Rolle s Theorem, the Mean Value Theorem, and L Hôpital s Rule

Transcription

1 Rolle s Theorem, the Mean Value Theorem, and L Hôpital s Rule 5. Rolle s Theorem In the following problems (a) Verify that the three conditions of Rolle s theorem have been met. (b) Find all values z that satisfy the conclusion of the theorem.. (a) f() = 7+0, on [, 5] (b) f() = 7+0, on [0, 7] (c) f() = 7+0, on [, 8]. (a) f() = 4, on [0, 4] (b) f() = 4, on [, 5] (c) f() = 4, on [ 4, 8] 3. (a) f() = , on 3 7 (b) f() = 3 6, on 4 4 (c) f() = 3 +, on 4. (a) f() = 6 7, on [, 7] (b) f() = 3 + 6, on [0, ] (c) f() =, on [, ] (d) g() = , on [ 3, 0] 5. (a) f() =sin, on [0,] (b) f() = cos, on [, 3] [ (c) f() =sin+ cos on 4, ] 3 4 ( Hint: 3 4 = 4 + )

2 In problems 6 to determine in what way the function fails to meet the conditions of Rolle s theorem. 6. f() =4 8 for f() = cos 4 + cos for 3 7. [ f ( 3 8. f() = 3 +4 for f() =sin+ cos for f() = 3+ ( ) for. ) 0 ]. f() = 3 for. [f (0) is undefined]. Find a function f() on the interval [, ] such that f() fails to meet the conditions of Rolle s theorem, but [f()] does meet them. In what way does f() fails to meet the conditions? 3. Use Rolle s theorem to prove that, regardless of the value of b, there is at most one point in the interval for which b =0. 5. Mean Value Theorem For problems numbered 4 to, (a) Verify that the conditions of the mean value theorem have been met. (b) Find all numbers z that satisfy the theorem. Eample: f() = 3 for 3. Solution. (a) f() is continuous (all polynomials are continuous) f () =3 4eists for (, 3). Thus the conditions of the mean value theorem have been met. (b) f(3)=7 8 = 9. f()= =. Therefore, f(b) f(a) b a = f(3) f() 3 f (z) = f(b) f(a) b a = 9 ( ) 3 3z 4z = 5 3z 4z 5 = 0 = 0 =5 (M.V.T.) z = 4± 6 4( 5)(3) 6 = 4 ± 76. or Since 0.78 is not in the interval discard it, but. is in the interval so that z = is the answer.

3 4. h() = + for 5. f() =3 +4 for 5 6. f() = 3 3 for 3 7. f() = 3 for 8. f() = 3 + for 0 [ ] z = g() = for r() = for 3. f() = 5+6 for 5 [z =3]. F () = for 0 3 For problems numbered 3 to 3 determine in what way each function fails to meet the conditions of the mean value theorem on the indicated interval. 3. f() = for 4. f() = 4 for 5 5. f() = sin for [not continuous at =0] 6. g() =3+ for << 7. h() = (+3) (3 ) for F () = ( ) (+) for 0 < 3 9. f() = for [f (0) does not eist] 30. f() = 3 for 3. g() = for 3 3. φ(t) =tant for 0 t For the problems numbered 33 to 55, (a) Decide whether the mean value theorem applies to the function on the given interval. (b) If it applies find z; ifnot,statewhy. 33. H(w) =w +3w 3 for 0 w 4 [z =] 34. f() = for f() = 5 for 36. y =tanθ for 4 θ y = sin(α) for 0 α 38. P () = for 4 [ z = 4 or z = ] 3 4

4 39. P () = + for 40. f() =(+3)/(3 ) for 4 4. h() = 3 for [h (0) does not eist] 4. f() = 3 for g() = + for 44. S() = + for r() = (+) ( ) for 3 [z =+ ] 46. H(w) =w w for w 47. f() = for φ(t) = (t+3) (t 3) for t F () = for 0 4 [F () doe snot eist] 50. F () = for 0 5. f() = 3 + for 3 5. H(y) = y 3 y for y 4 {, [0, ) 53. B() =, =. Interval for consideration is 0. 3, [0, ) 54. f() =. Interval for consideration is 0., [, ] 55. g() =4+5 for 3 [discontinuity at =] 56. Show that on the graph of any quadratic polynomial the chord joining the points for which = a and = b is parallel to the tangent line at the midpoint =(a+b)/. 57. Suppose you know that F is continuous on [a, b] and that it has a derivative on all of (a, b) for which F () = 0 for every. Show that F is a constant. (Hint: Take any (a, b]. Show F () =F(a).) 58. Suppose that in problem #57, we had said for every (a, b), F () = c where c is a nonzero constant. Show that the graph F is a straight line passing through ( a, F (a) ). (Hint: Show that F () =F(a)+c( a).) 59. Under the hypothesis of the Mean-Value Theorem with I =[a, a + ]( 0) prove there eists a number λ, 0<λ<, such that f(a + ) f(a)= f (a+λ ). 60. Determine λ, define in 59, in terms of a and for f() = and f() = 3. Find the limit of λ in each case as A function f, continuous on [a, b], has f everywhere in (a, b). The line joining the points ( a, f(a) ) and ( b, f(b) ) intersects the graph of f at a third point ( c, f(c) ), where a<c<b. Prove f (t) =0for at least one point t in (a, b). 6. Use theorems from this section to prove that an equation of the form a + b + c = 0, where a, b, c are real numbers, can have no more than two real solutions.

5 63. (a) Suppose that f is a differentiable function on an interval and suppose f ()isalwaysbetween and +. Show that if and y are in the interval, then f() f(y) y. (b) Show that for all real numbers and y. sin() sin(y) y 5.3 L Hospital s Rule The correct spelling is either L Hospital or L Hôpital. 64. List the 3 conditions which must be met in order that L Hospital s rule applies. Compute the following limits and show your steps. tan 65. lim lim lim +t t tan 0 t 68. lim 0 sin sin(5) cos 69. lim 0 sin(3) 70. lim 0 t sin t 7. lim t 0 cos t 7. lim (sin ) tan 73. lim lim lim lim sin +tan (sin )(tan ) 77. lim lim lim 0 sin +tan lim u 0 tan u u sec u From here on the problems will involve the indeterminate forms 0 0,,,0. L Hospital s rule works for only the indeterminate forms 0 0 or. Any questions that are of the form may be changed to a fraction by making a common denominator. Any questions that are of the 0 type may be rearranged by inverting part of it. e.g. lim 0 cot = lim 0 ( ) = lim 0 cot Now L Hospital s rule may be applied. 8. lim tan. (tan sec ) 8. lim

6 83. lim lim lim lim lim 3 3+ tan lim 0 sin 89. lim p a p sec a q a, a > lim 3 q sec 9. lim 4 sin lim 0 sin 93. lim sin 95. lim cos 94. lim 96. lim cos sin 97. lim cot 98. lim 0 sec 99. lim csc θ θ 0 θ sin 00. lim 0 ( cot ) 3 0. lim lim 03. lim lim lim cos ( ) (tan ) 07. lim 0 sin cos 09. lim 0 3 cos 06. lim 0 + sin cos 08. lim 0 sin sec 0. lim tan 4. lim 3 cos sin 3 sin. lim cos 0 sin csc 3. lim 0 4. lim tan lim (sin ) csc lim (sec 3 tan 3 ) 7. lim 0 (csc csc cot ) 8. lim a ( )tan where a = 9. lim sin +( 3 /6) 0 5 +cos. lim sin 3. lim cos nt 0. lim n+ (n+)t n + t (t ) sec +. lim tan 4. lim (sec tan )

7 5. L Hospital s Rule is not all-powerful! sin Show that the limit lim +sin eists but cannot be evaluated using L Hospital s Rule. ln 6. lim lim lim ln ln 9. lim + ln ln 30. lim 0 3. lim + n (n a positive integer) 3. lim 0 a b 33. lim 0 + ln e 34. lim e ln 0 4sin 35. lim + ln cos lim lim 0 sin 3 ( e ) 37. lim t t t ln t 39. lim 0 3 ln( e ) ln tan ln 40. lim 0 ln tan 4. lim 0 + e 4. lim 44. lim +ln 43. lim e 0 + ln(3+e ) 45. lim (Hint: use the substitution u = ) 3 e +