GATE Calculus and Optimization formula sheet

44 formulas in 3 topics, each with when to use it. Each is explained in the topic notes.

LIMIT EXISTS IFF BOTH ONE-SIDED LIMITS AGREE

x \to a lim f (x) = L ⟺ x \to a^{-} lim f (x) = x \to a^{+} lim f (x) = L

Use one-sided limits for $∣ x ∣$ , floor, piecewise functions and any function that changes formula at $a$ .

STANDARD TRIGONOMETRIC LIMITS

x \to 0 lim \frac{sin x}{x} = 1, x \to 0 lim \frac{tan x}{x} = 1, x \to 0 lim \frac{1 - cos x}{x ^{2}} = \frac{1}{2}

Only for $x$ in radians and $x \to 0$ . For $x \to \infty$ , $sin x / x \to 0$ instead.

Taylor series

All 14 formulas

TAYLOR SERIES ABOUT X = A

f (x) = n = 0 \sum \infty \frac{f ^{(n)} ( a )}{n !} (x - a)^{n} = f (a) + f^{'} (a) (x - a) + \frac{f ^{''} ( a )}{2 !} (x - a)^{2} + \dots

Valid for those $x$ where the series converges to $f (x)$ ; convergence is not automatic, so state the range of $x$ .

MACLAURIN SERIES (A = 0)

f (x) = n = 0 \sum \infty \frac{f ^{(n)} ( 0 )}{n !} x^{n}

The coefficient of $x^{n}$ is $f^{(n)} (0) / n!$ , so $f^{(n)} (0) = n! \times (coefficient of x^{n})$ .

Maxima, minima and single-variable optimisation

All 9 formulas

CRITICAL POINT

f^{'} (c) = 0 or f^{'} (c) does not exist, c interior to the domain

Necessary condition: if $f$ has a local extremum at an interior point $c$ and $f^{'} (c)$ exists, then $f^{'} (c) = 0$ . So every interior local extremum is a critical point. The converse is false: $f (x) = x^{3}$ has $f^{'} (0) = 0$ but no extremum at $0$ .

MONOTONICITY FROM THE SIGN OF THE DERIVATIVE

f^{'} (x) > 0 for all x \in I \Rightarrow f strictly increasing on I, f^{'} (x) < 0 for all x \in I \Rightarrow f strictly decreasing on I

$I$ is an interval, $f$ is continuous on $I$ and differentiable at its interior points. $f^{'} \geq 0$ on $I$ gives non-decreasing; if in addition $f^{'} = 0$ only at isolated points, $f$ is still strictly increasing. The converse of the first implication fails: $x^{3}$ is strictly increasing on $R$ although $f^{'} (0) = 0$ .

GATE Calculus and Optimization formula sheet

Limits, continuity and differentiability

Taylor series

Maxima, minima and single-variable optimisation