Differential Calculus Made Easy

Firewall Media, 2007 - 984 pàgines

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1	79

148	81

Ꮎ	83

20 0	84

lim	85

1+x+x	88

YA	6

Since cosec x is a periodic function with period	12

Sxx	428

15 Continuous	433

5	434

DIFFERENTIABILITY	435

292	438

f0+ hf0	446

fx is continuous at x	449

Solution i Let y tan x	454

Domain of f Dƒ R	13

The domain and range of various inverse trigonometric functions are	14

Note i If	17

Solution We have	18

and	19

3	21

33 THEOREM IF ANGLE IS MEASURED IN RADIANS THEN	22

Df Dg D	23

Example 12 Find the domain and range of the following	28

ii We have	37

if x 0	39

lim	41

and	45

Example 28 If fx	49

or	56

The graph of the given function is as shown in	60

12 Find the inverse of the function fx	62

2	65

CONTENTS	66

MADE EASY	66

PREFACE	66

SYMBOLS	66

1	66

14 TYPE OF FUNCTIONS	66

45	66

YA	66

The domain and range of various inverse trigonometric functions are	66

iii y tan¹x	93

and	94

and	94

lim	101

and	106

cos 3x 2	111

Solution We have	118

3	121

33 THEOREM IF ANGLE IS MEASURED IN RADIANS THEN	122

ff fx f	123

ii Let tan¹ x0	124

We have	126

The graph of the given function is as shown in	136

12 Find the inverse of the function fx	140

2	143

Thus if we go on decreasing this value of	145

ii Simplify the numerator and denominator of the	151

lim	155

or	156

sin x	157

h	162

vii We have	165

h	169

h	172

sin	177

1 lim	181

ii We have	192

1 x 13	202

ii We have lim x	211

介	219

Example 6 Find the derivative of the following functions wrt	221

4	223

This is the sum of a constant function a	229

and	230

Put x a + h	232

SOME SOLVED EXAMPLES	233

Also	238

Example 7 i For what value of k	239

x + x	240

lim 1 h²	246

Example 21 i Examine the continuity of the	254

On subtracting 1 and 2	258

介	261

7	268

sin 2xx0	272

lim fx does not exist	274

ie a is	281

And	284

5	288

Example 8 Prove that the function fx	295

RHD Rf2 lim	297

Also	298

334	300

lim	301

介	302

бу	306

Proof If	309

6	313

J	319

Proceeding to limits as dx 0 we have	322

Proof If	324

Sy 8x adx+2ax	331

у + бу	334

2	339

LIMITS CONTINUED	345

202	348

LIMITS CONTINUED	349

204	350

206	352

208	354

LIMITS CONTINUED	355

LIMITS CONTINUED	357

212	358

nn + 1	360

5	362

Differentiating both sides wrt x	363

7	365

1192	368

236	382

CONTINUITY	383

CONTINUITY	387

On subtracting 1 from 2	392

x + x	393

Example 19 i Discuss the continuity of the	397

256	402

Also	406

Proceeding to the limit as dx 0	409

f1k	410

Example 6 Find the derivative of the following functions wrt	412

介	414

介	416

CONTINUITY	417

CONTINUITY	421

sin x+dx sin x	424

v Let	456

介	466

iii Let	472

iii Let	473

sec² x 2tan x + 2 +	479

tan1	484

3 tan tan³	505

vi Let	506

1 x2	510

Ξα	511

ii Let	512

Solution i Let y log	514

b cos x	526

5	532

8	534

iv We have	542

Example 7 i If 1 x² +	550

iii We have y + x +	551

Solution i Let	565

Taking logarithms on both sides we get	566

Substituting the values of equations 3 and	567

log x	589

1	596

介	597

cos 20	599

iii We have	603

du	615

Also	618

v Let	625

Multiplying both sides by 1x2 we have	627

Differentiating again wrt x we get	631

1 dy	633

0+20	643

cos2 0	645

Example 46 If x tan	647

9	649

Differentiating 1 wrt t we get	657

dx	659

6y	660

ii Let r be the radius and V	665

dl	673

10	680

103 EQUATION OF TANGENT AT ANY POINT OF THE CURVE	681

Slope of normal at 1 1	684

viii We have x 1	686

dx	687

dy 3xx2a xx	688

The required points are 2n+	689

2	692

When x 1	694

介	705

ii We have	706

Solution We have 16x² + 9y² 144	711

Now the slope of normal at 3	718

Since the tangent to the given curve is parallel	723

Since the tangent to the given curve is parallel	728

2 1	737

y	738

11	743

y fx	744

When x 3	755

9	756

Solution i Let	757

fx	758

fx is strictly decreasing on 0	762

iv Let	765

fx is strictly increasing on 0	767

Example 15 Separate the interval 0	769

When	770

π	772

iv Let	776

12	779

y + dy 00036 +00001	783

Now Differentiating 1 wrt x we	785

When r 10 cm	793

Example 10 Find the percentage error in calculating the volume	794

13	801

vi Let	814

iii Let	824

f1sin² 2	828

cos c sin c	830

Put x c in equation 2	832

ffff	836

介	837

fc	847

1	848

133 1+57445	849

1 1	855

14	858

63660	862

For a maximum or minimum we have	863

2	865

fx changes sign from	875

3π	884

and	887

x	888

介介	893

Solution We have y	894

介	895

and	896

Since fx attains its maximum value	910

Example 17 i Determine two positive numbers whose	911

and	915

and	917

and	934

Put	935

Now	941

h2	942

Example 35 i Show that the height of	950

ii Let r and h be the radius	951

4	952

2	956

15	962

SOME SOLVED EXAMPLES	963

y x³ 4x which is same as	968

1111	969

ས	973

y is a maximum when x 0	974

AY	975

Put x 0 in 1 we get	978

Y	980

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Pàgina 75 - SYMBOLS Greek Alphabets Metric Weights and Measures LENGTH 10 millimetres 10 centimetres 10 decimetres 10 metres 10 dekametres 10 hectometres VOLUME 1000 cubic centimetres 1000 cubic decimetres...

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Pàgina 75 - Revision Exercise' has been incorporated. The purpose of this exercise is to give an opportunity to students to revise the entire chapter in the minimum possible time. It is hoped with all the above characteristics the book will be found really useful by teachers and students alike. A serious effort has been made to keep the book free from errors, but even then some errors might have crept in. I am grateful to M/s. Laxmi Publications (P) Ltd., New Delhi, for the keen interest and active co-operation....

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Informació bibliogràfica

Títol	Differential Calculus Made Easy
Autor	Deepak Bhardwaj
Editor	Firewall Media, 2007
ISBN	8131800938, 9788131800935
Nre. de pàgines	984 pàgines

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