Recapitulation of Mathematics

Transcription

1 Unit I Recapitulation of Mathematics Basics of Differentiation Rolle s an Lagrange s Theorem Tangent an Normal Inefinite an Definite Integral

2 Engineering Mathematics I

3 Basics of Differentiation CHAPTER Basics of Differentiation. Introuction f( a h) f( a) If f() is a function of, then Lim is efine as the ifferential coefficient of f() h0 h at a, provie this limit eists an is finite. The ifferential coefficient of f() with respect to is generally written as f() or f () or Df(). f( h) f( ) Note: Generally, f() Lim h0 h calculus. is known as the first principle of ifferential. Differential Coefficient of a Function at a Point The value of the erivative of f() obtaine by putting a is calle the ifferential coefficient of f() at a an is enote by f (a) or. List of Formulae. n n n. a loge. e e 4. a a log e a sin cos 7. cos sin 8. tan sec 9. cot cosec 0. sec sec tan. cosec cosec cot. C 0, where C is constant. loga loga e

4 4 Engineering Mathematics I Eample : Differentiate l Solution: y m n w.r.t.. l m n y l + m + n l + m + n l. + m. + n l m + n. Differentiation from First Principle f ( h) f ( ) Lim h0 h Eample : If y, fin by first principle Solution: Given f() by efinition Lim h0 f ( h) f( ) h Lim h0 h h Lim h0 h h ( ( h h ) ) Lim h 0 h h( h ) Lim h0 h Eample : Differentiate sin from first principle Solution: Given f() sin, f( + h) sin ( + h)

5 Basics of Differentiation 5 by efinition Lim h0 Lim h0 Lim h0 Lim h 0 Lim h0 sin ( h) sin h sin ( h ) sin ( h ) h (sin A sin B sin (A + B) sin (A B) sin ( h) sin h h (sin + h). Lim h0 sin ( + h) sin sin h h (i) Differential coefficient of the prouct of two functions [f().f()] f() f() f (ii) Differential coefficient of the quotient of two functions f() f () () [f ()] f () f () f () f () f () (prouct rule) (quotient rule) Eample 4: Differentiate log w.r.t.. Solution: (. log) log + log Eample 5: If y, 5 Solution: y. + log. + log ( + log ) then prove that y( y). ( 5) ( ( 5) ( 0) ( 5) +5 then ( 5). ( 5) ( 5) 5)

6 6 Engineering Mathematics I 5 then ( + 5) 5 ( + 5) y y ( y) +5.4 Differential Coefficient of a Function of Function If y f(z), where z F(), then by eliminating z, we get y f {F()} e.g., sin, e sin, cos are the function of function. Again, let y f(t), where t g() t Then., this is known as chain rule. t Eample 6: Differentiate log (log sin ) w.r.t.. Solution: Let y log (log sin ) Put log sin t y log t t. t log t. log sin t t Again, put sin u. t u. t u log sin u log u sin.. log sin sin. cot log sin cos cos m Eample 7: If y log, cos m Solution: y log then fin. cos m cos m

7 Basics of Differentiation 7 log m sin m cos log tan m Put tan m t log t. t t. tan m tan m Again put m u. tan u. m u tan m tan m m tan sec u. m m m cos.. m sin m cosec m. sec m m cos m.5 Differential Coefficient of Inverse Trigonometric Functions or Trigonometrical Transformation Formula.. sin tan. 4. cos cot

8 8 Engineering Mathematics I 5. sec Eample 8: Differentiate sin w.r.t.. Solution: Let y sin Put tan tan y sin tanθ tan θ 6. cosec sin tan θ (sin ) sin θ tan θ y y tan Eample 9: If y cot, tan then fin. Solution: y cot Put tan tan y cot tan θ tan θ cot secθ tan θ cot cos θ sin θ cos θ cot cos sin

9 Basics of Differentiation 9 θ cos cot θ θ sin cos cot cot y y tan. Eample 0: Differentiate sin Solution: Let y sin w.r.t.. Put sin A, sin B A sin, B sin y sin sin A sin B sin B sin A sin sin A cos B cos A sin B sin sin (A B) y A B y sin sin sin sin sin t t. t. t. put t

10 0 Engineering Mathematics I.6 Differentiation of Implicit Functions There may be a relation between an y when it is not possible to epress it in the form of y f() or to solve an equation for y in terms of, relation is calle implicit function. For by ifferentiating the given relation w.r.t. Eample : Fin, when + y a Solution: Given + y a Differentiating both the sies w.r.t., we get + y a + y 0 y Eample : Fin, when a + hy + by + g + fy + c 0 Solution: Given a + hy + by + g + fy + c 0 Differentiating both the sies w.r.t., we get a + h y + b y + g + f y + c 0 a + h y + by + g + f a + h + yh + by + g + f 0 (a + hy + g) + (h + by +f) 0 (h + by + f) (a + hy + g) ( a hy g) ( h by f )

11 .7 Logarithmic Differentiation Basics of Differentiation If the power of a function of is also a function of, then it woul be convenient for us if we first take logarithms w.r.t the base e an then ifferentiate both sies w.r.t. Let us recall the formulae (i) (ii) logm n n log m log(mn) log m + log n (iii) log n m log m log n Eample : If y, then prove that (+ log) Solution: Given y Taking log on both the sies, we get log y log log y. log Differentiating both the sies w.r.t, we get logy log log. log. y log y y( + log ) ( + log ).8 Differentiation of Parametric Equation When variables an y are epresse in terms of a thir variable like t or, then this thir variable is calle parameter. e.g., g(t) an y h(t), these are calle parametric equations. / t Formula / t Eample 4: If at, y at, then fin. Solution: Given at, y at

12 Engineering Mathematics I at, t t t t a t.9 Differentiation of Infinite Series When the function is given in infinite series. In this case we use the fact that if a term is elete from an infinite series it remains uneffecte.... Eample 5: If y, fin. Solution: Here y y log y y log Differentiating w.r.t, we have y log log y y log y log y ylog y y y ( ylog ) Eample 6: If y sin sin sin..., Prove that cos y Solution: Here y sin y y sin + y y cos cos y

13 Basics of Differentiation Eample 7: If Solution: Eample 8: If y e e e..., show that y y Here y e +y Taking log on both sies log y ( + y) log e ( loge ) log y + y Differentiating w.r.t, we get y y y y y y y... then prove that y Solution: Here y y..() Taking log on both sies log y log( + y) Differentiating w.r.t, we get. y y..() y ( y) ( y) From eq. () + y y Putting in eq. () we get y y y y y y y

14 4 Engineering Mathematics I.0 Successive Differentiation Differentiating the first erivative of a function we obtain secon erivative, ifferentiating the secon erivative, we get the thir erivative an so on. This process of fining erivatives of higher orer is known as successive ifferentiation. Eample 9: If y + tan, Prove that cos y + y Solution: Given y + tan Differentiating w.r.t, we get + sec Differentiating again, we get 0 + sec.sec tan y sec tan tan cos cos y tan cos y (y ) cos y y cos y + y Eample 0: If y ae m + be m y, show that m y Solution: Let y ae m + be m ifferentiating w.r.t., we get am e m bm e m

15 Basics of Differentiation 5 ifferentiating again w.r.t., we have y am e m + bm e m m (ae m + be m ) y m y y Eample : If y A cosn + B sin n, show that + n y 0 Solution: Let y A cosn + B sin n ifferentiating w.r.t. An sin n + Bncosn ifferentiating again w.r.t. y An cosn Bn sin n n (A cosn + B sin n) n y y + n y 0 Eample : If y A cos (log ) + B sin (log ) prove that y Solution: Let y A cos (log ) + B sin (log ) + + y 0 ifferentiating w.r.t., A sin (log ). + B cos (log ). A sin (log ) + B cos (log ) ifferentiating again w.r.t., we have y + A cos(log ). B sin (log ). y + [A cos(log ) +B sin (log )] y

16 6 Engineering Mathematics I y + + y 0 Eample : If y Ae p + Be q y show that Solution: Let y Ae p + Be q (p + q) + pqy 0 ifferentiating w.r.t., Ape p + Bqe q..(i) ifferentiating again w.r.t., we have y Ap e p + Bq e q..(ii) Now y (p + q) + pqy Ap e p + Bq e q (p + q) (Ape p + Bqe q ) + pq (Ae p + Be q ) Ap e p + Bq e q Ap e p Bq e q Apqe p Bpqe q + pqae p + pqbe q 0 Practice Problems Fin the ifferential coefficient of the following., ( 4) ( ) ( ) (4 5) log 7. e + n 8. ( sin + 7 ). 9. a b 0. e. log.e. e + 7 log +. e a sin b 4. log sin 5. log 6. ae sin + b n cos

17 Basics of Differentiation 7 7. ( + ) (e + sin ) 8. ( ) ( + ) 9. ( tan ) (5 + 4 sin ) n log a. cos log e a a 5 log log log log. ( ) sin cos sin cos tan tan e tan 6. n cot ( a) ( b) ( c) a a 5. tan 6. log tan 7. sin (tan ) 8. log 9. e a + b 40. log a b 4. log p q cot ( sin ) 44. sin sin s in e sin sin 45. log sin sin 47. log 46. log 48. cosec cot sin e 49. logloglog 50. n sin 5. sec 5. If y tan, 5. Differentiate sin ( 4 ) fin

18 8 Engineering Mathematics I 54. If y tan cos, cos 55. sin 57. sin fin ifferentiate the following: 56. tan 58. cos 59. tan 60. sin 6. cos (4 ) 6. tan a a 6. cos 65. If y + + +!!! 66. tan 64. sin + Then show that y 67. sin o o π Hint : 80 Fin the Differential coefficient of the following functions by First Principle cos (a + b) 70. tan a 7. cos 7. cos 7. e 74. sin 75. cos 76. tan 77. sin (a y) 78. If sin y sin (a + y), prove that sin a 79. If (e y e y ), prove that 80. If y sin sin sin... prove that (y ) cos. 8. If y + y y, prove that 0 Fin for implicit ifferentiation when;

19 Basics of Differentiation 9 8. y c y 8. a b 84. y a 85. y y 86. y sec + tan + y tan ( + y) + tan ( y) 88. If ( + y) m + n m y n, prove that y 89. Fin the value of at (, ) for the curve + y + y If y log log log... show that (y ) 9. If y a + y, show that 9. If cos(y), show that 9. If y log (y), fin 94. If + y sin ( + y), fin 95. If y log ( + y) + sin e, fin y log a y log a y sin (y) 0 sin (y) 96. If a.y b ( + y) a + b, prove that y provie that ay b Fin of the following functions; 97. (log ) 98. ( + ) (cos ) log ( ) e 0. e sin 0. sin + (log ) 04. (tan ) cot + (cot ) sin ( a) ( b) ( c) 08. y ( ) ( ) ( ) ( ) ( ) ( ) 06. tan tan tan tan 4

20 0 Engineering Mathematics I 0. ( 4) ( ) 5 ( ) Fin for the following parametric equations;. a sec, y b tan. a sin, y a cos. sin θ, y c os θ 4. at 4, y bt at at 5., y t t 7. log t, y e t + cos t Fin the erivative of the following functions 6. sin u, y tan u u u 8. sin w.r.t. 9. sin w.r.t. (sin ) 0. e tan w.r.t. tan. tan w.r.t. tan. If e cost an y e sint, prove that y log log y. If ae t (sin t + cos t), y ae t (sin t cos t) prove that tan t 4. If a t an y a t prove that. t t y 5. Fin secon erivative of a + b + c + 6. y If y a cosp + b sin p,prove that + p y 0 7. If y log ( + cos )then prove that y y + y 0 Hint : y y y, y, y y cos 8. If y tan + sec then prove that ( sin ) 9. If y log then prove that y 0. If y e a sin b then prove that y ay + (a + b )y 0. If y m cos e then prove that ( ) y y m y 0. If y sin (sin )then prove that y + y tan + y cos 0

21 Basics of Differentiation. If y t, t then prove that y 4. If y a cos (log ) + b sin (log ) then prove that y + y + y 0 5. If y a sin (log ) then prove that y + y + y 0 6. If y (sin ) then prove that ( y ) 7. If y + tan then prove that cos y y If y sin then prove that ( y ) y 0 9. If y Ae kt y cos (pt + c) then show that t k n y 0, t where n p + k. 40. If y tan e 4. If y log, thenprove that ( ) y y 0 y log then prove that 4. If y tan y then prove that 0 ( ) y 4. If y sin ( + y) then prove that y 0 y 44. If, then show that a b 4 b. a y. 5, Answers e + n n 8. cos + 7 +

22 Engineering Mathematics I 5 9. b a 0. e + e.. e e a (a sin b + b cosb) 4. sin cos log 5. ( + log ), 6.ae (cos + sin ) + b n ( sin + n cos ) 7. e ( + + ) + cos ( + ) + 4 sin 8. ( ) + ( ) ( + ) 9. 4 (cos sin ) sec (5 + 4 sin ) 4 ( 9 5) 0. ( 5) n n loga. (log ) a n log log (cos sin ) cos 4. (log ) 6. a a a e n. tan sec. ( sin cos ) 5. sin ( e sec ) (cot ) (cosec n ) (e tan ) 7. n (cot - ) (e 5 log 5 ) log (e 5 8. (log ) (a b c) (ab bc ca) 9. ( a) ( b) ( c) n log ) 0. ( log ).. ( ) ( 7 ) ( ) ( ) ( ) 5. (a ) 6. sec (a ) 7. cot. sec 8. cos (tan ) sec

23 Basics of Differentiation 9. log e cose 4. sin e 40. ae a+b 4. a p a b p q 4. cos sin sin cos sin sin 44. cosec ( sin ) ( sin + cos ) sin cosec (cosec n sin cosec. cot cot sin ) sin.e.cos e.cos 5. n sin ( ) n. n n 46. sec log.log log 5. sec tan 4 sec [( tan ) log ( sec ( tan ) )] 68. cos o a sin (a + b) 7. a sec a

24 4 Engineering Mathematics I 7. sin e 75. cos sin b a. y y y 86. y sec sec ( y) sec ( y) sec y ( log y) ( y) ( y) e cose y y y y 99. ( + ) 5 ( 5) log ( ) 77. sec y y y sec tan sec y, y y (sec cos ( y) cos ( y) ) y 98. (log ) log log log 00. (cos ) log.log cos tan log 0. e. ( + log ) 0. ( ) 7 0. e sin cot sin sin log 04. (log ) log log log 05. (tan ) cot cosec ( log tan ) + (cot ) tan. sec (log cot ) 06. ( a) ( b) ( c) a b c 07. tan tan tan tan 4 (cosec + cosec 4 + cosec cosec 8)

25 Basics of Differentiation ( log log 09. ( ) log 0. ( ) ( ) ( ) 4 6 ( ) ( ) ( ) ( ) 5. ( ) ( 4) ( ) ( 4) ( ) b.. cot a b 4. tan 5. 4at t ( t ) 6. t t (e t sin t) 9. cos sin sin cos log tan 0. e. (sin ) ( cot log sin ) cos. 6. 6a +