Description
This book follows on naturally from the first Introduction to Calculus book that showed how the subject can be taught in a very simple and approachable way.
So, here we cover the important extensions of basic Calculus: differentiations of products, quotients, function of a function etc., and integration techniques.
Series expansions, exponentials, solution of polynomial equations and Differential Equations are also covered.
And practical applications, like Simple Harmonic Motion and Projectile Motion are included.
Details
127 + ix pages.
Size: 24.5cm by 17cm.
Paperback. 2023
Author: Kenneth Williams
ISBN 978-1-902517-46-9.
Preface
PREFACE
This book is preceded by Book 1, which is aimed at the student and gives a carefully structured step by step introduction to Calculus. Book 2 follows on from Book 1 and shows various further applications of Vedic mathematics in Calculus. It does not follow a specific curriculum or syllabus. All Calculus concepts are however consistently followed through without gaps in the exposition, from the beginning of Book 1 to the end of Book 2. So the unusual and simple approach to this subject, introduced in Book 1 leads naturally into the development of these more advanced concepts.
Vedic mathematics is based on sixteen Sutras, or formulae. These do not have to be learnt since they are expressions of natural abilities which we all possess and apply. They are in use by us all the time. They form a structured and complete set of natural mental functions and it was the genius of Sri Bharati Krishna Tirthaji to penetrate the foundations of mathematics and formulate these universal principles.^{1}
The reader should not make the mistake of thinking that these easy methods are simply tricks to be memorised by the student to give fast results in competitive examinations. The Vedic system is complete, the methods here are fully explained and proved. Whether or not the student needs to go through all the proofs, everyone still wants to have the most efficient route from problem to answer. These efficient routes are also given here.
We make extensive use of patterns. Patterns not only make procedures easy to remember and carry out but they are attractive, and therefore memorable; they indicate that some simple law is at work.
Answers to the exercises are given below the exercise. The terms ‘differential and ‘derivative’ have equivalent meanings here.
Contents
Preface iii
Illustrative Examples vi
1 Functions 1
Notation 1
Composite functions 2
Inverse functions 3
Function of a function 4
Differentiation of function of a function 4
2 Products and Quotients 6
Differentiation of products 6
Quotients 9
Further simplifications 10
Turning points 13
3. Triples 17
Definitions 17
Types of triple 18
Angle in a triple 18
Variations of 3,4,5 20
Quadrant angles 21
Multiples of 30° 22
Differentiation of triples 23
4. Projectiles – 1 25
Projectile motion and triples 25
Projectile triple: 3,4,5 26
Velocity triple 26
Acceleration triple 27
General velocity triple 27
Distance triples 27
General formulae 28
5. Projectiles – 2 32
Algebraic approach 32
Time of flight, range, height 33
Negative elements 36
6. Calculus of Trig Functions 39
Sine and cosine 39
Tangent and secant 42
Cosecant and cotangent 43
Inverse functions 45
Summary 46
7. Simple Harmonic Motion 47
Cyclic motion 47
Derivatives of trig functions as triples 47
Modelling SHM 48
Maximum and minimum values 50
Alternative system 51
Standard formulae for SHM 53
8. Exponentials and Logarithms 55
Exponential functions 55
The exponential function 55
Function of a function 57
Logarithms 58
Integral of 1/x 61
9. Differentiation and Integration of Products 62
Derivatives of a product 62
Integration of products 64
Truncating 65
10. Series Expansions 67
Binomial theorem 67
Maclaurin’s theorem 69
Trig functions 70
Inverse functions 73
Other series 74
11. Factors and Calculus 76
Factors and differentials of polynomials 76
Repeated factors 78
12. Differential Equations 82
Perfect products 82
Sum of two solutions 86
2nd order equations 87
13. Square Root 91
Squaring 91
Observations 93
Square root 93
Non-terminating square root 96
Algebraic square root 97
14. Quadratic Equations 100
Basic quadratics 100
Coefficient of x2 103
Change of Roots 104
15. Cubing and Cube Root 108
Notation 108
Strategy 109
Derivation 110
Cubing 110
Cube root 112
Algebraic cube root 115
16. Cubic Equations 116
Introduction 116
The other solutions 117
Change of roots 120
Appendix 123
References 125
Index 126
Illustrative Examples
(These can be seen better here - click on 'read sample below book image)
1) From Chapter 2 – Differentiation of a Product
Differentiate y = (2x^{2} + x + 3)(3x^{2} + 5x – 3).
We do not need to multiply the contents of the brackets. We write the answer down in one line from left to right of right to left:
2x^{2} + x + 3
3x^{2} + 5x – 3
y’ = 24x^{3 }+ 39x^{2} + 16x + 12
2) From Chapter 2 – Locating Turning Points
Locate the turning points of the graph of y = x/(1+x^2) .
We do not need to use the formula for differentiating a quotient. We differentiate numerator and denominator, equate with the given fraction and solve:
x/(1+x^2) = 1/(2x), x = ±1, giving (1,½) and (–1,–½).
Extending to a 3^{rd} fraction by differentiating numerator and denominator again we can determine which are maxima and which are minima.
3) From Chapter 5 – Projectile Motion
Given a projection triple 30,40,50, fired from a horizontal plane, find the time the projectile takes to return to the plane (take g = 10 m/s^{2}).
From the general velocity and distance triples
30, 40 – 10t, —
30t, 40t – 5t^{2} ^{, }—
we solve 40t – 5t^{2} = 0 to get t = 8s.
4) From Chapter 7 – Simple Harmonic Motion
A particle oscillates in a horizontal line about a fixed point O such that its distance, x, from O at time t is given by x = 5sin2t.
Find the velocity and acceleration when x = 3.
| — 3 5 |
We do not need to remember and apply the standard formulae. We set up the chart for this motion, with
x = 3 as shown above.
| ±4 3 5 |
Then by simple proportion (see above) of the triples and Pythagoras’ theorem we obtain xdot = +/-8 and
xdotdot= -12.
5) From Chapter 9 – Successive Derivatives of a Product
Find the 2^{nd} derivative of x^{5 }× e^{2x}.
We differentiate the terms twice and cross-multiply:
We get D_{2} = x^{5}×4e^{2x} + 2×5x^{4}×2e^{2x} + 20x^{3}×e^{2x}.
6) From Chapter 9 – Integration by ‘Parts’ in one line
Find the integral of 3x^2 * e^(2x).
We differentiate and integrate and then cross-multiply to get:
I = 3x^{2} × ½e^{2x} – 6x × ¼e^{2x} + 6 × ⅛e^{2x} + c
7) From Chapter 10 – Binomial Theorem in one line
Expand (2 + 3x)^{-3}.
We again use a combination of differentiation and integration to generate the terms in sequence, each term generating the next term.
8) From Chapter 10 – Series Expansions in one line
Obtain a series expansion for e^{2x}.
The same process as above enables us to write successive terms of an infinite series like this one:
e^{2x} = 1 + 2x + 2x^2 + (4/3)x^3 + ...
9) From Chapter 11 – Factors and Derivatives
Find the 3^{rd} derivative of (x + 2)(x + 3)(x + 5)(x + 7).
We do not need to multiply the contents of the brackets here.
3!Sigmaa = 6[(x + 2) + (x + 3) + (x + 5) + (x + 7)]
= 6(4x + 17).
10) From Chapter 12 – Differential Equations
y x y_{1} 2 |
Solve 2y + xy_{1} = 3x.
The LHS is represented as shown above.
y x^{2} y_{1} 2x |
Using Proportionately we multiply the 2^{nd} column by x so that the bottom row is the differential of the top row.
Then 2xy + x^{2}y_{1} = 3x^{2} ∴ (yx^{2})_{1} = 3x^{2}
∴ yx^{2} = x^{3} + c.
11) From Chapter 14 – Solution of Quadratics
Solve x^{2} + x = 13.
7) 13 . _{1}0 _{3}0 _{1}0 _{2}0 _{4}0 _{5}0 _{5}0 |
We use the first differential of the LHS, with x equal to the first digit of the answer, as a divisor and start with x=3:
We then proceed as for finding a square root:
This gives x = 3.140055 and x = –4.140055 to 6 D.P.
12) From Chapter 16 – Solution of Cubic Equations
T_{1} T_{2} |
Find a solution to x^{3} – 4x^{2} + 12x = 32.
Similarly to the previous example we use the first differential of the LHS, with x equal to the first digit of the answer, as a divisor, then proceed as for finding a cube root.
We find x = 3.3013 to 5 figures.
Back Cover
This book follows on naturally from the first Introduction to Calculus book that showed how the subject can be taught in a very simple and approachable way.
So, here we cover the important extensions of basic Calculus: differentiations of products, quotients, function of a function etc.,
and integration techniques.
Series expansions, exponentials, solution of polynomial equations and Differential Equations are also covered.
Practical applications, like Simple Harmonic Motion and Projectile Motion are included too.
Kenneth Williams has been researching and teaching Vedic Mathematics for over 50 years. He has published many articles, books and DVDs and has been invited to many countries to give seminars and courses. His online courses are available at Math2Shine.com, including teacher training.