CALCULUS
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Syllabus

#### CALCULUS

FUNCTION & ITF

Definition of Function, Domain, Co-domain, Range, Mapping diagram, Graphical definition of function,
Rational (or Polynomial) Functions, Basic concepts, Rational inequalities, Steps to solve Rational-Inequalities.

Solving Rational-inequalities (Non-repeated and repeated linear factors), How to take square and reciprocal
in case of inequalities.

Modulus or Absolute value functions, Formulae of modulus-functions, Removal of Modulus-Functions, Graphs
of Modulus-Function, Modulus - Inequalities.

Modulus-Equations and Inequalities.

Irrational-functions, their domain and Range, Irrational Equations and inequalities, Determining domain of
irrational functions.

Irrational-Inequalities, Exponential & Logarithmic functions, their basic graphs, formulae.

Formulae of Log functions, Log and exponential equations.

Exponential and Log-inequalities when base is positive fractional or greater than one.

(a) Log-inequalities when base is variable
(b) Log-inequalities when base is variable. Determining domain of Log-functions.

Greatest integer function (GIF), Basic graph, Formulae, Fractional Part function (FPF), Basic Graph, Formulae,
Signum-function, Basic graph. Questions.

(a,b) Questions on GIF, FPF and Signum functions.

(a) Trigonometric equations, General Solutions, Fundamental and General period of Basic T-Ratios,
Questions
(b) Questions the determining General and Particular solutions of T-Equations.

(a) Questions, T-inequalities
(b) T-inequalities, Domain of T-Functions.

Inverse -trigonometric functions, condition for defining inverse of a function, classification of functions.
One-One (Injective) or many one functions, onto (Surjective) or into functions, bijective functions, Basic
Graphs of 6 inverse trigonometric - functions. Properties of ITF, Defining T (T–1(x)) or T–1 (T(x))

Finding basic values of ITF, Domain of all types of functions.

Domain of functions, Range of Functions
Method of determining Range of functions
M-1 Represent x or function of x in terms of y
M-2 Range by Using Monotonocity

M-3 Range of L / L, Q / L, L / Q,  Q / Q
M-4 Range of composite functions

Domain and Range of composite functions by defining them in one-interval or in different-different intervals.
(Using graphical method)

Composite functions in different intervals.
Types of functions: (1) one-one (injective function)
Condition of injectivity by differentiation
(2) Onto (surjective) functions.
(3) Bijective functions. Inverse of a function

Number of 1-1 mappings, number of surjective (onto) mapping, questions on classification of functions.

Questions on classification of functions and determining inverse of a function.

Inequalities of Inverse trigonometric functions, graphs of y = T (T–1 (x)) = x (Non-Periodic Functions)
Graphs of y = T–1 (T(x)) (Periodic Functions)

Graphs of y = T–1 (T(x)), Questions,
Inter-conversion between various ITF’s.

Equal or Identical functions; Simplification of Miscellaneous ITF’s, Graphs.

(a) Simplification of Miscellaneous ITF’s, Inverse-trigonometric functions of tan–1x ± tan–1y,
sin–1x ± sin–1y or cos–1x ± cos–1y, Questions
(b) Solving Inverse trigonometric equations.

Summation series of inverse-trigonometric functions, even or odd functions.

Even or odd functions, periodic functions, fundamental or general periods of basic functions, properties
related to periodicity of functions.

Determining the fundamental period of functions, Range by period of function, functional equations to
determining period

(a) Functional-Equations.
(b) Questions on functional equations,

Graphs:
Symmetry of graphs.
Transformation of curves
(G1) Graph of y = f(x) + a
(G2) Graph of y = a f (x)
(G3) Graph of y = f (x + a)
(G4) Graph of y = f (ax)
(G5) Graph of y = –f(x)
(G6) Graph of y = f (–x)
(G7) Graph of y = | f(x)|
(G8) Graph of y = f(|x|)
(G9) Graph of y = f (–|x|)
(G10) Graph of |y| = f (x)

Curve tracing using differential calculus.
Graph of maximum/minimum of functions between two or more than 2 functions.

Maximum-Minimum of a Curve, Miscellaneous graphs

LIMIT, CONTINUITY AND DIFFERENTIABILITY

(a) Concept of Limit, Left Hand Side Limit (LHL) and Right Hand Side Limit (RHL) , Algebra on limits
(b) 7 Indeterminant forms, Steps to determining limit of a function when x→a, where to evaluate LHL & RHL separately (Doubtful points)

(a) Identify type of indeterminant forms, Method of solving Limits
(i) Factorisation (ii) Rationalization
(b) Questions on factorisation and Rationalisation

(a) M-3- Evaluate of limit when x →∞ or x→ –∞
(b) Questions based on method no.3

(a) M-4- Series expansion by Maclaurin’s Series, Series Expansion of Basic functions,
(b) Determining unknown parameters by series expansion.
M-5- Standard - Limits

(a) Formulae of standard-limits, Questions based on standard limits.
(b) Standard limits using substitution method.
M-6- Limit in form of 1

(a) Questions on 1 form. L’Hospital’s rule (LH-Rule).
(b) Questions based on LH-Rule

(a) 0° or ∞° forms.
(b) Miscellaneous questions of limit

Sandwitch Theorem ( or Squeeze - Play Theorem)
Continuity of a function y = f(x) at point x = a
Types of discontinuity:
(1) First kind of discontinuity (removable discontinuity) (In this case limit exist)
(A) Missing point discontinuity.
(B) Isolated point discontinuity.
(2) Non-Removable Discontinuity (Limit does not exist)
(A) Finite Non-removable discontinuity, Jump of discontinuity = | RHL – LHL |
(B) Infinite Non-removable discontinuity.
(C) Oscillating discontinuity.
Jump of discontinuity = | RHL – LHL |

(a, b) Continuity at a point,
Continuity in an interval, determining unknown parameters using concept of continuity at a point.

(a, b) Differentiability of a function at a point, Equation of tangent at a point,
Questions to check continuity and differentiability at a point

(a) Determining unknown parameters using concepts of continuity and differentiability at a point.
Continuity and differentiability of higher order derivatives.
(b) Questions based on LH rule and differentiation.

(a, b) Differentiability in an interval, questions based to check continuity and differentiability in an interval.

(a) Graphical method to check differentiability,
Differentiability of maximum-minimum of two or more than 2 functions.
(b) Graphical method to check differentiability

(a) Determination of a function using differentiation
(b) Miscellaneous questions based on LCD.

(a, b) Miscellaneous questions based on LCD.

DIFFERENTIATION

(a) Some basic differentiation by using first principle (AB-Initio method), Rules of differentiation
(b) Formulae of differentiation, Properties of differentiation , Differentiation of Product of two functions,
Chain Rule, Differentiation of
u/v, Differentiation of composite functions,
Differentiation of Parametric functions, Differentiation of one function w.r.t. other functions.

Questions of Differentiation of functions.

(a, b) Differentiation of Log-functions.

(a) Derivative of inverse - functions.
(b) Derivative of inverse - functions by substitution method.

(a) Derivative of Inverse - Functions by substitution method
(b) Derivative of Inverse - Functions and derivative of higher order Inverse functions.
(c) Questions based on differentiation of ITFs, Parametric differentiation

(a,b) Parametric Differentiation, Differentiation of Implicit functions.

(a) Derivative of functions represented by infinite series, Differentiation of determinants.
(b) Higher order derivatives.

(a,b) Higher order derivatives.

APPLICATION OF DERIVATIVES

(a) Brief Revision of Straight Line and Tangent-Normal:
Equation of tangent and Normal to the curve y = f (x) at a point, Length of tangent,
Length of subtangent, Length of normal, Length of subnormal, Tangent to the curve at (0, 0)
(b) Questions based on concept of tangent and normal when point lies on the curve.

(a) Questions based on tangent and normal when curve given in parametric form.
(b) Tangent and normal from an external point.

(a) Questions based on tangents and normals from an external point.
(b) Tangent on the curve - intersecting the curve again.

(a) Common-tangents.
(b) Angle of intersection of two curves; shortest -distance between 2 non-intersecting curves.

(a) Rate of change
(b) Approximate value of a number, Monotonocity of a function, strictly increasing (SI),
Strictly decreasing (SD), Monotonically increasing (MI), Monotonically decreasing (MD) functions,
Monotonocity at a point and in an interval, Condition for monotonocity for differentiable functions,
Monotonocity of discontinuous functions.

(a, b) Questions on monotonicity of function at a point or in an interval.

(a) Questions of Monotonocity.
(b) Proving inequalities by using monotonocity.

(a) Concavity, Convexity and point of inflexion (POI) of curve.
(b) Curve tracing by using concept of differential calculus.

(a, b) Rolle’s theorem, Langrange’s Mean Value theorem (LMVT)

(a, b, c) Maxima and minima at a point, local maxima and local minima and absolute maxima and absolute
minima. Range of a function in an interval. Using concept of maxima and minima.

(a, b) Questions.

(a) Questions of Maxima and Minima based on location of roots.
Theory of equations using maxima and minima.
(b) Questions.
(c) Optimization of Geometrical problems by maxima and minima.

(a, b) Geometry Problems.

Geometry Problems.

INDEFINITE INTEGRATION

(a) Concept of integration, Standard formulae
(b) Defining all standard formulae.

(a, b) Basic integration directly formulae based.

(a) Substitution method; Formulae of some standard substitution.
(b) Questions based on substitution method.

(a) Integral in the form of : ∫sinm x cosn x dx ; ∫ tanm x secn x dx
(b) Integral in the form of : ∫ xm(a + bxn )dx , Questions on substitution method.

(a) Questions on substitution method in irrational functions.
(b) Questions on substitution method.

(a) Integration by parts.
(b) Integration by parts, Using
(A) ∫ex (f(x) + f '(x))dx = f(x)ex + C   OR   (B) ∫(f(x) + xf '(x))dx = xf(x) + C

(a) Questions based on integration by parts.
(b) Questions based on integration by parts, Integration of Rational function - by partial fraction method-
(i) When non-repeated linear factors in denominator
(ii) Repeated linear factors in denominator
(iii) Quadratic factors in denominator (D<0)

(a) Questions on partial fraction method
Integration in the form of : ∫ dx ÷ ax2 + bx + c

Integration in the form of : ∫ (px+q)dx ÷ ax2+bx+c

(b) Integration in the form of : ∫ (x2 ± a2)dx ÷ x4+kx2+a4 or ∫ dx ÷ x4+kx2+a4

Integration in the form of : (a) ∫ dx ÷ x(xn + 1) (b) ∫ dx ÷ xn (1+xn)1/n (c) ∫ dx ÷ x2(xn+1)n-1/n

(a) Integration of Irrational Functions
Integration in the form of : ∫ dx ÷ √ax2+bx+c OR ∫ √ax2+bx+c dx

Integration in the form of : ∫ (px+q)dx ÷ √ax2+bx+c OR ∫(px+q) √ax2+bx+c dx

(b) Integration in the form of :

(A) ∫ dx ÷ (px+q)√ax+b       (B)  ∫ dx ÷ (px2+qx+r)√ax+b

(C) ∫ dx ÷ (px+q)√ax2+bx+c (D)  ∫ dx ÷ (px2+qx+r)√ax2+bx+c

(c) Questions based on Integration of Irrational functions.
Integration in the form of : ∫ dx ÷ a + b sin2 x OR ∫ dx ÷ a + b cos2 x OR ∫ dx ÷ a cos2 x + b sin2 x OR ∫ dx ÷ a + b cos2 x + c sin2 x OR ∫ dx ÷ (a sin x + b cos x)2 OR ∫ f(tan x)dx ÷ a sin x + b sin x cos x + c cos2 x

(a) Integration in the form of : ∫ dx ÷ a + bsin x OR ∫ dx ÷ a + bcos x

∫ dx ÷ asinx ± bcos x OR ∫ dx ÷ a sinx ± b cos x + c OR ∫ (p sin x + qcos x + r) ÷ (a cos x + b sin x + c) * dx

Integration in the form of :

∫ (a sin x + b) dx ÷ (a+b sin x)2 OR ∫ (a cos x+b) dx ÷ (a+b cos x)2

Integration in the form of ∫(sinx + cos x)f(sin2x)dx

(b) Integration in the form of :

∫ f(eax )dx OR ∫ (aex + be-x ) ÷ (pex + qe-x )*dx , Reduction Formulae.

(a, b) Miscellaneous Questions

(a, b) Miscellaneous Questions

DEFINITE INTEGRATION

(a, b) Introduction of definite integral (DI), Geometrical interpretation of definite integral,
b              a
Property No. 1:  ∫ f(x)dx =- ∫ f(x)dx
a              b

b             b
Property No. 2:  ∫ f(x)dx = ∫ f(t)dt , Questions.
a             a

(a, b) Questions based on P1, P2 and Concepts of indefinite integration.

b             c          b
(a, b) Questions, property no. 3:  ∫ f(x)dx =  ∫ f(x)dx+∫ f(x)dx where a < c < b
a             b          c

b             b
Questions based on P-3, Property no. 4(King-Property): ∫ f(x)dx =  ∫ f(a+b-x)dx,
a             a

a             a
Modified property no. 4 : ∫ f(x)dx =  ∫ f(a-x)dx
0             0

Questions based on P4.

(a, b) Questions based on P4,

Questions based on P5, P6.

(a, b) Property No. 7 (Based on periodicity of function) :

nT            T
∫ f(x)dx = n ∫ f(x)dx (where T = Period of function y = f(x))
0              0

Walle’s formulae, Leibnitz theorem, Modified Leibnitz theorem.

(a) Questions based on Leibnitz theorem.
(b) Definite Integrals as the limit of a sum (AB-initio method).

Questions based on integral as Limit of a sum.

AREA UNDER THE CURVES

(a,b) Quadrature, How to evaluate area under the curve with x-axis or with y-axis, area bounded by the
two intersecting curves, area bounded by the curves in different-2 conditions.

(a, b, c) Questions based on area under the curves.

(a, b) Questions, Questions based on determining parameters.

(a, b) Questions based on determining the parameters, area under the curves using inequalities.

(a, b) Area under the curves using functional inequalities, area bounded with f(x) and its inverse f–1 (x).
Miscellaneous Questions.

DIFFERENTIAL EQUATIONS

(a, b, c) Introduction of DE, Ordinary Differential Equation (ODE) and Partial Differential Equations (PDE),
Order and degree of DE, about constants, arbitrary constants and essential arbitrary constants,
Formation of differential equations, Methods of solving differential equations.
General solutions and particular solutions of differential equations.
Method no.1 : Variable separable form, in the form of dy÷dx= f(x).g(y).

(a, b) Method no. 2: (a) Reduces to variable separable form, i.e. in the form of dy÷dx = f(ax+by+c).
(b) Substitution method: in x2 + y2 = r2 , put x = r cos θ, y = r sin θ,
and in x2 – y2 = r2 , put x = r sec θ, y = r tan θ,
Method no. 3: Solution of Homogeneous differential equations, in the form of dy÷dx = f(y÷x) or dx÷dy=f(x÷y), Questions

(a, b, c) Questions on method no. 3,
Method No. 4 :
Reduces to Homogeneous Differential equation, i.e. in the form of dy÷dx=ax+by+c÷Ax+By+k , Questions
Method no. 5 : Exact (direct) differential equations. Questions based on method no. 5. (a-25 Min., b-34

(a, b) Method no. 6 : Linear differential equation, i.e. in the form of dy÷dx+Py=Q OR dx÷dy+Px=Q Method No.7 : Reduces to linear differential equations (Bernoulli’s equations)

(a, b, c) Geometrical applications of differential equations,
Tangent and normal to the curve y = f(x) at point (x, y), length of tangent,
Length of subtangent, Length of Normal, Length of subnormal, Radius-vector,
Higher Degree & order of differential equations, orthogonal trajectory (OT) of curves,
Clairaut’s differential equations.