Chpter 3 Clculus in R 3.1 Di erentition Definition 3.1. Suppose U R is open. A function f : U! R is di erentible t x 2 U if there exists number m such tht lim y!0 pple f(x + y) f(x) my y =0. If f is di erentible t ech x 2 U then we sy tht f is di erentible on U. Lemm 3.2. Suppose U R is open nd f : U! R is di erentible on U. Then for ech x 2 U there exists unique number m such tht lim y!0 pple f(x + y) f(x) my y =0. Thus x 7! m is function U! R, which we denote f 0 nd cll the derivtive of f. Lemm 3.3. Suppose U R is open nd f : U! R is di erentible on U. Then for ech x 2 U we hve f 0 (x) =lim y!0 pple f(x + y) y f(x). Proposition 3.4 (Properties of di erentition). Suppose U R is open nd f,g: U! R re di erentible on U. Then 1. f : U! R is continuous; 22
CHAPTER 3. CALCULUS IN R 23 2. f + g is di erentible on U with (f + g) 0 = f 0 + g 0 ; 3. for ech 2 R the function f is di erentible on U with (f) 0 = f 0 ; 4. the function fg is di erentible on U with (fg) 0 = f 0 g + fg 0 ;nd 5. if g is di erentible on f(u) then f g is di erentible with (f g) 0 = (f 0 g)g 0. Definition 3.5. Let U R be open, nd let f : U! R be function. 1. Apointx 2 U is locl minimizer for f if there exists r>0 such tht f(x) pple f(y) for ll y 2 U \ B r (x) \{x}. If the inequlity is strict, then x is strict locl minimizer. 2. Apointx 2 U is locl mximizer for f if there exists r>0 such tht f(x) f(y) for ll y 2 U \ B r (x) \{x}. If the inequlity is strict, then x is strict locl mximizer. If x is either (strict) locl minimizer or (strict) locl mximizer, then we sy tht x is n (isolted) extremizer. Lemm 3.6. Suppose U R is open nd tht f : U! R is di erentible. If x 2 U is n extremizer of f, then f 0 (x) =0. Theorem 3.7 (Men vlue theorem). Let, b 2 R. Supposethtf :[, b]! R is continuous nd tht f is di erentible on (, b). Then there exists x 2 (, b) such tht f 0 (x )= f(b) f(). b Corollry 3.8 (Uniqueness of nti-derivtives). Suppose f :[, b]! R is continuous nd is di erentible on (, b). Then f is constnt if nd only if f 0 =0on (, b). Definition 3.9. Suppose U R is open nd f : U! R is di erentible on U. Wesythtf is continuously di erentible if the function f 0 : U! R is continuous. The set of continuously di erentible functions U! R is denoted by C 1 (U; R) or simply by C 1 (U). Lemm 3.10 (Locl monotonicity lemm). Let U R is open nd let f 2 C 1 (U). Suppose x 2 U is such tht f 0 (x ) > 0. Then there exists r>0 such tht if x, y 2 B r (x ) with x<ythen f(x) <f(y).
CHAPTER 3. CALCULUS IN R 24 Definition 3.11. If the derivtive f 0 of function f is itself di erentible, then we sy tht f is twice di erentible. The derivtive (f 0 ) 0 is clled the second derivtive nd is denoted f 00. More generlly, we denote the k th derivtive of f by f (k). The collection of functions f : U! R for which ll derivtives up to order k re continuous is denoted C k (U; R) or simply C k (U). Proposition 3.12 (Second derivtive test). Suppose tht U R is open, tht f 2 C 2 (U), ndx 2 U is such tht f 0 (x )=0. Then if f 00 (x ) > 0 then x is strict minimizer, nd if f 00 (x ) < 0 then x is strict mximizer. Theorem 3.13 (Tylor expnsion). Suppose f 2 C n+1 (B r (x )). Then for ech x 2 B r (x ) there exists between x nd x such tht f(x) = nx k=0 3.2 Integrtion 1 k! f (k) (x )(x x ) k 1 + (n + 1)! f (n+1) ( )(x x ) n+1. Definition 3.14. Let [, b] R. 1. A prtition of [, b] is finite set P = {x i } n i=0 = x 0 pple x 1 pple pplex n = b. [, b] such tht 2. The dimeter of prtition P is defined by kp k = mx i x i x i 1. 3. A refinement P 0 of prtition P is itself prtition of [, b] nd must hve the property tht P P 0. 4. If P nd Q re prtitions of [, b] we cll P [ Q their common refinement. The collection of ll prtitions of [, b] is denoted P([, b]). Definition 3.15. Suppose f :[, b]! R is bounded function nd P = {x i } n i=0 is prtition of [, b]. 1. The upper Drboux sum of f with respect to P, which we denote U(P, f), isthesum! nx U(P, f) = sup f(x) (x i x i 1 ) [x i 1,x i ] i=1
CHAPTER 3. CALCULUS IN R 25 2. The lower Drboux sum of f with respect to P, which we denote L(P, f), isthesum nx L(P, f) = inf f(x) (x i x i 1 ) [x i 1,x i ] i=1 Lemm 3.16. Suppose P is prtition of [, b] R nd tht f :[, b]! R is bounded. 1. There exists constnt M such tht M(b ) pple L(P, f) pple U(P, f) pple M(b ). 2. If P 0 is refinement of P, then L(P, f) pple L(P 0,f) pple U(P 0,f) pple U(P, f). 3. If Q is nother prtition of [, b], then L(P, f) pple U(Q, f). Definition 3.17. Suppose f :[, b]! R is bounded. 1. The upper Drboux integrl of f on [, b] is defined by f(x) dx =inf{u(p, f) P 2 P([, b])} 2. The lower Drboux integrl of f on [, b] is defined by f(x) dx =sup{l(p, f) P 2 P([, b])} 3. If the upper nd lower Drboux integrls re the sme, then we sy tht f is integrble nd define the Drboux integrl of f on [, b] by f(x) dx = f(x) dx = f(x) dx Lemm 3.18. Let f :[, b]! R be bounded function. Then f(x) dx pple f(x) dx
CHAPTER 3. CALCULUS IN R 26 Theorem 3.19 (Drboux integrbility criterion). A bounded function f :[, b]! R is integrble if nd only if for ech ">0 there exists prtition P of [, b] such tht U(P, f) L(P, f) <". Exmple 3.20. The following functions re integrble on ny domin [, b]: the function f(x) =c, where c is ny constnt, nd the function f(x) =x. The following function is not integrble on ny domin [, b]: the function f(x) = ( 1 if x 2 Q, 0 if x/2 Q. Proposition 3.21. Suppose f :[, b]! R is continuous. Then f is integrble. Definition 3.22. A function f :[, b]! R is monotone if either f(x) pple f(y) whenever x pple y, in which cse f is clled monotone incresing, or f(x) f(y) whenever x pple y, in which cse f is clled monotone decresing. Proposition 3.23. Suppose f :[, b]! R is monotone. Then f is integrble. Theorem 3.24 (Drboux convergence criterion). Suppose f :[, b]! R is bounded nd I 2 R. Then the following re equivlent. 1. The function f is integrble nd R b f(x) dx = I. 2. Whenever {P k } 1 k=1 is sequence of prtitions of [, b] such tht kp kk! 0 then L(P k,f)! I nd U(P k,f)! I. Definition 3.25. Let [, b] R. 1. A pointed prtition P b of intervl [, b] consists of prtition P = {x i } n i=0 of [, b] together with set of points C = {c i} n i=1 such tht x i 1 pple c i pple x i for ech i =1,...,n.
CHAPTER 3. CALCULUS IN R 27 2. Suppose b P =(P, C) is pointed prtition of [, b]. The dimeter of bp, which we denote k b P k, is defined s the dimeter of the prtition P. 3. Suppose b P is pointed prtition of [, b]. The Riemnn sum of f with respect to b P, which we denote R( b P,f), isthesum R( b P,f)= nx f(c i )(x i x i 1 ). i=1 The collection of ll pointed prtitions of [, b] is denoted c P([, b]). Lemm 3.26. Suppose b P = (P, C) is pointed prtition of [, b] nd f :[, b]! R is bounded function. Then L(P, f) pple R( b P,f) pple U(P, f). Theorem 3.27 (Riemnn convergence criterion). Suppose f :[, b]! R nd I 2 R. Then the following re equivlent. 1. f is integrble nd R b f(x) dx = I. 2. Whenever b P k is sequence of pointed prtitions of [, b] such tht k b P k k!0 then R( b P k,f)! I. Remrk 3.28. The limit is clled the Riemnn integrl of f. lim R( P b k,f) kp b k k!0 Definition 3.29. Suppose f :[, b]! R is integrble. We define Z b f(x) dx = f(x) dx. Proposition 3.30 (Properties of integrls). 1. (Linerity of integrtion I) If f,g re integrble on [, b], then so is f + g nd (f(x)+g(x)) dx = f(x) dx + g(x) dx.
CHAPTER 3. CALCULUS IN R 28 2. (Linerity of integrtion II) If f is integrble on [, b] nd c 2 R, then cf is Drboux integrble nd cf(x) dx = c f(x) dx. 3. (Monotonicity of integrtion) If f,g re integrble on [, b] with f(x) pple g(x) for ech x 2 [, b] then f(x) dx pple g(x) dx. 4. (Additivity of intervls) Function f is integrble on intervls [, b] nd [b, c] if nd only if it is Drboux integrble on [, c], in which cse f(x) dx + Z c b f(x) dx = Z c f(x) dx. Lemm 3.31. Suppose f :[, b]! R is integrble, nd g : R! R is continuous. Then g f :[, b]! R is integrble. Corollry 3.32. If f is integrble on [, b] then so is f, with f(x) dx pple f(x) dx. Proposition 3.33 (Men vlue theorem for integrls). Suppose f :[, b]! R is continuous. Then there exists c 2 [, b] such tht f(x) dx = f(c)(b ). Theorem 3.34 (Fundmentl theorem of clculus). Suppose f :[, b]! R is integrble on ny closed intervl contined in [, b]. 1. The function F :[, b]! R defined by F (x) = Z x f(t) dt is continuous. Furthermore, if f is continuous then F is di erentible with F 0 = f.
CHAPTER 3. CALCULUS IN R 29 2. Suppose F :[, b]! R is di erentible on (, b) with F 0 = f. Then f(x) dx = F (b) F (). Proposition 3.35 (Integrtion by prts). Suppose f,g: [, b]! R re continuous functions tht re continuously di erentible on (, b). Then f(x) g 0 (x) dx = f(b)g(b) f()g() g(x)f 0 (x) dx. Proposition 3.36 (Chnge of vribles). Suppose g :[, b]! R is continuous, is di erentible on (, b), ndthtg 0 extends to continuous function on [, b]. Supposelsothtf is continuous on g([, b]). Then (f g)(x) g 0 (x) dx = Z g(b) g() f(x) dx. Proposition 3.37. Suppose tht for ech n =1, 2, 3,... we hve n integrble function f n :[, b]! R nd suppose tht f n converges to f :[, b]! R with respect to the L 1 ([, b]) norm. Then f is integrble nd f(x) dx = lim n!1 f n (x) dx.