13 Lecture 13 L Hospital s Rule and Taylor s Theorem

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1 3 Lecture 3 L Hospital s Rule and Taylor s Theorem 3 L Hospital s Rule Theorem 3 L Hospital s Rule) Let a R or a = and functions f,g : a,b) R satisfy the following properties ) f,g are differentiable on a,b); ) lim f) g) = or lim g) = + ; a+ a+ a+ 3) g ) for all a,b); f 4) there eists lim ) a+ g ) =: L R f) Then there eists lim a+ g) = L Proof We will only give a proof for the case a R and lim f) g) = For the general a+ a+ case see eg [, p4-44] We first etend the functions f and g to the interval [a,b), setting fa) = ga) := According to assumption ), f and g are continuous at the point a Since f,g are differentiable on a,b), they are continuous also at each point of a,b), by Theorem Thus, f,g are continuous on [a,b) Net, we note that g) for all a,b) Indeed, if g ) = for some a,b), then applying Rolle s theorem see Theorem 3) to the function g : [a, ] R, we obtain that there eists c a, ) such that g c) =, that is impossible by assumption 3) f) g) Net, to show that lim a+ Theorem 77 and assumption 4), = L, we are going to use Theorem 77 Let ε > be fied By δ > a,a+δ) : f ) g ) L < ε Applying the Cauchy theorem see Theorem 5) to the functions f,g : [a,] R, we have for all a,a+δ) f) g) L = f) fa) g) ga) L = f c) g c) L < ε, where c a,) a,a+δ) Remark 3 A similar statement is true for the left-sided limit as goes to b Eample 3 Using L Hospital s Rule, we compute the following limits: sin sin) a) lim ) = ; b) lim ln ln c) lim cos) We compute lim cos ln) ) e lncos lncos lncos lncos) ) ) = lim = ; sin cos 5 = lim sin cos = lim sin lim cos =

2 Thus, by the continuity of the function f) = e, R, we have lim e lncos = e lim lncos = e = e See [, p45-48] for more eamples of the application of L Hospital s Rule Eercise 3 Using L Hospital s Rule, show that sin = ; c) lim ln)α e) β e e cos ln+) a) lim = ; b) lim d) lim 4 π arctan)α ln = α π g) lim, α R; e) lim + ln+) ) = α β e, where α,β are some real numbers; = e ; f) lim = ; ln ε = for all ε > ; h) lim + ε ln = for all ε > ; i) lim + ln+)) = Eercise 3 Compute the following limits: a) lim ; b) lim ; c) lim π arctan)) ; d) lim ln+) e) lim e e sin sin sin + ) ; f) lim 3 Higher Order Derivatives sin + ) ; g) lim +) e ; h) lim ln+) ln ) + ; ln ln) We assume that a function f : a,b) R is differentiable on a,b) We denote its derivative f by g, that is g) = f ), a,b) Definition 3 If there eists a derivative g ) of the function g at a point, then this derivative is called the second derivative of f at the point and is denoted by f ) or d f d ) Let the n-th derivative f n) be defined on a,b) Then the n+)-th derivative of f at a,b) is defined as f n+) ) = dfn) ) d ), if it eists Eample 3 Let a > Then for each R we obtain a ) = a lna, a ) = a ln a, a ) = a ln 3 a,, a ) n) = a ln n a In particular, e ) n) = e, R Eercise 33 Let α R Show that α ) n = αα )α )α n+) α n for all > and n N Eample 33 Let α R Then +) α ) n) = αα )α )α n + ) + ) α n for all > and n N Indeed, +) α ) = α+) α, +) α ) = α+) α ) = αα )+) α and so on Eercise 34 Show that ln+)) n) = )n n )! +) n for all > and n N Eample 34 For each R sin) n) = sin +n π ) and cos) n) = cos +n π ) Indeed, sin) = cos = sin + π ), sin) = cos) = sin = sin + π ), sin) = sin) = cos = sin +3 π ) and so on The same computation for cos) n) Eercise 35 Compute the n-th derivative of the following functions: a) f) =, R; b) f) = +, > ; c) f) = arctan, R Theorem 3 Let functions f,g : a,b) R have n-th derivatives on a,b) Then the following equalities are true 5

3 ) for all k {,,n} f n k) ) k) = f k) ) n k) = f n), where f ) = f; ) for all c R cf) n) = cf n) ; 3) f +g) n) = f n) +g n) Theorem 33 Leibniz Formula) For a number n N let g,f : a,b) R have n-th derivatives on a,b) Then f g has the n-th derivative on a,b) and where C k n = n! n k)! f g) n) = Cnf k k) g n k), Eercise 36 Compute the following derivatives: a) e ) n), R; b) 3 sin) n), R; c) n ln) n), > 33 Taylor s Formula 33 Taylor s Formula for a Polynomial Let n N and {a,a,a,,a n } R For any point R a polynomial can be written in the form k= P) = a +a +a a n n, R, P) = b +b )+b ) b n ) n, R, ) where{b,b,b,,b n }aresomerealnumbers, whichcanbecomputedbythefollowingway Inserting = into ), we obtain b = P ) Net we compute P So, P ) = b +b )+3b 3 ) nb n ) n, R ) Inserting = into ), we get b = P ) Net, we compute the second derivative of P P ) = b +3 b 3 )nn )b n ) n, R 3) Inserting = into 3), we obtain b = P ) Similarly, we obtain Thus, for each R b k = Pk) ), k P) = P )+ P )! )+ P )! ) Pn) ) ) n 4) n! We see that any polynomial can be completely defined only by its value and values of its derivatives at a point Formula 4) does not hold if P is not a polynomial, but it turns out that values of a function are close to the right hand side of 4) if is close to 53

4 33 Taylor s Formula with Peano Remainder Term Let f,g : A R be some functions and be a limit point of A If f) g),, then we will write f) = og)),, or f = og), Eercise 37 Show that a) = o), ; b) 3 = o ), + ; c) ln = o ), + ; d) sin = o), Theorem 34 Let n N and let a function f : a,b) R and a point a,b) satisfy the following conditions: ) there eists f n ) ) for all a,b); ) there eists f n) ) Then f) = k= The term o ) n ) is called the Peano remainder term Proof We recall that! = and set R n ) := f) f k) ) ) k +o ) n ), 5) k= f k) ) ) k, a,b) According to assumptions ) and ), there eists R n ) ) for all a,b) and R n) ) Moreover it is easy to see that R n ) = R n ) = R n ) = = R n) n ) = Assuming > and applying the Lagrange theorem see Theorem 4), we have R n ) ) n = R n ) R n ) ) n = R nc ) ) ) n = R nc ) R n ) ) n = R nc )c ) ) n R nc ) ) n = R nc ) R n ) ) n = R nc 3 )c ) ) n R n n ) c n ) R n n ) ) R n) n ) =, +, where < c n < c n < < c < c < Moreover c n as + One can similarly obtain that Rn) ), n Consequently, by Theorem 78 Eample 35 For every n N R n ) = o ) n ),, e = ++! n n! +on ), The formula follows from Theorem 34 applying to f) = e, R, and the fact that f k) ) = e = see Eample 3) 54

5 Eample 36 For all n N ln+) = )n n n +on ), The formula follows from Theorem 34 applying to f) = ln + ), >, and the fact that f k) ) = )k k )! +) k = ) k k )! see Eample 34) Eample 37 For each α R and n N +) α = +α+ αα )! αα )α n+)n n! +o n ), The formula follows from Theorem 34 applying to f) = + ) α, >, and the fact that f k) ) = αα )α )α k+)+) α k = αα )α )α k+) see Eample 33) Eercise 38 Show that for every n N {} sin = 3 3! + 5 n+ + )n 5! n+)! +on+ ),, cos =! + 4 n + )n 4! n)! +on+ ), Eercise 39 Show that for every n N {} sinh = e e cosh = e +e = + 3 3! + 5 n+ 5! n+)! +on+ ),, = +! + 4 n 4! n)! +on+ ), Eercise 3 Use Taylor s formula to compute the limits: e a) lim ; b) lim ; c) lim sin sin e ln++ ) ln ) cose ; d) lim ) cose ) 3 55