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Chapter 4 of 6

Calculus

In the UPSC ESE Civil syllabus under Engineering Mathematics · 3 parts

Differential Calculus — Limits, Continuity, Derivatives & Mean Value Theorems · Maxima & Minima · Integral Calculus

CIVILGYAN TEST SERIES · civilgyantests.com — free study notes for civil engineering exams

📑 Contents (25 sections)

Part 1 of 3

Differential Calculus — Limits, Continuity, Derivatives & Mean Value Theorems

Last reviewed 16 Sept 2026 · 7 min read

Limits

$lim_{x \to a} f (x) = L$ means $f (x)$ approaches $L$ as $x$ approaches $a$ (from both sides). The limit exists iff the left-hand limit (LHL) and right-hand limit (RHL) exist and are equal.

Standard limits

Limit	Value
$lim_{x \to 0} \frac{sin x}{x}$	1 ( $x$ in radians)
$lim_{x \to 0} \frac{tan x}{x}$	1
$lim_{x \to 0} \frac{1 - cos x}{x ^{2}}$	1/2
$lim_{x \to 0} \frac{e ^{x} - 1}{x}$	1
$lim_{x \to 0} \frac{a ^{x} - 1}{x}$	$ln a$
$lim_{x \to 0} \frac{ln ( 1 + x )}{x}$	1
$lim_{x \to a} \frac{x ^{n} - a ^{n}}{x - a}$	$n a^{n - 1}$
$lim_{x \to \infty} (1 + \frac{1}{x})^{x}$	e
$lim_{x \to 0} (1 + x)^{1/ x}$	e
$lim_{x \to \infty} (1 + \frac{a}{x})^{x}$	$e^{a}$

Indeterminate forms and L'Hôpital's rule

Forms $\frac{0}{0}$ , $\frac{\infty}{\infty}$ , $0 \cdot \infty$ , $\infty - \infty$ , $1^{\infty}$ , $0^{0}$ , $\infty^{0}$ .

L'Hôpital's rule: for $\frac{0}{0}$ or $\frac{\infty}{\infty}$ ,

lim \frac{f ( x )}{g ( x )} = lim \frac{f ^{'} ( x )}{g ^{'} ( x )}

(repeat if still indeterminate). Other forms are converted: $0 \cdot \infty$ to a quotient; $1^{\infty}$ , $0^{0}$ , $\infty^{0}$ by taking logarithms. For $1^{\infty}$ : $lim f^{g} = e^{l i m g (f - 1)}$ .

Continuity

$f$ is continuous at $x = a$ if $lim_{x \to a} f (x) = f (a)$ (limit exists, function defined, and they are equal).

Discontinuity	Description
Removable	Limit exists but ≠ $f (a)$ or $f (a)$ undefined
Jump	LHL and RHL exist but are unequal
Infinite	Function tends to ±∞
Oscillatory	e.g. $sin (1/ x)$ at 0

Polynomials, $e^{x}$ , $sin x$ , $cos x$ are continuous everywhere; rational functions except where the denominator is zero.

Differentiability

f^{'} (a) = h \to 0 lim \frac{f ( a + h ) - f ( a )}{h}

exists (left and right derivatives equal).

Differentiable ⇒ continuous, but continuous does not imply differentiable — e.g. $f (x) = ∣ x ∣$ is continuous at 0 but not differentiable (corner: left derivative −1, right derivative +1).

Derivatives

Rules

$(u \pm v)^{'} = u^{'} \pm v^{'}$ ; $(c u)^{'} = c u^{'}$
Product: $(uv)^{'} = u^{'} v + u v^{'}$
Quotient: $(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$
Chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$

Standard derivatives

$f (x)$	$f^{'} (x)$	$f (x)$	$f^{'} (x)$
$x^{n}$	$n x^{n - 1}$	$sin x$	$cos x$
$e^{x}$	$e^{x}$	$cos x$	$- sin x$
$a^{x}$	$a^{x} ln a$	$tan x$	$sec^{2} x$
$ln x$	$1/ x$	$sec x$	$sec x tan x$
$lo g_{a} x$	$\frac{1}{x ln a}$	$csc x$	$- csc x cot x$
$sin^{- 1} x$	$\frac{1}{1 - x ^{2}}$	$cot x$	$- csc^{2} x$
$cos^{- 1} x$	$\frac{- 1}{1 - x ^{2}}$	$sinh x$	$cosh x$
$tan^{- 1} x$	$\frac{1}{1 + x ^{2}}$	$cosh x$	$sinh x$

Special techniques

Implicit differentiation — differentiate both sides treating $y$ as a function of $x$ .
Parametric: $x = x (t)$ , $y = y (t)$ → $\frac{d y}{d x} = \frac{d y / d t}{d x / d t}$ ; $\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d t} (\frac{d y}{d x}) / \frac{d x}{d t}$ .
Logarithmic differentiation — for $u^{v}$ or long products: take $ln$ first.

Leibniz theorem (nth derivative of a product)

(uv)_{n} = r = 0 \sum n^{n} C_{r} u_{n - r} v_{r}

Useful nth derivatives: $(e^{a x})_{n} = a^{n} e^{a x}$ ; $(sin a x)_{n} = a^{n} sin (a x + \frac{nπ}{2})$ ; $(\frac{1}{x + a})_{n} = \frac{( - 1 ) ^{n} n !}{( x + a ) ^{n + 1}}$ .

Part 2 of 3

Maxima & Minima

Last reviewed 16 Sept 2026 · 7 min read

Increasing and decreasing functions

$f^{'} (x) > 0$ on an interval → $f$ is increasing; $f^{'} (x) < 0$ → decreasing.
Critical (stationary) points — where $f^{'} (x) = 0$ (or $f^{'}$ does not exist).

Local and absolute extrema

Local (relative) maximum at $c$ : $f (c) \geq f (x)$ for $x$ near $c$ ; local minimum similarly.
Absolute (global) maximum/minimum — largest/smallest value over the whole domain.

First derivative test

At a critical point $c$ :

$f^{'}$ changes from + to − → local maximum.
$f^{'}$ changes from − to + → local minimum.
No sign change → neither (possible inflection).

Second derivative test

At $c$ with $f^{'} (c) = 0$ :

$f^{''} (c) < 0$ → local maximum.
$f^{''} (c) > 0$ → local minimum.
$f^{''} (c) = 0$ → test fails — use the first derivative test or higher derivatives (if the first non-zero derivative at $c$ is of even order, extremum: max if negative, min if positive; if odd order, inflection point).

Concavity and inflection

$f^{''} > 0$ → concave upward (convex); $f^{''} < 0$ → concave downward.
Point of inflection — where concavity changes ( $f^{''} = 0$ and changes sign).

Absolute extrema on a closed interval [a, b]

Evaluate $f$ at all critical points in (a, b) and at the end points a and b; the largest is the absolute maximum, the smallest the absolute minimum.

Functions of two variables

For $z = f (x, y)$ :

Stationary points: solve $f_{x} = 0$ and $f_{y} = 0$ .
At each point compute $r = f_{xx}$ , $s = f_{x y}$ , $t = f_{y y}$ .

FormulaSecond derivative test (two variables)

Condition	Nature
$r t - s^{2} > 0$ and $r < 0$	Maximum
$r t - s^{2} > 0$ and $r > 0$	Minimum
$r t - s^{2} < 0$	Saddle point (neither)
$r t - s^{2} = 0$	Inconclusive — further investigation

Lagrange multipliers (constrained optimisation)

To find extrema of $f (x, y, z)$ subject to $ϕ (x, y, z) = 0$ :

Form $F = f + λ ϕ$ .
Solve $F_{x} = 0$ , $F_{y} = 0$ , $F_{z} = 0$ and $ϕ = 0$ for $x, y, z, λ$ .
Evaluate $f$ at the solutions to identify maxima/minima (by physical reasoning or further tests).

(Equivalent condition: $\nabla f = - λ \nabla ϕ$ — gradients are parallel at the optimum.)

Engineering applications

Problem	Result
Rectangle of given perimeter with maximum area	Square
Rectangle of given area with minimum perimeter	Square
Open box from a square sheet of side $a$ by cutting squares of side $x$ from corners	Maximum volume at $x = a /6$ , $V_{ma x} = 2 a^{3} /27$
Cylinder of given volume with minimum total surface area (closed)	Height = diameter
Strongest rectangular beam (maximum section modulus $b d^{2} /6$ ) cut from a circular log of diameter $D$	$b = D / 3$ , $d = D 2/3$ → $d / b = 2$
Stiffest rectangular beam (maximum $b d^{3}$ ) from a log	$d / b = 3$
Most economical rectangular channel (max flow for given area)	Width = 2 × depth (see Open Channel Flow)
Bending moment maximum	Where shear force = 0 ( $d M / d x = V = 0$ )

Worked examples

Worked ExampleExample 1 — local extrema

Find the local maxima and minima of $f (x) = x^{3} - 6 x^{2} + 9 x + 1$ .

Solution. $f^{'} (x) = 3 x^{2} - 12 x + 9 = 3 (x - 1) (x - 3)$ → critical points $x = 1, 3$ $f^{''} (x) = 6 x - 12$ : $f^{''} (1) = - 6 < 0$ → maximum $f (1) = 5$ ; $f^{''} (3) = 6 > 0$ → minimum $f (3) = 1$ Inflection at $x = 2$ ( $f^{''} = 0$ , sign change).

Worked ExampleExample 2 — absolute extrema on an interval

Find the absolute extrema of the same function on $[0, 4]$ .

Solution. $f (0) = 1$ , $f (1) = 5$ , $f (3) = 1$ , $f (4) = 64 - 96 + 36 + 1 = 5$ Absolute maximum 5 (at $x = 1$ and $x = 4$ ); absolute minimum 1 (at $x = 0$ and $x = 3$ )

Worked ExampleExample 3 — two variables

Find and classify the stationary points of $f (x, y) = x^{3} + y^{3} - 3 x y$ .

Solution. $f_{x} = 3 x^{2} - 3 y = 0$ , $f_{y} = 3 y^{2} - 3 x = 0$ → $y = x^{2}$ , $x = y^{2}$ → points (0, 0) and (1, 1) $r = 6 x$ , $s = - 3$ , $t = 6 y$ At (0, 0): $r t - s^{2} = - 9 < 0$ → saddle point At (1, 1): $r t - s^{2} = 36 - 9 = 27 > 0$ , $r = 6 > 0$ → minimum, $f = - 1$

Worked ExampleExample 4 — simple minimum

Find the minimum of $f (x, y) = x^{2} + y^{2} - 4 x + 6 y + 5$ .

Solution. $f_{x} = 2 x - 4 = 0$ , $f_{y} = 2 y + 6 = 0$ → (2, −3); $r = 2$ , $t = 2$ , $s = 0$ → $r t - s^{2} = 4 > 0$ , $r > 0$ → minimum $f = 4 + 9 - 8 - 18 + 5 =$ −8

Worked ExampleExample 5 — open box

Squares are cut from the corners of a 12 cm × 12 cm sheet to make an open box. Find the maximum volume.

Solution. $V = x (12 - 2 x)^{2}$ → $V^{'} = (12 - 2 x) (12 - 6 x) = 0$ → $x = 2$ (x = 6 gives zero volume) $V_{ma x} = 2 \times 8^{2} =$ 128 cm³ (check $2 a^{3} /27 = 2 \times 1728/27 = 128$ ✓)

Worked ExampleExample 6 — Lagrange multipliers

Maximise $x y$ subject to $x + y = 10$ .

Solution. $F = x y + λ (x + y - 10)$ : $F_{x} = y + λ = 0$ , $F_{y} = x + λ = 0$ → $x = y = 5$ → maximum 25

Worked ExampleExample 7 — strongest beam from a log

A rectangular beam is cut from a log of 300 mm diameter. Find $b$ and $d$ for maximum bending strength.

Solution. Maximise $S = b d^{2} = b (D^{2} - b^{2})$ : $d S / d b = D^{2} - 3 b^{2} = 0$ → $b = D / 3 =$ 173.2 mm, $d = D^{2} - b^{2} = 90000 - 30000 =$ 244.9 mm ( $d / b = 2$ )

Worked ExampleExample 8 — minimum surface cylinder

A closed cylindrical tank must hold 1000 m³. Find the dimensions for minimum surface area.

Solution. $A = 2 π r^{2} + 2 π r h$ with $π r^{2} h = 1000$ → $A = 2 π r^{2} + \frac{2000}{r}$ → $A^{'} = 4 π r - \frac{2000}{r ^{2}} = 0$ → $r^{3} = \frac{500}{π}$ → $r =$ 5.42 m; $h = \frac{1000}{π r ^{2}} =$ 10.84 m ( $h = 2 r$ )

Part 3 of 3

Integral Calculus

Last reviewed 16 Sept 2026 · 6 min read

Indefinite integrals

$\int f (x) d x = F (x) + C$ where $F^{'} (x) = f (x)$ .

Standard integrals

$f (x)$	$\int f (x) d x$
$x^{n}$ ( $n \neq = - 1$ )	$\frac{x ^{n + 1}}{n + 1}$
$\frac{1}{x}$	$ln ∣ x ∣$
$e^{a x}$	$\frac{e ^{a x}}{a}$
$a^{x}$	$\frac{a ^{x}}{ln a}$
$sin x$ / $cos x$	$- cos x$ / $sin x$
$sec^{2} x$ / $csc^{2} x$	$tan x$ / $- cot x$
$tan x$	$ln ∣ sec x ∣$
$sec x$	$ln ∣ sec x + tan x ∣$
$\frac{1}{a ^{2} + x ^{2}}$	$\frac{1}{a} tan^{- 1} \frac{x}{a}$
$\frac{1}{a ^{2} - x ^{2}}$	$sin^{- 1} \frac{x}{a}$
$\frac{1}{x ^{2} - a ^{2}}$	$\frac{1}{2 a} ln \frac{x - a}{x + a}$
$\frac{1}{x ^{2} + a ^{2}}$	$ln x + x^{2} + a^{2}$
$a^{2} - x^{2}$	$\frac{x}{2} a^{2} - x^{2} + \frac{a ^{2}}{2} sin^{- 1} \frac{x}{a}$
$e^{a x} sin b x$	$\frac{e ^{a x}}{a ^{2} + b ^{2}} (a sin b x - b cos b x)$

Methods of integration

Substitution

Replace a function and its derivative: $\int f (g (x)) g^{'} (x) d x = \int f (u) d u$ . In particular $\int \frac{f ^{'} ( x )}{f ( x )} d x = ln ∣ f (x) ∣$ .

Integration by parts

\int u v d x = u \int v d x - \int (u^{'} \int v d x) d x

Choose $u$ in the order ILATE — Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential.

Partial fractions

Rational functions $\frac{P ( x )}{Q ( x )}$ (proper) are split: linear factors → $\frac{A}{x - a}$ ; repeated → $\frac{A}{x - a} + \frac{B}{( x - a ) ^{2}}$ ; quadratic factors → $\frac{A x + B}{x ^{2} + p x + q}$ .

Trigonometric integrals

Use identities: $sin^{2} x = \frac{1 - cos 2 x}{2}$ , $cos^{2} x = \frac{1 + cos 2 x}{2}$ , product-to-sum formulas, and substitutions $t = tan (x /2)$ for rational functions of $sin x$ , $cos x$ .

Definite integrals

Fundamental theorem of calculus: $\int_{a}^{b} f (x) d x = F (b) - F (a)$ .

Properties

$\int_{a}^{b} f d x = - \int_{b}^{a} f d x$ ; $\int_{a}^{a} f d x = 0$ .
$\int_{a}^{b} f d x = \int_{a}^{c} f d x + \int_{c}^{b} f d x$ .
$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$ — very useful.
$\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ if $f$ is even; $= 0$ if $f$ is odd.
$\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ if $f (2 a - x) = f (x)$ ; $= 0$ if $f (2 a - x) = - f (x)$ .
For a periodic function with period $T$ : $\int_{0}^{n T} f d x = n \int_{0}^{T} f d x$ .

Wallis' formulas

FormulaWallis' formulas

\int_{0}^{π /2} sin^{n} x d x = \int_{0}^{π /2} cos^{n} x d x = \frac{( n - 1 ) ( n - 3 ) \dots}{n ( n - 2 ) \dots} \times k

$k = \frac{π}{2}$ if $n$ is even; $k = 1$ if $n$ is odd.

\int_{0}^{π /2} sin^{m} x cos^{n} x d x = \frac{[( m - 1 ) ( m - 3 ) \dots ] [( n - 1 ) ( n - 3 ) \dots ]}{( m + n ) ( m + n - 2 ) \dots} \times k

$k = \frac{π}{2}$ if both $m$ and $n$ are even; otherwise $k = 1$ .

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