatives — sum, product, quotient and chain rules all carry over:

d(cf) = c\,f'\,dx,\quad d(f+g) = df + dg,\quad d(fg) = f\,dg + g\,df,\quad d(f\circ g) = f'(g)\,g'\,dx

ExampleA differential from the product rule

Find the differential of

g(x) = x^2 + \sin x

.

ExampleShrinking sphere

A sphere of radius

10\text{ cm}

has its radius decrease by

0.1\text{ cm}

. Approximately how much does its volume change?

4. Functions of several variables

Most real quantities depend on more than one input: the volume of a cylinder depends on radius and height, temperature on position and time. A function of two variables assigns a single number $z = f(x,y)$ to each point $(x,y)$ in a region of the plane.

DefinitionFunction of two variables

A function $f$ of two variables is a rule assigning to each ordered pair $(x,y)$ in a domain $D \subseteq \mathbb{R}^2$ a unique real number $z = f(x,y)$ . Its graph is a surface sitting above the $xy$ -plane.

For example $g(x,y) = 30 - x^2 - y^2$ is a paraboloid. Fixing $y = 3$ gives $g(x,3) = 21 - x^2$ , a parabola — the curve where the vertical plane $y = 3$ slices the surface. Fixing $x = 2$ gives $g(2,y) = 26 - y^2$ , another parabola. Slicing to hold one variable fixed is exactly what partial derivatives will formalise.

Key idea: Open the Visualize tab to slice a surface: freeze $y = b$ to get an $x$-curve, freeze $x = a$ to get a $y$-curve. The partial derivatives are the slopes of these two slice-curves.

5. Limits and continuity in two variables

We say $f(x,y) \to L$ as $(x,y) \to (a,b)$ if $f$ can be made arbitrarily close to $L$ by taking $(x,y)$ close enough to $(a,b)$ . The crucial new feature in two dimensions is that a point can be approached from infinitely many directions — along any path in the plane, not just from left and right.

Watch out: Path dependence: for a limit to exist, $f$ must approach the same value along every path. If two paths give different values, the limit does not exist.

ExampleA limit that fails to exist

Show that

f(x,y) = \dfrac{xy}{x^2 + y^2}

has no limit at

(0,0)

.

DefinitionContinuity

$f$ is continuous at $(a,b)$ if $\displaystyle\lim_{(x,y)\to(a,b)} f(x,y) = f(a,b)$ . Polynomials in $x$ and $y$ are continuous everywhere; a quotient is continuous wherever its denominator is non-zero.

ExampleContinuity by the squeeze

Show

g(x,y) = \dfrac{x^2 y}{x^2 + y^2}

(with

g(0,0)=0

) is continuous at the origin.

6. Partial derivatives

To measure how $f(x,y)$ changes, we vary one input at a time and hold the other fixed. Freezing $y$ turns $f$ into a function of $x$ alone; its ordinary derivative is the partial derivative with respect to $x$ .

DefinitionPartial derivatives

The partial derivatives of $f(x,y)$ are $\dfrac{\partial f}{\partial x} = \lim_{h\to 0}\dfrac{f(x+h,\,y) - f(x,y)}{h}$ and $\dfrac{\partial f}{\partial y} = \lim_{h\to 0}\dfrac{f(x,\,y+h) - f(x,y)}{h}$ , when these limits exist. They are also written $f_x$ and $f_y$ .

Key idea: To compute $f_x$, differentiate as usual in $x$ while treating every $y$ as a constant. For $f_y$, treat every $x$ as constant. Geometrically, $f_x(a,b)$ is the slope of the surface in the $x$-direction — the slope of the curve cut by the plane $y = b$.

ExampleBoth partials of a polynomial

Find

f_x

and

f_y

for

f(x,y) = x^2 y + 3xy^2

.

ExampleEvaluating at a point

For

F(x,y) = x^3 y + y^2 x + 7

, find

F_x(-1,3)

and

F_y(2,-1)

.

ExamplePartial of an exponential

For

u(x,y) = e^{x^2 + y^2}

, find

u_x

.

ExamplePartial of a power tower

For

w(x,y) = x^y

with

x > 0

, find

w_x

and

w_y

.

7. Higher-order partials and Clairaut's theorem

Each partial derivative is itself a function of $x$ and $y$ , so we can differentiate again. This gives four second-order partial derivatives:

f_{xx} = \dfrac{\partial^2 f}{\partial x^2},\quad f_{yy} = \dfrac{\partial^2 f}{\partial y^2},\quad f_{xy} = \dfrac{\partial}{\partial y}\!\left(\dfrac{\partial f}{\partial x}\right),\quad f_{yx} = \dfrac{\partial}{\partial x}\!\left(\dfrac{\partial f}{\partial y}\right)

The two mixed partials $f_{xy}$ (by $x$ , then $y$ ) and $f_{yx}$ (the other order) are, remarkably, almost always equal.

DefinitionClairaut's theorem (equality of mixed partials)

If $f_{xy}$ and $f_{yx}$ both exist and are continuous on a region, then $f_{xy} = f_{yx}$ there. The order of differentiation does not matter.

ExampleVerifying $f_{xy} = f_{yx}$

Verify Clairaut's theorem for

f(x,y) = x^2 y + 3xy^2

.

ExampleA harmonic function (Laplace's equation)

Show

u(x,y) = e^{-2y}\cos(2x)

satisfies

u_{xx} + u_{yy} = 0

.

Note: A function with $u_{xx} + u_{yy} = 0$ is called harmonic; that equation is Laplace's equation, central to heat flow, electrostatics and fluid flow.

8. The total differential and error propagation

With one variable, $dy = f'(x)\,dx$ . With two variables, a small change in $f$ comes from both inputs moving, so the two contributions add:

DefinitionTotal differential

For $z = f(x,y)$ , the total differential is $dz = \dfrac{\partial f}{\partial x}\,dx + \dfrac{\partial f}{\partial y}\,dy = f_x\,dx + f_y\,dy$ . It gives the tangent-plane (linear) approximation $\Delta z \approx f_x(a,b)\,\Delta x + f_y(a,b)\,\Delta y$ .

ExampleApproximating a change

For

f(x,y) = x^2 + y^2

, estimate the change as

(x,y)

moves from

(3,4)

to

(3.1,\,3.9)

.

ExampleTotal differential of a 3-variable function

Find

dw

for

w(x,y,z) = x^2 y + y^2 z + z^2 x

.

ExamplePercentage error in a product

The dimensions of a rectangle are measured as length

x = 10

and width

y = 5

, each with a possible error of

1\%

. Estimate the percentage error in the area

A = xy

.

9. Chain rule for partial derivatives

When $x$ and $y$ are themselves functions of another variable $t$ , then $w = f(x,y)$ depends only on $t$ . Its rate of change combines both channels through which $t$ influences $w$ :

DefinitionChain rule (one parameter)

If $w = f(x,y)$ with $x = x(t)$ and $y = y(t)$ differentiable, then $\dfrac{dw}{dt} = \dfrac{\partial w}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial w}{\partial y}\dfrac{dy}{dt}$ .

ExampleChain rule along a curve

Let

g(x,y) = x^2 - yx

, with

x(t) = e^t

and

y(t) = t^2

. Find

\dfrac{dg}{dt}

.

When $x$ and $y$ depend on two variables $s$ and $t$ , the same idea gives partial chain rules:

\dfrac{\partial w}{\partial s} = \dfrac{\partial w}{\partial x}\dfrac{\partial x}{\partial s} + \dfrac{\partial w}{\partial y}\dfrac{\partial y}{\partial s},\qquad \dfrac{\partial w}{\partial t} = \dfrac{\partial w}{\partial x}\dfrac{\partial x}{\partial t} + \dfrac{\partial w}{\partial y}\dfrac{\partial y}{\partial t}

ExampleTwo-variable chain rule

Let

g(x,y) = 2x^2 + y

, with

x = r - s

and

y = 2r + s

. Find

\dfrac{\partial g}{\partial r}

.

10. Homogeneous functions and Euler's theorem

Many functions in geometry and physics scale in a simple way when all inputs are stretched by the same factor. Such functions are called homogeneous.

DefinitionHomogeneous function of degree $n$

A function $f(x,y)$ is homogeneous of degree $n$ if $f(tx, ty) = t^n f(x,y)$ for all admissible $t > 0$ . Equivalently, every term of a polynomial $f$ has the same total degree $n$ ; a ratio is homogeneous of degree (numerator degree $-$ denominator degree).

For instance $f(x,y) = x^3 + x^2 y + y^3$ is homogeneous of degree $3$ , and $f(x,y) = \dfrac{x^2 + y^2}{x + y}$ is homogeneous of degree $1$ . But $e^{x} + y$ is not homogeneous of any degree.

DefinitionEuler's theorem

If $f(x,y)$ is homogeneous of degree $n$ with continuous partials, then $x\,f_x + y\,f_y = n f$ . Differentiating once more gives the second-order form $x^2 f_{xx} + 2xy\,f_{xy} + y^2 f_{yy} = n(n-1)f$ .

Quick facts

Read the degree by summing exponents in each term (they must all match).
Euler's relation $x f_x + y f_y = n f$ is a fast check and a shortcut in many proofs.
If $u$ is not homogeneous but $\phi(u)$ is (of degree $n$ ), apply Euler to $\phi(u)$ and then substitute back.

ExampleVerifying Euler's theorem

Verify Euler's theorem for

f(x,y) = x^3 + x^2 y + y^3

.

ExampleEuler via a substitution

If

u = \sin^{-1}\!\left(\dfrac{x + y}{\sqrt{x} + \sqrt{y}}\right)

, show that

x u_x + y u_y = \tfrac12 \tan u

.

Key idea: In the Visualize tab, fix a point $(a,b)$ and read $f_x$ as the tangent slope of the $x$-slice and $f_y$ as the tangent slope of the $y$-slice — the two numbers that Euler's theorem combines.