Skills in Mathematics - Differential Calculus for JEE Main and Advanced

2021·7.65 MB·english

by  Amit M Agarwal

Di�erential
Calculus
With Sessionwise Theory & Exercises

Di�erential
Calculus
With Sessionwise Theory & Exercises
Amit M. Agarwal
ARIHANT PRAKASHAN (Series), MEERUT
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P R E FAC E
“YOU CAN DO ANYTHING IF YOU SET YOUR MIND TO IT, I TEACH CALCULUS
TO JEE ASPIRANTS BUT BELIEVE THE MOST IMPORTANT FORMULA IS
COURAGE + DREAMS = SUCCESS”
Iit is a matter of great pride and honour for me to have received such an overwhelming response to
the previous editions of this book from the readers. In a way, this has inspired me to revise this book
thoroughly as per the changed pattern of JEE Main & Advanced. I have tried to make the contents
more relevant as per the needs of students, many topics have been re-written, a lot of new problems
of new types have been added in etcetc. All possible efforts are made to remove all the printing
errors that had crept in previous editions. The book is now in such a shape that the students would
feel at ease while going through the problems, which will in turn clear their concepts too.
A Summary of changes that have been made in Revised & Enlarged Edition
— Theory has been completely updated so as to accommodate all the changes made in JEE Syllabus &
Pattern in recent years.
— The most important point about this new edition is, now the whole text matter of each chapter has
been divided into small sessions with exercise in each session. In this way the reader will be able to go
through the whole chapter in a systematic way.
— Just after completion of theory, Solved Examples of all JEE types have been given, providing the
students a complete understanding of all the formats of JEE questions & the level of difficulty of
questions generally asked in JEE.
— Along with exercises given with each session, a complete cumulative exercises have been given at the
end of each chapter so as to give the students complete practice for JEE along with the assessment of
knowledge that they have gained with the study of the chapter.
— Last 10 Years questions asked in JEE Main & Adv, IIT-JEE & AIEEE have been covered in all
the chapters.
However I have made the best efforts and put my all calculus teaching experience in revising this
book. Still I am looking forward to get the valuable suggestions and criticism from my own
fraternity i.e. the fraternity of JEE teachers.
I would also like to motivate the students to send their suggestions or the changes that they want
to be incorporated in this book. All the suggestions given by you all will be kept in prime focus at
the time of next revision of the book.
Amit M. Agarwal
CONTENTS
1. ESSENTIAL MATHEMATICAL TOOLS 1-24
LEARNING PART Session 4
Session 1 — Quadratic Expression
— Basic Definitions — Non-Negative Function
Session 2
PRACTICE PART
— Intervals
— Chapter Exercises
— Modulus or Absolute Value Function
Session 3
— Number Line Rule
— Wavy Curve Method
2. DIFFERENTIATION 25-96
LEARNING PART Session 6
Session 1 — Logarithmic Differentiation
— Geometrical Meaning of the Derivative — Differentiation of Infinite Series
— Derivative of f (x) from the First Principle or ab- Session 7
Initio Method
— Differentiation of a Function w.r.t. Another
— Rules of Differentiation Function
Session 2 Session 8
— Chain Rule — Higher Derivatives of a Function
Session 3 Session 9
— Differentiation of Implicit Functions — Differentiation of a function given in the form
Session 4 of a Determinant
— Differentiation of Inverse Trigonometric Session 10
Functions — Derivation form inverse function
— Graphical Approach for Differentiation of
Inverse PRACTICE PART
— JEE Type Examples
Session 5
— Chapter Exercises
— Differentiation of a function in
Parametric Form
3. FUNCTIONS 97-200
LEARNING PART Session 7
Session 1 — Periodic Functions
— Functions
Session 8
Session 2 — Mapping Functions
— Domain
Session 9
— Algebraic Functions
— Identical Functions
Session 3
Session 10
— Transcendental Functions
— Composite Functions
Session 4
Session 11
— Piecewise Functions
— Inverse of a Function
Session 5
Session 12
— Range
— Miscellaneous Problems of Functions
Session 6
PRACTICE PART
— Odd and Even Functions
— JEE Type Examples
— Chapter Exercises
4. GRAPHICAL TRANSFORMATIONS 201-244
LEARNING PART — f (x) Transforms to ïf (ïxï), i.e.
Session 1 f (x) ®ïf (ïxï)ï
— When f (x) Vertically transforms to f (x) ± a — To draw the graph of ïyï=f (x) when
y=f (x) is given
Session 2
Session 4
— When f (x) horizontally transforms to
f (x±a), where a is any positive constant — To draw y=f ([x]) when the graph of
— When f (x) transforms to a f (x) or — 1 f (x) y=f (x) is given
a
— When f (x) transforms to f (ax) or f (x/a) — When f (x) transforms to y =f ({x}) where
— When f (x) Transforms to f (–x) {×} denotes fractional part of x, i.e. {x} =
x–[x] or f (x)® f ({x})
— When f (x) Transforms to –f (x)
— When f (x) and g (x) are two functions and
Session 3
are transformed to their sum
— To draw y =÷ f (x)÷ , when y = f (x) is given 1
— When f (x) transforms to f (x)® — — =h (x)
f (x)
— To draw y = f (÷ x÷), when f (x) is given
— To draw the graph for y=f (x) × sin x when PRACTICE PART
graph of y = f (x) is given — JEE Type Examples
— When f (x) and g (x) are given, then find h(x) — Chapter Exercises
= max (f (x), g (x)) or h(x) = min (f (x), g (x))
5. LIMITS 245-310
LEARNING PART Session 4
Session 1 — Miscellaneous Form
— Definition of Limits Session 5
— Indeterminant Form — Left Hand and Right Hand Limit
— L’Hospital’s Rule Session 6
— Evaluation of Limit — Use of Standard Theorems/Results
Session 2 — Use of Newton-Leibnitz’s Formula in Evaluating
— Trigonometric Limits the Limit
Session 3
PRACTICE PART
— Logarithmic Limits
— JEE Type Examples
— Exponential Limits
— Chapter Exercises
6. CONTINUITY AND DIFFERENTIABILITY 311-396
LEARNING PART Session 5
Session 1 — Intermediate Value Theorem
— Continuous Function Session 6
— Continuity of a Function at a Point — Differentiability : Meaning of Derivative
Session 2 Session 7
— Continuity in an Interval or Continuity at — Differentiability in an Interval
End Points
PRACTICE PART
Session 3
— Discontinuity of a Function — JEE Type Examples
— Chapter Exercises
Session 4
— Theorems on Continuity
— Continuity of Composite Function
7. dy/dx AS A RATE MEASURER & TANGENTS, NORMALS 397-458
LEARNING PART Session 4
Session 1 — Angle of Intersection of Two Curves
— Derivative as the Rate of Change — Length of Tangent, Subtangent, Normal and
Subnormal
— Velocity and Acceleration in Rectilinear Motion
Session 5
Session 2
— Rolle’s Theorem
— Differential and Approximation
— Lagrange’s Mean Value Theorem
— Geometrical Meaning of Dx, Dy, dx and dt
Session 6
Session 3
— Application of Cubic Functions
— Slope of Tangent and Normal
— Equation of Tangent PRACTICE PART
— Equation of Normal — JEE Type Examples
— Chapter Exercises
8. MONOTONICITY, MAXIMA AND MINIMA 459-550
LEARNING PART — Methods of Finding Extrema of Continuous
Functions
Session 1
— Convexity/Concavity and Point of Inflection
— Monotonicity
— Concept of Global Maximum/Minimum
Session 2
Session 5
— Critical Points
— Maxima and Minima of Discontinuous
Session 3
Functions
— Comparison of Functions Using Calculus
PRACTICE PART
Session 4
— Introduction to Maxima and Minima — JEE Type Examples
— Chapter Exercises