TN 12th Maths — Learn, Visualize, Practice

1. Linear approximation and differentials

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

To make this exact as a definition, we introduce the differentials $dx$ and $dy$ . We are free to treat $dx$ as an independent variable (the size of the step), and then $dy$ is defined to be the corresponding rise along the tangent line.

DefinitionDifferential

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

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

Differentials give a quick way to estimate small changes and to propagate measurement errors: if a quantity $x$ is measured with a small error $dx$ , the induced error in $f(x)$ is about $dy = f'(x)\,dx$ .

ExampleEstimating a root

Use differentials to approximate

\sqrt{36.6}

.

ExamplePropagating an error

The side of a cube is measured as

5\text{ cm}

with a possible error of

0.02\text{ cm}

. Estimate the error in the computed volume.

2. 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 that assigns 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.

Examples include $f(x,y) = x^2 + y^2$ (a bowl-shaped paraboloid) and $f(x,y) = xy$ (a saddle). The same ideas extend to three or more variables, though we can no longer draw the graph.

3. Limits and continuity

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.

Watch out: Path dependence: for a limit to exist, $f$ must approach the same value along every path. If two different paths give different values, the limit does not exist. For example, $f(x,y) = \dfrac{xy}{x^2 + y^2}$ tends to $0$ along the $x$-axis but to $\tfrac12$ along the line $y = x$, so it has no limit at the origin.

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; quotients are continuous wherever the denominator is non-zero.

4. 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 the limits $\dfrac{\partial f}{\partial x} = \lim_{h\to 0}\dfrac{f(x+h,\,y) - f(x,y)}{h}$ 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 $\dfrac{\partial f}{\partial x}$, differentiate as usual in $x$ while treating every $y$ as a constant. To compute $\dfrac{\partial f}{\partial y}$, treat every $x$ as constant. Geometrically, $f_x(a,b)$ is the slope of the surface in the $x$-direction at $(a,b)$ — the slope of the curve you get by slicing the surface with the plane $y = b$.

ExampleBoth partials

Find

\dfrac{\partial f}{\partial x}

and

\dfrac{\partial f}{\partial y}

for

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

.

5. 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^2 f}{\partial y\,\partial x},\quad f_{yx} = \dfrac{\partial^2 f}{\partial x\,\partial y}

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

DefinitionClairaut's theorem (equality of mixed partials)

If $f_{xy}$ and $f_{yx}$ are both continuous on a region, then they are equal there: $f_{xy} = f_{yx}$ . 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

.

6. The total differential and linear approximation

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

DefinitionTotal differential

For $z = f(x,y)$ , the total differential is $dz = \dfrac{\partial f}{\partial x}\,dx + \dfrac{\partial f}{\partial y}\,dy = f_x\,dx + f_y\,dy$ .

This gives the linear (tangent-plane) approximation to the change in $f$ when both $x$ and $y$ change by small amounts $\Delta x$ and $\Delta y$ :

\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)

.

7. 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 $t > 0$ . Equivalently, every term of $f$ has the same total degree $n$ in $x$ and $y$ .

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 $1$ .

DefinitionEuler's theorem

If $f(x,y)$ is homogeneous of degree $n$ , then $x\dfrac{\partial f}{\partial x} + y\dfrac{\partial f}{\partial y} = n f$ .

Quick facts

Degree is read off by summing the exponents in each term (they must all match).
A ratio of homogeneous functions is homogeneous of degree (numerator degree $-$ denominator degree).
Euler's relation $x f_x + y f_y = n f$ is a fast check and a shortcut in many proofs.

ExampleApplying Euler's theorem

Verify Euler's theorem for

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

.

Key idea: Open the Visualize tab to see partial derivatives directly: fix a point $(a,b)$, then read $f_x$ as the tangent slope of the $x$-slice and $f_y$ as the tangent slope of the $y$-slice.