Calculus

Syllabus: Mean value theorems, theorems of integral calculus, evaluation of definite and improper integrals, partial derivatives, maxima and minima, multiple integrals, line, surface and volume integrals, Taylor series.

9 hand-picked visual resources on this topic, 3 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Maxima and minima on an interval

Find stationary points from $f'(x)=0$ , then classify with $f''$ . On a closed interval the extremum can sit at an endpoint, so always evaluate there too — a concave function takes its minimum at an endpoint, never inside.

$f'(x)=0,\quad f''(x)<0\ \text{max},\ f''(x)>0\ \text{min}$

Where marks are lost

Finding the stationary point and stopping is the classic loss of marks — check both endpoints.

See it

▶Second partial derivative test intuitionhidden gemKhan Academy · 11 minGrant Sanderson (3Blue1Brown) on Khan Academy: why f_xx·f_yy − f_xy² decides maximum, minimum or saddle, shown on 3-D surfaces.

Partial derivatives

Differentiate in one variable while holding the others fixed. Three results do most of the work: the chain rule through intermediate variables, implicit differentiation, and Euler's theorem for homogeneous functions. For smooth functions the mixed partials agree, so $f_{xy}=f_{yx}$ .

$\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}$

Taylor and standard expansions

Worth memorising rather than deriving; most questions need only the first three terms.

$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$

Multiple integrals

Set the limits from the region, not from habit. Over a rectangle the variables separate and the integral factorises; over a triangle or circle one limit depends on the other, and polar coordinates usually shorten circular regions.

$\iint_R f\,dA=\int_{x_1}^{x_2}\!\!\int_{y_1(x)}^{y_2(x)} f\,dy\,dx$

Mean value theorems

Rolle's is the special case of Lagrange's with equal endpoint values. GATE usually uses them to bound a function given a bound on its derivative.

$f'(c)=\frac{f(b)-f(a)}{b-a}$

Improper and standard integrals

A few standard results cover most of what is asked, including the Gaussian and the sinc integral.

$\int_{-\infty}^{\infty}e^{-y^2}dy=\sqrt{\pi}$

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