Calculus

1. Limits

The concept of limit is the foundation of calculus. For a function f(x), we say lim_x→a f(x) = L if f(x) gets arbitrarily close to L when x is sufficiently close to a (but not equal to a). Limits are used to define derivatives, continuity, and integrals.

Basic Limits & Indeterminate Forms

When evaluating limits, we may encounter forms that are not immediately determinate. These require algebraic manipulation, standard limit formulas, or L'Hopital's Rule.

Indeterminate Forms

The following seven forms are indeterminate. Further analysis is required to evaluate limits that produce these forms. When you directly substitute the limiting value and obtain one of these forms, it means the limit cannot be determined without additional work.

Rationalization Method

When the limit involves square roots, rationalizing the numerator or denominator often resolves the 0/0 form. Multiply by the conjugate to eliminate the radical.

Indeterminate Form	Example	Technique
0/0	limx→0 (sin x)/x	Factorisation, standard limits, L'Hopital
∞/∞	limx→∞ ex/x	Divide by highest power, L'Hopital
0 × ∞	limx→0+ x ln x	Rewrite as 0/0 or ∞/∞
∞ − ∞	limx→0 (1/x − 1/sin x)	Combine fractions, rationalise
00	limx→0+ xx	Take ln, use L'Hopital
∞0	limx→∞ x1/x	Take ln, use L'Hopital
1∞	limx→0 (1 + x)1/x	Use standard exponential limit

Standard Limits

Memorising standard limits is essential for quick evaluation in NDA exams. The following tables list the most frequently tested limits.

Trigonometric Limits

Limit Formula	Result
limx→0 sin x / x	1
limx→0 tan x / x	1
limx→0 (1 − cos x) / x2	1/2
limx→0 sin−1 x / x	1
limx→0 tan−1 x / x	1
limx→a (sin x − sin a) / (x − a)	cos a

Exponential & Logarithmic Limits

Limit Formula	Result
limx→0 (ex − 1) / x	1
limx→0 (ax − 1) / x	ln a
limx→0 (1 + x)1/x	e
limx→∞ (1 + 1/x)x	e
limx→0 ln(1 + x) / x	1
limx→0 (eax − 1) / x	a

Series Expansions for Limits

Standard series expansions are powerful tools for evaluating limits, especially when L'Hopital's rule becomes tedious. These expansions represent functions as infinite polynomials around x = 0.

Function	Series Expansion (Maclaurin)
ex	1 + x + x2/2! + x3/3! + ...
sin x	x − x3/3! + x5/5! − ...
cos x	1 − x2/2! + x4/4! − ...
tan x	x + x3/3 + 2x5/15 + ...
ln(1 + x)	x − x2/2 + x3/3 − x4/4 + ...
(1 + x)n	1 + nx + n(n−1)x2/2! + ...
sin−1 x	x + x3/6 + 3x5/40 + ...
tan−1 x	x − x3/3 + x5/5 − ...

L'Hopital's Rule

If lim_x→a f(x)/g(x) gives 0/0 or ∞/∞, then lim_x→a f(x)/g(x) = lim_x→a f'(x)/g'(x), provided the limit exists. The rule can be applied repeatedly until a determinate form is obtained.

Condition	Action	Example
0/0 form	Differentiate numerator and denominator separately	limx→0 (ex − 1)/x = limx→0 ex/1 = 1
∞/∞ form	Differentiate numerator and denominator separately	limx→∞ x/ex = limx→∞ 1/ex = 0
0 × ∞ form	Rewrite as 0/0 or ∞/∞ first	limx→0+ x ln x = limx→0+ (ln x)/(1/x) = 0

2. Continuity & Differentiability

Continuity at a Point

A function f(x) is continuous at x = a if lim_x→a f(x) = f(a). This requires three conditions:

1. f(a) is defined (a belongs to the domain)
2. lim_x→a f(x) exists (LHL = RHL)
3. lim_x→a f(x) = f(a)

If any condition fails, f is discontinuous at x = a.

Properties of Continuous Functions

Continuous functions have several important properties that are frequently tested in NDA exams:

• The sum, difference, product, and quotient (with non-zero denominator) of continuous functions are continuous.
• Polynomials, trigonometric functions, exponential functions, and logarithmic functions are continuous on their domains.
• The composition of two continuous functions is continuous.
• If f is continuous on [a, b] and f(a) · f(b) < 0, then there exists c ∈ (a, b) such that f(c) = 0 (Intermediate Value Theorem).
• A continuous function on a closed interval attains its maximum and minimum values (Extreme Value Theorem).

Types of Discontinuities

Type	Description	Example
Removable	limx→a f(x) exists but ≠ f(a) or f(a) undefined	f(x) = (x2 − 1)/(x − 1) at x = 1
Jump	LHL and RHL exist but are not equal	f(x) = { x, x < 0; x + 1, x ≥ 0 } at x = 0
Infinite	At least one one-sided limit is ∞ or −∞	f(x) = 1/x at x = 0
Oscillatory	Limit does not exist due to oscillation	f(x) = sin(1/x) at x = 0

Differentiability

A function f is differentiable at x = a if the derivative f'(a) = lim_h→0 [f(a + h) − f(a)]/h exists (LHD = RHD). Differentiability implies continuity, but the converse is not true.

3. Differentiation

Basic Rules & Formulas

The derivative represents the rate of change. Geometrically, f'(x) is the slope of the tangent at point (x, f(x)).

Function	Derivative
xn	nxn−1
ex	ex
ax	ax ln a
ln x	1/x
loga x	1/(x ln a)
sin x	cos x
cos x	−sin x
tan x	sec2 x
sec x	sec x tan x
cosec x	−cosec x cot x
cot x	−cosec2 x
sin−1 x	1/√(1 − x2)
cos−1 x	−1/√(1 − x2)
tan−1 x	1/(1 + x2)

Chain Rule, Product Rule, Quotient Rule

Chain Rule: If y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx).

Product Rule: (uv)' = u'v + uv'.

Quotient Rule: (u/v)' = (u'v − uv')/v².

Implicit Differentiation

When y cannot be expressed explicitly as a function of x, differentiate both sides of the equation with respect to x, treating y as a function of x and using the chain rule (dy/dx appears).

Logarithmic Differentiation

Used when the function is of the form [f(x)]^g(x) or a product/quotient of many functions. Take natural log on both sides and then differentiate.

Second Order Derivatives

The second derivative f''(x) or d²y/dx² is the derivative of the first derivative. It represents the rate of change of slope and is used to determine concavity and classify critical points.

If f''(x) > 0 on an interval, the function is concave up (holds water). If f''(x) < 0, the function is concave down (spills water). Points where concavity changes are called points of inflection (f''(x) = 0).

Successive Differentiation (n^th Derivatives)

Successive differentiation involves finding higher-order derivatives. Some standard formulas for n^th derivatives are useful in NDA exams:

f(x)	f(n)(x)
xm	m(m−1)...(m−n+1) xm−n
eax	an eax
sin(ax + b)	an sin(ax + b + nπ/2)
cos(ax + b)	an cos(ax + b + nπ/2)
ln(ax + b)	(−1)n−1 (n−1)! an/(ax + b)n

4. Applications of Derivatives

Rate of Change

The derivative dy/dx gives the rate of change of y with respect to x. In NDA problems, this is often used with time as the independent variable: dy/dt = (dy/dx)(dx/dt). Related rates problems involve finding the rate at which one quantity changes with respect to time given the rate of another related quantity.

Increasing/Decreasing Functions

A function f(x) is increasing if f'(x) > 0 and decreasing if f'(x) < 0. If f'(x) = 0, the function may have a stationary point.

Tangents & Normals

Tangent: Equation at (x₀, y₀) is y − y₀ = f'(x₀)(x − x₀). The slope of the tangent gives the instantaneous rate of change at that point.

Normal: Equation is y − y₀ = −1/f'(x₀) × (x − x₀), provided f'(x₀) ≠ 0. The normal is perpendicular to the tangent (product of slopes = −1).

Lengths: The length of tangent = |y₀ √(1 + (f'(x₀))²)/f'(x₀)|. The length of normal = |y₀ √(1 + (f'(x₀))²)|. The sub-tangent = |y₀/f'(x₀)|. The sub-normal = |y₀ f'(x₀)|.

Mean Value Theorem & Rolle's Theorem

Rolle's Theorem: If f is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there exists c ∈ (a, b) such that f'(c) = 0.

Lagrange's Mean Value Theorem (LMVT): If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that f'(c) = [f(b) − f(a)]/(b − a).

Maxima & Minima

First Derivative Test: Find critical points (f'(x) = 0). If f' changes from + to −, it is a local maximum; if − to +, local minimum.

Second Derivative Test: At a critical point, if f''(x) > 0, local minima; if f''(x) < 0, local maxima; if f''(x) = 0, test fails.

Condition	Conclusion
f'(c) = 0 and f''(c) > 0	Local minimum at x = c
f'(c) = 0 and f''(c) < 0	Local maximum at x = c
f'(c) = 0 and f''(c) = 0	Test inconclusive (use first derivative test)
f'(x) > 0 on both sides of c	Point of inflection

Approximations & Differentials

Using differentials: f(x + Δx) ≈ f(x) + f'(x)Δx. This is useful for approximate calculations when Δx is small. The error in approximation is proportional to (Δx)².

Absolute Error: |Δy| ≈ |f'(x) Δx|. Relative Error: |Δy/y| ≈ |f'(x)/f(x) Δx|. Percentage Error: (|Δy/y|) × 100%.

5. Integration

Indefinite Integrals

Integration is the inverse process of differentiation. The indefinite integral ∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration. Every differentiation formula gives rise to a corresponding integration formula.

Method	When to Use	Example
Direct Formula	Standard forms (power, trig, exponential)	∫ x2 dx = x3/3 + C
Substitution	Composite functions, integrand contains f'(x)	∫ 2x cos(x2) dx
Integration by Parts	Product of two dissimilar functions	∫ x ex dx
Partial Fractions	Rational functions P(x)/Q(x)	∫ 1/(x2−4) dx
Trigonometric Identities	Powers of sin, cos, tan	∫ sin2x dx = ∫ (1−cos 2x)/2 dx

Basic Formulas

Integral	Result
∫ xn dx	xn+1/(n+1) + C, n ≠ −1
∫ (1/x) dx	ln |x| + C
∫ ex dx	ex + C
∫ ax dx	ax/ln a + C
∫ sin x dx	−cos x + C
∫ cos x dx	sin x + C
∫ sec2 x dx	tan x + C
∫ cosec2 x dx	−cot x + C
∫ sec x tan x dx	sec x + C
∫ cosec x cot x dx	−cosec x + C
∫ 1/(1 + x2) dx	tan−1 x + C
∫ 1/√(1 − x2) dx	sin−1 x + C

Substitution Method

If the integrand contains a composite function, substitute u = g(x) so that du = g'(x) dx and the integral simplifies to ∫ f(u) du. Common substitutions include u = sin x, cos x, tan x, e^x, or algebraic expressions under radicals.

Integrand Contains	Substitution	Result
√(a2 − x2)	x = a sin θ	a cos θ dθ
√(x2 + a2)	x = a tan θ	a sec2θ dθ
√(x2 − a2)	x = a sec θ	a sec θ tan θ dθ
a2 − x2 in denominator	x = a sin θ	a cos θ dθ
a2 + x2 in denominator	x = a tan θ	a sec2θ dθ
√(a − x)/(a + x)	x = a cos 2θ	Trigonometric simplification

Integration by Parts

Using the formula: ∫ u dv = uv − ∫ v du. Choose u and dv using the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) where u is chosen from the first category.

Partial Fractions

Used to integrate rational functions P(x)/Q(x). Decompose the fraction into simpler terms based on the factors of Q(x).

Factor in Denominator	Partial Fraction Form
(ax + b)	A/(ax + b)
(ax + b)n	A1/(ax+b) + A2/(ax+b)2 + ... + An/(ax+b)n
ax2 + bx + c (non-repeated)	(Ax + B)/(ax2 + bx + c)

Integration Using Trigonometric Identities

Trigonometric identities simplify integrals involving products of sine and cosine, powers of trigonometric functions, and rational trigonometric expressions.

• sin²x = (1 − cos 2x)/2, cos²x = (1 + cos 2x)/2
• sin 3x = 3 sin x − 4 sin³x, cos 3x = 4 cos³x − 3 cos x
• 2 sin A cos B = sin(A+B) + sin(A−B)
• 2 cos A cos B = cos(A+B) + cos(A−B)
• 2 sin A sin B = cos(A−B) − cos(A+B)

Integration of Special Functions

Integral	Result
∫ 1/(x2 + a2) dx	(1/a) tan−1(x/a) + C
∫ 1/(x2 − a2) dx	(1/2a) ln |(x−a)/(x+a)| + C
∫ 1/√(a2 − x2) dx	sin−1(x/a) + C
∫ 1/√(x2 + a2) dx	sinh−1(x/a) + C
∫ 1/√(x2 − a2) dx	ln |x + √(x2 − a2)| + C
∫ √(a2 − x2) dx	(x/2)√(a2−x2) + (a2/2) sin−1(x/a) + C
∫ tan x dx	ln |sec x| + C
∫ cot x dx	ln |sin x| + C
∫ sec x dx	ln |sec x + tan x| + C
∫ cosec x dx	ln |cosec x − cot x| + C
∫ eax sin(bx) dx	eax(a sin bx − b cos bx)/(a2 + b2) + C

Reduction Formulas (for NDA Advanced Problems)

Reduction formulas express an integral involving powers of functions in terms of lower powers. These are useful for integrals like ∫ sinⁿx dx and ∫ cosⁿx dx.

Reduction Formula	Result
∫ sinnx dx	−(1/n) sinn−1x cos x + ((n−1)/n) ∫ sinn−2x dx
∫ cosnx dx	(1/n) cosn−1x sin x + ((n−1)/n) ∫ cosn−2x dx
∫ tannx dx	tann−1x/(n−1) − ∫ tann−2x dx
∫ secnx dx	(1/(n−1)) secn−2x tan x + ((n−2)/(n−1)) ∫ secn−2x dx

Definite Integrals

The definite integral ∫_a^b f(x) dx = F(b) − F(a), where F'(x) = f(x). It represents the signed area under the curve y = f(x) from x = a to x = b.

Properties of Definite Integrals

Property	Formula
Reversal of limits	∫ab f(x) dx = −∫ba f(x) dx
Zero width	∫aa f(x) dx = 0
Additivity	∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx
King's property	∫ab f(x) dx = ∫ab f(a + b − x) dx
Odd function	∫−aa f(x) dx = 0 if f is odd
Even function	∫−aa f(x) dx = 2 ∫0a f(x) dx if f is even
Periodic function	∫0nT f(x) dx = n ∫0T f(x) dx if period = T
I = ∫ab f(x) dx = ∫ab f(a+b−x) dx	Adding: 2I = ∫ab [f(x)+f(a+b−x)] dx

Inequalities & Estimation of Definite Integrals

Several important inequalities help estimate definite integrals without evaluating them exactly:

• If m ≤ f(x) ≤ M on [a, b], then m(b − a) ≤ ∫_a^b f(x) dx ≤ M(b − a).
• Cauchy-Schwarz Inequality: (∫_a^b f(x)g(x) dx)² ≤ ∫_a^b [f(x)]² dx · ∫_a^b [g(x)]² dx.
• Triangle Inequality: |∫_a^b f(x) dx| ≤ ∫_a^b |f(x)| dx.
• Mean Value Theorem for Integrals: There exists c ∈ [a, b] such that ∫_a^b f(x) dx = f(c)(b − a).

Area Under Curves

The area bounded by y = f(x), x-axis, x = a, x = b is given by ∫_a^b |f(x)| dx. The area between two curves y = f(x) and y = g(x) is ∫_a^b |f(x) − g(x)| dx.

6. Differential Equations

Order & Degree

Order is the highest derivative present. Degree is the power of the highest derivative after removing radicals and fractions. A differential equation is linear if the dependent variable and its derivatives appear in the first degree. If the DE involves products or powers of the dependent variable or its derivatives, it is non-linear.

Variable Separable

A DE of the form dy/dx = f(x)g(y) can be solved by separating variables: ∫ dy/g(y) = ∫ f(x) dx.

Homogeneous DE

A DE of the form dy/dx = F(y/x) is homogeneous. Substitute y = vx, then dy/dx = v + x(dv/dx) to convert it to a variable separable equation.

Linear DE

A first-order linear DE is of the form dy/dx + P(x)y = Q(x). The integrating factor (IF) is e^{∫ P dx}. The solution is y(IF) = ∫ Q(IF) dx + C. Linear DEs appear frequently in NDA exams and can be identified by the dependent variable y and its derivative dy/dx appearing only in the first degree without any product terms.

Type of DE	Standard Form	Solution Method	Example
Variable Separable	dy/dx = f(x)g(y)	∫ dy/g(y) = ∫ f(x) dx	dy/dx = xy
Homogeneous	dy/dx = F(y/x)	y = vx, separate variables	(x+y) dx = x dy
Linear (first order)	dy/dx + Py = Q	IF = e∫ P dx	dy/dx + y = x
Bernoulli	dy/dx + Py = Qyn	Substitute v = y1−n	dy/dx + y = xy2
Exact	M dx + N dy = 0	∂M/∂y = ∂N/∂x	(y)dx + (x)dy = 0

Bernoulli's Differential Equation

A Bernoulli DE has the form dy/dx + P(x)y = Q(x)yⁿ where n ≠ 0, 1. To solve, substitute v = y¹⁻ⁿ. This reduces it to a linear DE in v.

Exact Differential Equations

A DE of the form M(x,y) dx + N(x,y) dy = 0 is exact if ∂M/∂y = ∂N/∂x. The solution is ∫ M dx + ∫ (terms of N not containing x) dy = C.

Applications of Differential Equations

Differential equations model real-world phenomena including growth, decay, cooling, and motion. In NDA exams, application problems often involve Newton's Law of Cooling, population growth, and radioactive decay.