Mathematics for economists

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1 Mthemtics for economists Peter Jochumzen September 26, 2016 Contents 1 Logic 3 2 Set theory 4 3 Rel number system: Axioms 4 4 Rel number system: Definitions 5 5 Rel numbers: Results 5 6 Rel numbers: Powers with rtionl exponent 6 7 e 6 8 Rel numbers: Logrithms 7 9 Rel numbers: inequlities 7 10 Equtions, nottion 8 11 Solving equtions 8 12 Qudrtic equtions 9 13 Two liner equtions in two unknowns Sums Functions Infinite sequences Infinite sums Liner functions Qudrtic functions Polynomils nd rtionl functions Power functions Exponentil functions Logrithmic functions Shifting grphs Composite functions 16 1

2 26 Injective, surjective nd bijective functions Inverse functions Piecewise functions Limits Limits from one side Limits t infinity Infinite its Continuity Continuity from one side Limit Lws Continuity lws Derivtives Rules for differentition Derivtives, results Higher order derivtives Differentible convex nd concve functions Tylors formul Derivtive with respect to function Elsticities Single-vrible optimiztion, definitions Single-vrible optimiztion, results Inflection points Definite integrls Indefinite integrls Indefinite integrls of specil functions Definite integrls, rules R n Mtrices, definitions Sclr multipliction, mtrix ddition nd trnspose Vectors Mtrix multipliction Mtrix Inverse 36 2

3 58 Systems of liner equtions Determinnts Definite mtrices Functions of severl vribles Derivtives of functions of severl vribles Higher order prtil derivtives Convex nd concve functions of severl vribles Chin rule for functions of severl vribles Implicit function theorem Multivrible optimiztion, 2 vribles Multivrible optimiztion, n vribles Constrined optimiztion Topics tht my be dded lter 43 1 Logic A sttement or proposition is sentence tht is either true or flse but not mbiguous. If A nd B re two sttements, then the sttement If A, then B is clled n impliction. A is clled the hypothesis nd B the conclusion of the sttement. Alterntive nottion: A = B An impliction is true unless A is true nd B is flse. Exmple: The sttement If x = 0 then x 2 = 0 (the hypothesis is x = 0 nd the conclusion is x 2 = 0 ) is true impliction nd we cn lso write x = 0 = x 2 = 0 If the hypothesis is flse then we sy tht the impliction is vcuously true. Exmple: If penguins fly then the moon is mde of cheese is vcuously true. Exmple: If x 2 = 4 then x = 2 is flse s it does not hold for x = 2. Alterntive expressions for If A, then B A is sufficient for B B is necessry for A A only if B Two sttements cn be combined in severl wys. The most importnt ones: A nd B, the conjunction of A nd B A or B, the disjunction of A nd B Not A, the negtion of A A if nd only if B, the equivlence of A nd B. It is denoted by A B (if nd only if is often bbrevited iff). A B A = B A nd B A or B not A A B T T T T T F T T F F F T F F F T T F T T F F F T F F T T 3

4 B = A is clled the converse of A = B. It is possible for sttement to be true nd the converse to be flse nd vice vers. Note tht A B = (A = B) nd (B = A) 2 Set theory If A = {, b, c} then A is set nd, b, c re elements or members of the set. is the empty set. A mens tht belongs to the set A. / A mens tht does not belong to the set A. If A nd B re sets nd every element of A is lso n element of B, then A is subset of (or is included in) B, denoted by A B. If A is subset of B, but A is not equl to B then A is lso proper (or strict) subset of B, denoted by A B. A B is the collection of elements in A or B (or both). A B is the collection of elements in A nd B. A \ B is the collection of elements in A but not in B. A C or Ā or Ã, is the complement of A, is Ω \ A where Ω is the universl set. If A B = then we sy tht A nd B re disjoint. 3 Rel number system: Axioms We ssume the existence of the rel number system consisting of the set of rel numbers R, two binry opertions + nd on R (ddition nd multipliction) nd binry reltion on R stisfying the following 16 properties (, b nd c re rbitrry rel numbers): RNA 1. Addition is commuttive, + b = b + 2. Multipliction is commuttive, b = b 3. Addition is ssocitive, ( + b) + c = + (b + c) 4. Multipliction is ssocitive, ( b) c = (b c) 5. Existence of zero, + 0 = 6. Existence of one different from zero, 1 = 7. Existence of negtive, + ( ) = 0 8. Existence of reciprocl, 1 = 1 for 0 9. Distributive lw, (b + c) = b + c 10. Reflexivity, 11. Antisymmetry, if b nd b then = b 12. Trnsitivity, if b nd b c then c 13. Totlness, either b or b (or both) 14. Preservtion of order under ddition, if b then + c b + c 15. Preservtion of order under multipliction, if 0 nd 0 b then 0 b 16. The order is complete 4

5 4 Rel number system: Definitions Subtrction: b is defined s + ( b) Division: /b is defined s b 1 for b 0. is clled the numertor, b the denomintor : b if nd only if b <: < b if nd only if b nd b >: > b if nd only if b nd b is positive if > 0 is negtive if < 0 is nonnegtive if 0 Nturl numbers: N = {1, 2, } Integers: Z = {0, ±1, ±2, } Rtionl numbers: Q. A rel number is rtionl number if it cn be written s = m/n where m, n re integers nd n 0 Irrtionl numbers: A rel number which is not rtionl number is clled n irrtionl number. Integer powers: n : If is rel number nd n nturl number then the power n is defined s } {{ }. is n terms clled the bse, nd n is the exponent. n : If is rel number, 0 nd n nturl number then n is defined s 1/ n. 0 : For 0, 0 is defined s 0 = is not defined. Fctoril: For nturl numbers n! = n. 0! is defined s 1. 5 Rel numbers: Results Some importnt results on rel numbers: R 1. ( ) = 2. ( + b) = ( ) + ( b) = b 3. 0 = 0 4. b = 0 if nd only if = 0 or b = 0 (or both) 5. b 0 if nd only if 0 nd b 0 6. if 0 then 1 0 nd ( 1 ) 1 = 7. ( b) = ( b) ( ) b = ( b) ( ) ( b) = b 8. ( + b)c = c + bc 9. ( + b)(c + d) = c + d + bc + bd 10. ( + b)( b) = 2 b ( + b) 2 = 2 + 2b + b 2 ( b) 2 = 2 2b + b /1 = / = 1 ( 0) 13. b c d = c bd (b 0, d 0) 5

6 14. b = c bc (b 0, c 0) 15. c + b c = +b c (c 0) 16. b = b = b b = b (b 0) 17. /b c/d = b d c = d bc (b 0, c 0, d 0) 18. b + c d = d+bc bd (b 0, d 0) 19. n m = n+m ( 0, n Z, m Z) 20. n m = n m ( 0, n Z, m Z) 21. ( n ) m = nm ( 0, n Z, m Z) 22. (b) n = n b n ( 0, b 0, n Z, m Z) 23. Adding the sme number to n eqution: = b if nd only if + c = b + c 24. Multiplying the sme number to n eqution ( 0): if c 0 then = b if nd only if c = b c 6 Rel numbers: Powers with rtionl exponent 1/n : If is rel nonnegtive number nd n nturl number then the eqution x n = hs unique nonnegtive solution. The solution is denoted by 1/n. Nottion: 1/2 = 1/n = n m/n : If is rel nonnegtive number nd n, m nturl numbers then m/n is defined s ( 1/n ) m m/n : If is rel positive number nd n, m nturl numbers then m/n is defined s 1/ m/n b defined b = 1, 2, b = 0, 1, b > 0 not integer b < 0 not integer > 0 OK (> 0) OK (> 0) OK (> 0) OK (> 0) = 0 OK (0) NO OK (0) NO < 0 OK OK NO NO Note 1: Sometimes 1/n is defined even when < 0 if n is odd. For exmple, 2 s ( 2) 3 = my be defined s Note 2: The power rules re not necessrily vlid when exponents re rtionl (eg 21 with = 1, n = 2, m = 1/2 would c tht 1 = 1). However, they re vlid if the bse is positive. 7 e 1. The number e: e = is n irrtionl number. 2. Power rules with bse e re vlid even with rtionl exponents : PE 3. e = e b if nd only if = b e > e b if nd only if > b Nottion: e is the sme s exp () 6

7 8 Rel numbers: Logrithms The eqution e x = hs unique solution if > 0. This solution is denoted x = log or x = ln. Logrithm rules (with bse e): L 1. e ln = for > 0. ln e = for ll. 2. ln b = ln + ln b for > 0, b > ln b = ln ln b for > 0, b > ln b = b ln for > ln = ln b if nd only if = b ln > ln b if nd only if > b More generlly, for b > 0, > 0 the eqution b x = hs unique solution. This solution is denoted x = log b. Logrithm rules (generl): LL 1. b log b = for > 0. log b b = for ll. 2. log b c = log b + log b c for > 0, c > log b c = log b log b c for > 0, c > log b c = c log b for > log b = 1 log b log 9 Rel numbers: inequlities Four inequlity symbols: < (less thn or strictly less thn) (less thn or equl to) > (greter thn or strictly greter thn) (greter thn or equl to). The first two hve the sme direction. The lst two hve the sme direction, opposite to the first two. I1. The direction of n inequlity is unchnged if the sme number is dded or subtrcted from both sides I2. The direction of n inequlity is unchnged if both sides re multiplied or divided by the sme positive number I3. The direction of n inequlity is reversed if both sides re multiplied or divided by the sme negtive number Double inequlity: < x < b mens < x nd x < b. All four inequlity symbols my be used but the direction of both must be the sme. (, b) The open intervl from to b < x < b [, b] The closed intervl from to b x b (, b] The intervl hlf open on the left from to b < x b [, b) The intervl hlf open on the right from to b x < b (, ) The unbounded intervl less thn x < (, ) The unbounded intervl greter thn x > The interior of the first four intervls is the open intervl (, b). The first four intervls re clled bounded intervls while the lst two re unbounded. Any point next to ] or { [ is clled boundry point. if 0 Absolute vlue : = if < 0 x < is equivlent to < x < x is equivlent to x 7

8 10 Equtions, nottion An eqution is n equlity contining one or more vribles, for exmple 2x + 1 = 5 is n eqution contining one vrible. Unknowns is nother word for vribles. A system of equtions is set of simultneous equtions contining one or more vribles, for exmple x + y = 2 nd x y = 0 is system of two equtions with two vribles. The vlues of the vribles tht mke the equlity or equlities true re clled solutions to the eqution(s). x = 2 is the only solution to the eqution 2x + 1 = 5 nd x = 1, y = 1 is the only solutions to the system x + y = 2 nd x y = 0. An eqution my hve no solution (e.g. x 2 = 1 mong the rel numbers), one solution (e.g. 2x + 1 = 5), n solutions where n is ny nturl number (e.g. x 2 = 1 hs two solutions, x = 1 nd x = 1) or n infinite number of solutions (e.g. x + y = 0). An identity is sttement resembling n eqution which is true for ll possible vlues of the vrible(s) it contins (e.g. (x + y)(x y) = x 2 y 2 ). An eqution my, in ddition to vribles, contin constnts, lso known s prmeters or coefficients, which re ssumed to be known. The solutions will then often be expressed in terms of the constnts. For exmple, 2x + = 1 hs the solution x = (1 )/2 if x is the vrible nd constnt. Usully, the unknowns re denoted by letters t the end of the lphbet, x, y, z, w while coefficients re denoted by letters t the beginning,, b, c, d. If, b re constnts nd x vrible then the expression + bx is sid to be liner expression in the x-vrible. If, b, c re constnts nd x, y re vribles then the expression + bx + cy is sid to be liner expression in ll the vribles. This definition my be extended to n rbitrry number of vribles. An eqution or system of equtions is sid to be liner if ll sides of the eqution(s) re liner in ll the vribles. The eqution x 2 + bx + c = 0 is clled qudrtic eqution if, b, c re constnts, 0 nd x is vrible. Two equtions or two systems of equtions re equivlent if they hve the sme solutions. For exmple, 2x + 1 = 5 is equivlent to 2x = Solving equtions Numericl nd nlyticl solutions: Finding solutions by trying different vlues for the vribles usully with computer: Numericl solutions. Finding solutions by successively replcing n eqution with simpler equivlent eqution: Anlyticl solutions. Most equtions cnnot be solved nlyticlly! Importnt exceptions: liner equtions, liner systems of equtions nd qudrtic equtions. Methods for creting equivlent equtions: Adding or subtrcting the sme quntity to both sides of n eqution. For exmple, 2x + 1 = 5 is equivlent to 2x = 4. Multiplying or dividing both sides of n eqution by non-zero constnt. For exmple, 2x = 4 is equivlent to x = 2 nd the eqution is solved. Cre must be tken when multiplying by n expression involving vrible. For exmple, if we multiply both sides of 1 x = 1 x + 1 by x we get 1 = 1 + x or x = 0 which re equivlent only if x 0. Therefore, x = 0 is not solution, in fct, the eqution hs no solution. 8

9 Applying n identity to trnsform one side of the eqution. For exmple, 2(x + 1/2) = 5 my be trnsformed into 2x + 1 = 5 using the distributive lw. For system of equtions: dding to both sides of n eqution the corresponding side of nother eqution, multiplied by the sme quntity. For exmple, for the system x + y = 2 nd x y = 0 we cn dd the second eqution to the first: 2x = 2 or x = 1. Squre both sides of n eqution if you know tht both sides hve the sme sign. For exmple, x = 2 is equivlent to x = 4. (x = 2 is not equivlent to x 2 = 4 unless you know tht x 0). Tke the nturl exponent of both sides. For exmple, ln x = 4 is equivlent to e ln x = e 4 or x = e 4. Tke the logrithm of both sides if they re positive. For exmple, e x = 2 is equivlent to ln e x = ln 2 or x = ln 2. Another useful ide when solving equtions is to use R4: b = 0 if nd only if = 0 or b = 0 (or both). For exmple, if x(x + 1) = 0 then either x = 0 or x + 1 = 0 nd the solutions re x = 0, x = Qudrtic equtions A qudrtic eqution cn be written s x 2 + bx + c = 0 where, b, c re constnts, 0 nd x is vrible. A reduced qudrtic eqution cn be written s where p, q re constnts nd x is vrible. x 2 + px + q = 0 The discriminnt of qudrtic eqution is defined s d = b 2 4c The discriminnt of reduced qudrtic eqution is defined s d = p 2 4q If the discriminnt is positive then the qudrtic eqution hs two distinct roots (solutions): nd the reduced qudrtic eqution: b ± d 2 p ± d 2 If the discriminnt is zero then the qudrtic eqution hs exctly one root (clled double root): nd the reduced qudrtic eqution: b 2 If the discriminnt is negtive, then there re no rel roots. p 2 For b = 0, the eqution x 2 + c = 0 hs solutions x = ± c/ s long s c/ 0 (otherwise there re no rel roots). For p = 0, the eqution x 2 + q = 0 hs solutions x = ± q s long s q 0 (otherwise there re no rel roots). 9

10 For c = 0, the eqution x 2 + bx = 0 is equivlent to x(x + b) = 0 with solutions x = 0 nd x = b/. For q = 0, the eqution x 2 + px = 0 is equivlent to x(x + p) = 0 with solutions x = 0 nd x = p. If the discriminnt is positive then x 2 + bx + c = (x x 1 )(x x 2 ) where x 1, x 2 re the distinct solutions to x 2 + bx + c = 0. Also x 2 + px + q = (x x 1 )(x x 2 ) where x 1, x 2 re the distinct solutions to x 2 + px + q = 0. If the discriminnt is zero then x 2 + bx + c = (x x 1 ) 2 where x 1 is the unique solution to x 2 + bx + c = 0. Also x 2 + px + q = (x x 1 ) 2 where x 1 is the unique solution to x 2 + px + q = 0. Completing the squre : x 2 + bx + c = (x + b 2 )2 + c b2 4 x 2 + px + q = (x + p 2 )2 + q p Two liner equtions in two unknowns Eqution 1: x + by = c. Eqution 2: dx + ey = f where x, y re vribles nd, b, c, d, e, f re constnts. The system cn hve no solution, exctly one solution or n infinite number of solutions. Two primry methods of solving the system: Method of substitution: Solve one of the equtions for one of the vribles in terms of the other nd substitute the solution into the other eqution. Method of eintion: Adding to both sides of one eqution the corresponding side of the other eqution, multiplied by the sme quntity to einte one vrible. 14 Sums n numbers x 1, x 2,, x n is clled sequence of numbers, sometimes denoted by {x i } for i = 1, 2,, n If x 1, x 2,, x n is sequence of n numbers then x 1 + x x n is denoted by n i=1 x i Summtion rules: S1: The sum of constnt S2: Homogeneity n c = nc i=1 n n cx i = c i=1 i=1 x i 10

11 S3: Additivity S4: Breking up sum (1 < m < n) n n (x i + y i ) = x i + n i=1 i=1 i=1 y i n m n x i = x i + i=1 i=1 i=m+1 x i Some specil sums The verge or smple men of x 1, x 2,, x n is denoted by x nd is defined by x = 1 n The smple vrince of x 1, x 2,, x n is defined by 1 n 1 n i=1 x i n (x i x) 2 i=1 Other importnt sums: n (x i x) 2 = i=1 n i=1 n i=0 n i=1 n x 2 i n x 2 i=1 i = 1 n(n + 1) 2 i 2 = 1 n(n + 1)(2n + 1) 6 r i = 1 rn+1 1 r r 0 15 Functions A function with domin A nd codomin B is rule tht for ech element in A ssigns unique element in B. We denote function by f : A B. If x A is n rbitrry element in the domin then f(x) denotes the element in B ssigned by the function. This element is often denoted by y, y = f(x). The rnge of f is the set of ll elements in the codomin B reched by the function. Tht is, y is in the rnge of f if f(x) = y for some x A. If the codomin B is R or subset of R we sy tht the function is rel-vlued. If the domin A is R or subset of R we sy tht we hve function of rel vrible. For exmple, f : R R is rel-vlued function of rel vrible. In the remining bullets, ll functions re rel-vlued of rel vrible defined by formul such s f(x) = x 2. If the domin is not specified, it is given by the nturl domin, which is the subset of R for which the formul is vlid. For exmple, the nturl domin of f defined by f(x) = x is [0, ) while it is ll of R if f(x) = x 2. If the codomin is not specified, it is given by R. 11

12 The grph of function f is the collection of ll ordered pirs (x, f(x)) where x is in the domin of f. (See lso section 39) If x 1, x 2 re rbitrry numbers in the domin of f with x 1 < x 2 then we sy tht f is incresing if f(x 1 ) f(x 2 ) strictly incresing if f(x 1 ) < f(x 2 ) decresing if f(x 1 ) f(x 2 ) strictly decresing if f(x 1 ) > f(x 2 ) (See lso section 39) If f is defined on some intervl [, b] nd x 1, x 2 re rbitrry numbers in the intervl with x 1 < x 2 nd g is liner function (see section 18) tht goes through x 1 nd x 2 then we sy tht f is convex on the intervl if g(x) f(x) on (x 1, x 2 ) strictly convex on the intervl if g(x) > f(x) on (x 1, x 2 ) concve on the intervl if g(x) f(x) on (x 1, x 2 ) strictly concve on the intervl if g(x) < f(x) on (x 1, x 2 ) If f, g re functions, we cn define new functions, f + g, f g, f g nd f/g. The nturl domin for these functions is the intersection of the domin for f nd g. h = f + g is defined by h(x) = f(x) + g(x). h = f g is defined by h(x) = f(x) g(x). h = f g or h = fg is defined by h(x) = f(x) g(x). h = f/g is defined by h(x) = f(x)/g(x). Here, ll vlues for which g(x) = 0 must be excluded from the domin. Symmetry: If f( x) = f(x) for ll x in the domin we sy tht f is even nd tht the grph is symmetric bout the y-xis or bout x = 0. If f( x) = f(x) for ll x in the domin we sy tht f is odd nd tht the grph is symmetric bout the origin. If f( x) = f( + x) for ll x in the domin we sy tht the grph is symmetric bout x =. In sections 14 to XX, ll functions re rel-vlued functions of rel vrible 16 Infinite sequences An infinite sequence is function s : N R. If n N then s(n) is often denoted by s n. An infinite sequence is lso denoted by {s n } i=1 or {s n}. The domin of s my be djusted, for exmple by including 0. Exmple. If we define s n = 1 n for ll n N then {s n} is n infinite sequence. 12

13 Roughly, we sy tht {s n } converges to number s if s n cn be mde rbitrrily close to s by choosing n sufficiently lrge. We then write n s n = s or s n s s n Formlly, s n s s n if for ll ε > 0 there exists n N such tht s ε < s n < s + ε for ll n > N. Some importnt its: 1 n 0 s n (1 + 1/n) n e s n (this often tken s the definition of e) r n 0 s n if r < 1 17 Infinite sums If {x n } is n infinite sequence then we define s where Exmple: For r < 1, since s n = n i=1 i=1 x i n s n s n = i=0 r i = 1 rn+1 1 r n i=1 x i r i = 1 1 r 1 1 r s n 18 Liner functions A function is clled liner if f(x) = x + b where, b re rbitrry constnts. This is lso written s y = x + b. The nturl domin is ll of R. The grph of liner function is stright line nd y = x + b is clled the eqution of stright line. Any stright line except the verticl lines is the grph of liner function. If f(x) = x + b then b is clled the intercept of the function nd the slope of the function. If > 0 then f is strictly incresing nd the line slnts upwrds to the right. The lrger the vlue of, the steeper is the line. If = 0 then f(x) = b nd the line is horizontl. If < 0 then f is strictly decresing nd the line slnts downwrds to the right. The lrger the bsolute vlue of, the steeper is the line. f is liner function. Two different points on the line re known: (x 1, y 1 ) nd (y 2, x 2 ) (typiclly with x 2 > x 1 ). Define x s x = x 2 x 1 nd y s y = y 2 y 1 13

14 Then the slope of the function is given by = y x For x = 1, = y so tht y increses by units s x increses by 1 unit. The eqution of the stright line pssing through (x 1, y 1 ) with slope is given by 19 Qudrtic functions y y 1 = (x x 1 ) f(x) = x 2 + bx + c with 0 then f is qudrtic function. Nturl domin is R. The grph of qudrtic function is clled prbol. The shpe is vlley if > 0 nd hill if < 0. f(x) = (x + b 2 )2 + c b2 b 4. For > 0, f hs minimum nd for < 0, f hs mximum t x = 2 for which y = c b2 b 4. The point ( 2, c b2 4 ) is clled the vertex of the prbol. The grph of qudrtic function is symmetric bout x = b Polynomils nd rtionl functions If f(x) = x 3 + bx 2 + cx + d with 0 then f is cubic function. If f(x) = n x n + n 1 x n x + 0 with n 0 then f is polynomil of degree n. If f, g re two polynomils then h = f/g is rtionl function. The nturl domin for h is ll of R except where g(x) = 0. For exmple, f(x) = 1 x is rtionl function with domin R \ Power functions if f(x) = cx r then f is power function. The nturl domin is: r is nturl number: R r is zero or negtive integer: R \ 0 r > 0, not n integer: [0, ) r < 0, not n integer: (0, ) 14

15 If f(x) = x 2 then f is power function s well s qudrtic function. If f(x) = 1 x then f is power function s well s rtionl function. For the domin (0, ), the power function is strictly incresing if r > 0 nd strictly decresing if r < Exponentil functions If f(x) = c x with > 0 then f is n exponentil function. The nturl domin is R. It is strictly incresing if > 1 (red) nd strictly decresing if 0 < < 1 (green). If f is n exponentil function then f(x + 1) = f(x) for ll x. y increses by ( 1) 100% for ech unit increse in x. If f(x) = ce x where e = then f is the nturl exponentil function. e x nd exp (x) mens the sme thing. The functions f(x) = e x (red), f(x) = e x (green) nd f(x) = e x2 (blue). 15

16 23 Logrithmic functions If f(x) = log x or f(x) = ln x then f is the nturl logrithm function. The nturl domin is (0, ) log x is strictly incresing over the nturl domin. log x < 0 for 0 < x < 1, log x = 0 for x = 1 nd log x > 0 for x > Shifting grphs The grph of f(x) + c is the grph of f(x) moved c units upwrds (downwrds if c < 0) The grph of f(x + c) is the grph of f(x) moved c units to the left (to the right if c < 0) The grph of cf(x) is the grph of f(x) stretched verticlly (stretched verticlly nd reflected bout the x-xis if c < 0) The grph of f( x) is the grph of f(x) reflected bout the y-xis. 25 Composite functions Given two functions g : A B nd f : B C where the domin of f is the sme s the codomin of g we cn define new function h : A C clled the composite of f nd g by y = h(x) = f(g(x)). h is denoted by f g, tht is, (f g)(x) = f(g(x)). f is clled the outer or exterior function nd g the inner or interior function. We will typiclly denote g(x) by u such tht h(x) = f(u). 16

17 Exmple: y = h(x) = e x2 then h = f g where u = g(x) = x 2 nd y = f(u) = e u. 26 Injective, surjective nd bijective functions f : A B is given function. f is sid to injective or one-to-one if it never mps distinct elements of its domin to the sme element of its codomin. A function which is strictly incresing or strictly decresing over its entire domin is lwys injective (the opposite is not true). Exmples: f : R R given by f(x) = x 2 is not injective. It mps 1 nd 1 to the sme number 1. f : R R given by f(x) = e x is injective s it is strictly incresing. f : [0, ) R given by f(x) = x 2 is injective. f is strictly incresing when x 0. f is sid to surjective or onto if every element y in B hs corresponding element x in A such tht f(x) = y. f reches every element in the codomin. If the codomin is selected to be the rnge of f then f is lwys surjective. Exmples: f : R R given by f(x) = 2x is surjective. It cn rech ny y simply by letting x = y/2. f : (, ) R given by f(x) = log x is lso surjective. It cn rech ny y by letting x = e y. f : R R given by f(x) = x 2 is not surjective s it cnnot rech negtive numbers. There is no x such tht x 2 = 1. The sme is true for f : R R given by f(x) = e x. However, f : R [0, ) given by f(x) = x 2 is surjective. It cn rech ny non-negtive y by letting x = y. Similrly, f : R (0, ) given by f(x) = e x is surjective. If f is both injective nd surjective we sy tht f is bijective or one-to-one correspondence. Exmples: f : R R given by f(x) = x + b is bijective whenever 0. f : [0, ] [0, ] given by f(x) = x 2 is bijective. f : [0, ] [0, ] given by f(x) = x is bijective. f : R (0, ) given by f(x) = e x is bijective. f : (0, ) R given by f(x) = log x is bijective. 27 Inverse functions If f : A B is bijective then there exists function f 1 : B A clled the inverse of f. For y B, x = f 1 (y) is the unique vlue x A which is mpped to y by f, f(x) = y. f 1 (f(x)) = x for ll x A. f(f 1 (y)) = y for ll y B. f 1 is bijective nd its inverse is f. 17

18 Exmples: f : R R given by y = x + b for 0 hs the inverse x = y b or f 1 (y) = y b. f : [0, ] [0, ] given by y = x 2 hs the inverse x = y or f 1 (y) = y. Similrly, y = x hs the inverse x = y 2 (sme domin nd codomin). f : R (0, ) given by y = e x hs the inverse x = log y or f 1 (y) = log y. f : (0, ) R given by f(x) = log x hs the inverse x = e y or f 1 (y) = e y. If f is strictly incresing or strictly decresing over its entire domin nd the codomin is the rnge of f then f is bijective invertible. 28 Piecewise functions A piecewise-defined function is function which is defined by multiple sub-functions, ech sub-function pplying to certin sub-domin of the min function s domin. Piecewise is wy of expressing the function, rther thn chrcteristic of the function itself, but it cn lso describe the nture of the function. Exmple: The following function is piecewise liner { x + 1 if 4 x 1 f(x) = x if 1 < x 3 At the endpoints of sections of the grph we drw n open circles for <, > nd closed circles for,. In the exmple bove, the first sub-function pplies when x = 1 nd f(1) = 2 s indicted by the closed circle. 29 Limits Rough introduction: Suppose tht f is defined for x ner x 0 (but not necessrily t x 0 ). We then sy tht f(x) pproches the it L s x pproches x 0 nd write f(x) = L if f(x) tends to L s x tends to x 0. We lso sy tht f hs the it L t x = x 0. Equivlent nottion: f(x) L s x x 0 Remember: even if f is defined t x 0 it is not necessrily true tht f hs the it f(x 0 ) t x = x 0 (unless f is continuous t x = x 0, see section 33). Exmples: 18

19 f(x) = x2 1 x 1 = (x+1)(x 1) x 1 nd x 0 = 1. x 0 is not in the domin of f but f is defined for ll x ner 1. For x 1, f(x) = x + 1. We drw the grph of f with n open circle t x = 1 to signl tht f is not defined t x = 1. (continued): x 2 1 x 1 2 s x 1. If f is defined s in section 28 then f hs no it s x tends to 1. f(x) = 1 for x 0 nd f(x) = 0 for x = 0. Then x 0 f(x) = 1 even though f(0) = 0. Forml definition: x 0 is sid to be it point of the domin A of function f if for every r > 0 the open intervl (x 0 r, x 0 + r) contins point from A which is distinct from x 0. x 0 my or my not be in the domin A. Suppose tht x 0 is it point of A. We then sy tht f(x) pproches the it L s x pproches x 0 if for every ε > 0, there exists δ > 0 such tht f(x) L < ε for ll x A such tht x x 0 < δ. We write f(x) = L 30 Limits from one side Rough introduction: Suppose tht f is defined for ll x < x 0 ner x 0 (but not necessrily t x 0 ). We then sy tht f(x) pproches the left-hnd it L s x pproches x 0 from the left nd write f(x) = L if f(x) tends to L s x tends to x 0 from the left. We use the nottion f(x 0 ) for the left-hnd it. Suppose tht f is defined for ll x > x 0 ner x 0 (but not necessrily t x 0 ). We then sy tht f(x) pproches the right-hnd it L s x pproches x 0 from the right nd write f(x) = L + if f(x) tends to L s x tends to x 0 from the right. We use the nottion f(x 0 +) for the right-hnd it. If f hs left-hnd it nd right-hnd it x = x 0 then f hs it t x = x 0 if nd only if its left-hnd it is equl to its right hnd it. Exmples: f defined s in section 28 hs left-hnd it 2 nd right-hnd it 1 t x = 1. x hs right hnd it 0 t x = 0 but the left hnd it cnnot be defined. It lso hs the it 0 t x = 0 (Note: some uthors will define its in different wy nd sy tht x hs no it t x = 0.) 19

20 Forml definitions: Suppose tht there is n open intervl (, x 0 ) such tht f is defined for ll x in this intervl. Then f(x) pproches the left-hnd it L s x pproches x 0 from the left if for every number ε > 0, there exists δ > 0 such tht L ε < f(x) < L + ε. whenever x 0 δ < x < x 0. Suppose tht there is n open intervl (x 0, b) such tht f is defined for ll x in this intervl. Then f(x) pproches the right-hnd it L s x pproches x 0 from the right if for every number ε > 0, there exists δ > 0 such tht L ε < f(x) < L + ε. whenever x 0 < x < x 0 + δ. 31 Limits t infinity Rough introduction: Suppose tht f is defined for rbitrry lrge x-vlues. We then sy tht f(x) pproches the it L s x tends to infinity nd write x f(x) = L if f(x) cn be mde rbitrrily close to L by mking x sufficiently lrge. is defined similrly Exmples: 1 x 0 s x e x 0 s x Forml definition: f(x) = L x Suppose tht f is defined for ll x in (, ) for some. Then f(x) pproches the it L s x pproches if for every number ε > 0, there exists c such tht L ε < f(x) < L + ε. whenever x > c. 32 Infinite its Rough introduction: Suppose tht f is defined for ll x ner x 0 (but not necessrily t x 0 ). We then sy tht f(x) hs the it s x pproches x 0 nd write f(x) = if f(x) cn be mde rbitrrily lrge s x tends to x 0. f(x) s x x 0 is defined similrly. f(x) ± s x ± is defined similrly. Exmples: 1 x 2 s x 0 log x s x 0+ 1 x s x 0+ while 1 x s x 0 e x s x Forml definition: Suppose tht there is n open intervl (, b) contining x 0 such tht f is defined for ll x in this intervl possibly with the exception of x 0. Then f(x) pproches the it s x pproches x 0 if for every c, there exists δ > 0 such tht f(x) > c whenever x 0 δ < x < x 0 + δ. 20

21 33 Continuity Rough introduction: Suppose tht f is defined for ll x ner x 0 (including x 0 ). f is continuous t x = x 0 if smll chnge in x round x = x 0 results in smll chnges in y. f is continuous if it is continuous t every point in its domin. If the domin of f is closed intervl then f is continuous if its grph is connected (no breks or jumps). Exmples: f(x) = 2x + 1 is continuous t x = 0. Further, it is continuous t every point in R so it is continuous. f defined s in section 28 is not continuous t x 0 = 1. f(1) = 2 while f(1.001) = 1.001; smll chnge in x leds to lrge chnge in y. The function is continuous t the remining points in the domin. If f(x) = x2 1 x 1 then f is not continuous t x 0 = 1 since x 0 is not in the domin of f. Forml definition (using its): f : A B is continuous t x = x 0 if it is defined t x = x 0 nd x x0 f(x) = f(x 0 ) Forml definition (Weierstrss). f : A B is continuous t x = x 0 A if x 0 is it point of A nd for ech ε > 0, there exists δ > 0 such tht f(x)f(x 0 ) < ε for ll x A such tht x x 0 < δ. 34 Continuity from one side Rough introduction: Suppose tht f is defined for ll x x 0 ner x 0. Then f is continuous from the left t x = x 0 if smll decrese in x from x = x 0 results in smll chnge in y. Suppose tht f is defined for ll x x 0 ner x 0. Then f is continuous from the right t x = x 0 if smll increse in x from x = x 0 results in smll chnge in y. If f is defined for ll x ner x 0 (bove nd below) then f is continuous t x = x 0 if nd only if it is continuous from the left s well s from the right. Exmples: f(x) = 2x + 1 is continuous from the left s well from the right t ny x = x 0. f defined s in section 28 is continuous from the left t x 0 = 1. f(1) = 2 nd f(0.999) = 1.999; smll decrese in x leds to smll chnge in y. The function is not continuous from the right t x 0 = 1. f : [0, ) R defined by f(x) = x is continuous t x = 0. Only continuity from the right mtters (Note: some uthors will define continuity in different wy nd sy tht x is not continuous t x = 0.) Forml definitions (using its): f is continuous from the left t x = x 0 if it is defined on n intervl (, x 0 ] nd x x0 f(x) = f(x 0 ) f is continuous from the right t x = x 0 if it is defined on n intervl [x 0, b) nd x x0+ f(x) = f(x 0 ) Forml definitions (Weierstrss): f is continuous from the left t x = x 0 if it is defined on n intervl (, x 0 ] nd for ech ε > 0, there exists δ > 0 such tht f(x 0 ) ε < f(x) < f(x 0 ) + ε. whenever x 0 δ < x x 0. 21

22 f is continuous from the right t x = x 0 if if it is defined on n intervl [x 0, b) nd for ech ε > 0, there exists δ > 0 such tht f(x 0 ) ε < f(x) < f(x 0 ) + ε. whenever x 0 x < x 0 + δ. Continuity, forml: A function f is continuous on n open intervl (, b) if it is continuous t every point in (, b) A function f is continuous on [, b) if it is continuous t every point in (, b) nd f(+) = f() A function f is continuous on (, b] if it is continuous t every point in (, b) nd f(b ) = f(b) A function f is continuous on [, b] if it is continuous t every point in (, b) nd f(+) = f() nd f(b ) = f(b) 35 Limit Lws In lws LM2 - LM11, c is constnt nd Limit lws LM: f(x) = L 1 nd g(x) = L 2 1. (Uniqueness of it). If exists, it is unique. 2. (Limit of constnt). 3. (Limit of x). 4. (Addition lw). 5. (Subtrction lw). 6. (Constnt lw). 7. (Multipliction lw). 8. (Division lw). If L 2 0, f(x) c = c x = x 0 (f(x) + g(x)) = L 1 + L 2 (f(x) g(x)) = L 1 L 2 cf(x) = cl 1 (f(x) g(x)) = L 1 L 2 (f(x)/g(x)) = L 1 /L 2 9. (Positive Power lw). If n is positive integer then [f(x)] n = L n (Negtive Power lw). If n is positive integer nd L 1 0 then [f(x)] n = L n x x

23 11. (Root lw). If n is positive integer nd L 1 > 0 then For the rest of the lws, f, g re rbitrry functions. n f(x) = n L (Squeeze or sndwich lw 1). If f(x) g(x) h(x) for ll x in n open intervl (, b) tht contins x 0 except possibly t x = x 0 nd f(x) = L nd h(x) = L then g(x) = L 13. (Squeeze or sndwich lw 2). If f(x) = g(x) for ll x in n open intervl (, b) tht contins x 0 except possibly t x = x 0 then f(x) = g(x) 14. (Composition lw). g : A B, f : B C nd f g : A C is the composition, (f g)(x) = f(g(x)). If g(x) = L nd f is continuous t u = L then f(g(x)) = f(l) 15. (Chnge of vribles). g : A B, f : B C nd f g : A C is the composition, (f g)(x) = f(g(x)) nd g(x) = u 0 If g does not tke the vlue u 0 in n open intervl (, b) contining x 0 except possibly t x 0 or f is continuous t u = u 0 then f(g(x)) = f(u) u u 0 36 Continuity lws In lws C3 - C11 f nd g re both continuous t x = x 0 Continuity lws C: 1. (Constnts re continuous). If c is constnt then f(x) = c is continuous on R 2. (x is continuous). f(x) = x is continuous on R 3. (Addition lw). f + g is continuous t x = x 0 4. (Subtrction lw). f g is continuous t x = x 0 5. (Constnt lw). cf is continuous t x = x 0 6. (Multipliction lw). f g is continuous t x = x 0 7. (Division lw). If g(x 0 ) 0 then f/g is continuous t x = x 0 8. (Positive Power lw). If n is positive integer then [f(x)] n is continuous t x = x 0 9. (Negtive Power lw). If n is positive integer nd f(x 0 ) 0 then [f(x)] n is continuous t x = x (Root lw). If n is positive integer nd f(x 0 ) > 0 then n f(x) is continuous t x = x (Exponentil lw). If > 0 then f(x) is continuous t x = x 0 (e x is continuous on R) For the rest of the lws, f, g re rbitrry functions. 23

24 12. (Composition lw). g : A B, f : B C nd f g : A C is the composition, (f g)(x) = f(g(x)). If g is continuous t x = x 0 nd f is continuous t u = g(x 0 ) then (f g) is continuous t x = x (Inverse lw). f : A B is bijective nd f 1 : B A is its inverse. If f is continuous t x = x 0 then f 1 is continuous t y = f(x 0 ) (log x is continuous on (0, )). 37 Derivtives If nd b with b > re two points in the domin of function f such tht (, f()) nd (b, f(b)) re two points on the grph then the stright line through these points is clled secnt. The slope of the secnt through (, f()) nd (b, f(b)) is given by f(b) f() b If you define x 0 = nd h = b then the slope of the secnt through (x 0, f(x 0 )) nd (x 0 +h, f(x 0 +h)) is given by f(x 0 + h) f(x 0 ) h The slope of the secnt is lso clled the Newton quotient. If f(x 0 + h) f(x 0 ) h 0 h exists then we sy tht f is differentible t x = x 0 nd we denote the it by f (x 0 ) clled the derivtive of f t x = x 0. The derivtive t x = x 0 cn equivlently be defined s f(x) f(x 0 ) x x 0 If f is differentible t x = x 0 then the stright line through (x 0, f(x 0 )) with slope f (x 0 ) is clled the tngent of f t x = x 0. The tngent hs eqution T 1 (x) = f(x 0 ) + f (x 0 )(x ) If f : A B we sy tht f is differentible on C A if f is differentible t every point of C. If f is differentible on C, then f : C R is function on C clled the derivtive of f on C. We typiclly use the sme nme for the vribles for f nd f ; if f(x) = x 2 we write f (x) = 2x. Alterntive nottion for f (x): df(x) dx df dx df(x)/dx d dx f(x) dy dx dy/dx Alterntive nottion for f (x 0 ), the derivtive evluted t the point x = x 0 : df(x) dx x=x0 24

25 38 Rules for differentition Derivtives of specil functions: DSF 1. (Derivtive of constnt). If f(x) = c then f is differentible on R with derivtive f (x) = 0 for ll constnts c, f(x) = c = f (x) = 0 2. (Derivtive of x). If f(x) = x then f is differentible on R with derivtive f (x) = 1, f(x) = x = f (x) = 1 3. (Derivtive of liner function) The liner function f(x) = x+b is differentible on R with derivtive f (x) = for ll constnts, b, f(x) = x + b = f (x) = 4. (Nturl power rule) If f(x) = x n where n is nturl number then f is differentible on R with derivtive f (x) = nx n 1, f(x) = x n = f (x) = nx n 1 5. (Derivtive of qudrtic function) The qudrtic function f(x) = x 2 + bx + c is differentible on R with derivtive f (x) = 2x + b for ll constnts 0, b, c, f(x) = x 2 + bx + c = f (x) = 2x + b 6. (Derivtive of 1 x ) If f(x) = 1 x then f is differentible on R \ 0 with derivtive f (x) = 1 x 2, f(x) = 1 x = f (x) = 1 x 2 7. (Integer power rule) If f(x) = 1 derivtive f (x) = n x n+1, x n where n is nturl number then f is differentible on R \ 0 with f(x) = 1 x n = f (x) = n x n+1 8. (Derivtive of x). If f(x) = x then f is differentible on (0, ) (f is not differentible t x = 0) with derivtive f (x) = 1 2 x, f(x) = x = f (x) = 1 9. (Rel power rule) If f(x) = x r where r is rel number, r 1 then f (x) = rx r 1. The domin of f is [0, ) if r 1 nd (0, ] if r < 1. 2 x f(x) = x r = f (x) = rx r (Derivtive of e x ) If f(x) = e x then f is differentible on R with derivtive f (x) = e x, f(x) = e x = f (x) = e x 11. (Derivtive of log x) If f(x) = log x then f is differentible on (0, ) with derivtive f (x) = 1 x, f(x) = log x = f (x) = 1 x Derivtives of generl functions: DGF 1. (Additive constnt). c + f is differentible t x with derivtive f (x), d dx (c + f(x)) = f (x) 25

26 2. (Multiplictive constnt). cf is differentible t x with derivtive cf (x), d dx (cf(x)) = cf (x) 3. (Addition lw). f + g is differentible t x with derivtive f (x) + g (x), d dx (f(x) + g(x)) = f (x) + g (x) 4. (Subtrction lw). f g is differentible t x with derivtive f (x) g (x), d dx (f(x) g(x)) = f (x) g (x) 5. (Multipliction lw). f g is differentible t x with derivtive f (x) g(x) + f(x) g (x), d dx [f(x)g(x)] = f (x)g(x) + f(x)g (x) 6. (Division lw). If g(x) 0 then f/g is differentible t x with derivtive [f (x) g(x) f(x) g (x)]/(g(x) 2 ), d f(x) dx g(x) = f (x) g(x) f(x) g (x) g(x) 2 7. (Composition lw or chin rule). g : A B, f : B C nd f g : A C is the composition, (f g)(x) = f(g(x)). If g is differentible t x nd f is differentible t u = g(x) then (f g) is differentible t x with derivtive f (u)g (x) = f (g(x))g (x). Alterntive nottion: If y = f(u) nd u = g(x) then dy dx = dy du du dx 8. (Logrithmic differentition). f is function such tht f(x) > 0 on the domin of f. Then by the chin rule, d ln f(x) = f(x)f (x) dx or f d ln f(x) (x) = f(x) dx 9. (Inverse function theorem). Suppose the following holds: x 0 is n rbitrry number f(x 0 ) is defined f is continuous t x 0 f is differentible t x 0 nd f (x 0 ) 0 The derivtive f is continuous t x 0 Then there exists n open intervl (, b) contining x 0 such tht the following holds: f is continuous nd differentible on (, b) f is strictly incresing or strictly decresing on (, b) f (x) 0 on (, b) f is invertible on (, b) The domin of the inverse function of f, denoted by g, is the rnge of f over (, b) g is continuous on its domin g is differentible on its domin If y 0 = f(x 0 ) then g (y 0 ) = 1 f (x 0 ) 26

27 39 Derivtives, results 1. If f is differentible t x then f is continuous t x (the opposite does not hold) 2. Suppose tht f is continuous on the closed intervl I c = [, b] nd differentible on the open intervl I o = (, b) (it my or my not be differentible t nd b). Then f (x) 0 for ll x I o if nd only if f is incresing in I c f (x) 0 for ll x I o if nd only if f is decresing in I c If f (x) > 0 for ll x I o then f is strictly incresing in I c (opposite is not true) If f (x) < 0 for ll x I o then f is strictly decresing in I c (opposite is not true) 40 Higher order derivtives If f : A B nd x A then we denote the derivtive of f t x by f (x) clled the second derivtive of f t x. Alterntive nottion for f (x): f (2) (x) d 2 f(x) dx 2 d 2 f(x)/dx 2 d 2 dx 2 f(x) d 2 y dx 2 d 2 y/dx 2 Higher order derivtives re defined similrly. The n th derivtive of f is denoted f (n) (x) d n f(x) dx n d n f(x)/dx n d n dx n f(x) d n y dx n d n y/dx n For convenience, the zero th derivtive f (0) (x) is defined s f(x). 41 Differentible convex nd concve functions Suppose tht f is continuous on n intervl I nd twice differentible on the interior (, b) (it my or my not be differentible t nd b). Then f (x) 0 for ll x (, b) if nd only if f is convex in I f (x) 0 for ll x (, b) if nd only if f is concve in I If f (x) > 0 for ll x (, b) then f is strictly convex in I (opposite is not true) If f (x) < 0 for ll x (, b) then f is strictly concve in I (opposite is not true) Exmples: For f(x) = x 2, f (x) = 2 nd the function is strictly convex on ny intervl. For f(x) = x 2, f (x) = 2 nd the function is strictly concve on ny intervl. For f(x) = x, f (x) = 0 nd the function is concve nd convex on ny intervl. f(x) = x 4 is strictly convex on ny intervl. However, f (0) = Tylors formul If f is differentible t x = x 0 then the tngent of f t x 0, T 1 (x) = f(x 0 ) + f (x 0 )(x ) is liner pproximtion to f bout x = x 0. Roughly, f(x) T 1 (x) for x close to x 0. If f is twice differentible t x = x 0 then the qudrtic pproximtion to f bout x = x 0 is given by T 2 (x) = f(x 0 ) + f (x 0 )(x ) f (x 0 )(x x 0 ) 2 The qudrtic pproximtion is typiclly better pproximtion thn the liner pproximtion for x close to x 0. 27

28 If f is n times differentible t x = x 0 then the nth-order polynomil pproximtion to f bout x = x 0, lso clled the nth-order Tylor polynomil is given by, T n (x) = f(x 0 ) + f (1) (x 0 ) 1! More? Infinite? Reminder? (x ) + f (2) (x 0 ) 2! (x x 0 ) f (n) (x 0 ) (x x 0 ) n = n! 43 Derivtive with respect to function If f nd g re two functions which re differentible t x, then we define df(x)/dg(x) s For exmple, df(x) df(x) dg(x) = dx dg(x) dx df(x) d ln x = xf (x) In prticulr ( mens chnge in nd % mens the percentge chnge in ) df(x) f(x) f(x 0 ) = f(x 0) dx x x 0 x 0 d ln f(x) dx df(x) d ln x d ln f(x) d ln x 44 Elsticities x=x0 x=x0 x=x0 x=x0 1 f(x) f(x 0 ) = = f(x 0 ) x x 0 n i=0 f(x) f(x 0) f(x 0) x x 0 % f(x 0) x 0 f(x) f(x 0 ) f(x) f(x 0 ) = x 0 = x x 0 x x x x 0 f(x 0) 0 x 0 % x 0 x 0 f(x) f(x 0 ) = = f(x 0 ) x x 0 f(x) f(x 0) f(x 0) x x 0 % f(x 0) x 0 % x 0 f (i) (x 0 ) (x x 0 ) i i! f : (0, ) (0, ) is given differentible function. The elsticity of f is new function clled Ef defined by Ef(x) = x f(x) f (x) = x dy y dx We cn lso write We use the terminology If Ef(x 0 ) > 1 then f is elstic t x = x 0 Ef(x) = If Ef(x 0 ) < 1 then f is inelstic t x = x 0 If Ef(x 0 ) = 1 then f is unit elstic t x = x 0 d ln f(x) d ln x Exmple: If f(x) = x b then ln f(x) = ln + b ln x nd Ef(x) = b. At x = x 0, Ef(x 0 ) % f(x 0) % x 0 where % mens the percentge chnge in. If in ddition the conditions for inverse function theorem holds nd g is the inverse of f then Eg(y 0 ) = 1 Ef(x 0 ) 28

29 45 Single-vrible optimiztion, definitions Minimum nd mximum points. f : A B nd x 0 A x 0 is clled mximum point for f if f(x) f(x 0 ) for ll x A. x 0 is clled strict mximum point for f if f(x) < f(x 0 ) for ll x A, x x 0. x 0 is clled minimum point for f if f(x) f(x 0 ) for ll x A. x 0 is clled strict minimum point for f if f(x) > f(x 0 ) for ll x A, x x 0. Exmples: If f(x) = x 2 then x min = 0 is strict minimum point of f If f(x) = x 2 then x mx = 0 is strict mximum point of f If f(x) = 1 then every point is mximum point nd every point is minimum point of f but none of them re strict. If f(x) = x then f hs no mximum points nd no minimum points If f(x) = x with domin (0, 1) then f hs no mximum points nd no minimum points If f(x) = x with domin [0, 1] then x min = 0 is strict minimum point of f nd x mx = 1 is strict mximum point of f A point tht is either mximum point or minimum point is clled extreme point or n optiml point of f. Minimum nd mximum vlues. f : A B nd x 0 A If x 0 is mximum point of f then f(x 0 ) is clled the mximum vlue of f. If x 0 is minimum point of f then f(x 0 ) is clled the minimum vlue of f. Exmples: If f(x) = x 2 then the minimum vlue of f is 0. If f(x) = x 2 then the mximum vlue of f is 0. If f(x) = 1 then the minimum vlue of f nd the mximum vlue of f is 1. If f(x) = x then f hs no minimum or mximum vlues. If f(x) = x with domin (0, 1) then f hs hs no minimum or mximum vlues. If f(x) = x with domin [0, 1] then the minimum vlue of f is 0 nd the mximum vlue of f is 1. Locl minimum nd mximum points. f : A B nd x 0 A x 0 is clled locl mximum point for f if there exits n open intervl (, b) contining x 0 such tht f(x) f(x 0 ) for ll x (, b) A. x 0 is clled strict locl mximum point for f if there exits n open intervl (, b) contining x 0 such tht f(x) < f(x 0 ) for ll x (, b) A, x x 0. x 0 is clled locl minimum point for f if there exits n open intervl (, b) contining x 0 such tht f(x) f(x 0 ) for ll x (, b) A. x 0 is clled strict locl minimum point for f if there exits n open intervl (, b) contining x 0 such tht f(x) > f(x 0 ) for ll x (, b), x x 0 A. Every mximum (minimum) point is locl mximum (minimum) point but the opposite is not true. 29

30 A point tht is either locl mximum point or locl minimum point is clled locl extreme point or locl optiml point of f. Locl minimum nd mximum vlues. f : A B nd x 0 A If x 0 is locl mximum point of f then f(x 0 ) is clled locl mximum vlue of f. If x 0 is locl minimum point of f then f(x 0 ) is clled locl minimum vlue of f. If f (x 0 ) = 0 then x 0 is clled sttionry or criticl point for f Exmples: If f(x) = x 2 or f(x) = x 2 then x 0 = 0 is criticl point for f. if f(x) = 1 then every point is criticl point for f. If f(x) = x then f hs no criticl points, no mtter the domin. If f(x) = x then x min = 0 is strict minimum point for f but it is not criticl point. 46 Single-vrible optimiztion, results 1. First derivtive test (necessry conditions for extreme vlues): f : I R where I is n intervl. Let (, b) be the interior of I. If f is differentible on (, b) nd x 0 (, b) is locl extreme point then x 0 must be sttionry point. Exmples: If f(x) = x 2 on [ 1, 1] then f is differentible on the interior ( 1, 1) nd the extreme point x 0 = 0 ( 1, 1) must be sttionry point which it is s f (0) = 0. If f(x) = x on [0, 1] then f is differentible on (0, 1) nd x 0 = 0 is n extreme point but since x 0 / (0, 1) it need not be sttionry point (nd it is not). If f(x) = x on [ 1, 1] then x 0 = 0 ( 1, 1) is n extreme point but since f is not differentible on ( 1, 1) it need not be criticl point (nd it is not). The opposite is not true. If f(x) = x 3 on [ 1, 1] then f is differentible on ( 1, 1) but the sttionry point x = 0 is not locl extreme point. 2. (Second derivtive test) f : I R where I is n intervl. Let (, b) be the interior of I. If f is twice differentible on (, b) nd x 0 (, b) is sttionry point then if f (x 0 ) < 0 then x 0 is strict locl mximum point f (x 0 ) > 0 then x 0 is strict locl minimum point f (x 0 ) = 0 then no conclusion my be drwn Exmples: If f(x) = x 2 then x 0 = 0 is strict locl minimum point since f (0) = 2 If f(x) = x 2 then x 0 = 0 is strict locl mximum point since f (0) = 2 If f(x) = x 3 then f (0) = 0 nd no conclusion my be drwn bout the sttionry point x 0 = 0. In this cse, it is not locl extreme point (it is clled sddle-point). If f(x) = x 4 then f (0) = 0 nd no conclusion my be drwn bout the sttionry point x 0 = 0. In this cse, it is strict locl mximum point. Similrly for f(x) = x 4 where x 0 = 0 is strict locl minimum point. If f(x) = 1 then ll points re sttionry. Since f (0) = 0 for ll x the second derivtive test is inconclusive. In this cse, every point is locl minimum nd locl mximum point, lthough not strict. 3. (Extreme vlue theorem). If f : I R is continuous nd I is closed nd bounded intervl, I = [, b] then f hs (globl) mximum nd (globl) minimum in I. Tht is, there exists c I where f hs mximum nd d I where f hs minimum, f(d) f(x) f(c) for ll x I 30

31 c nd/or d need not be unique. For f : [0, 1] R given by f(x) = 2 every point in [0, 1] is mximum point nd minimum point. f need not be differentible. 4. (Concve nd convex functions). f : I R where I is n intervl. Let (, b) be the interior of I. If f is differentible on (, b) nd f is convex (concve) on I nd x 0 (, b) is sttionry point then x 0 is minimum (mximum) point for f in I. If, in ddition, f is strictly convex (strictly concve) on I then x 0 is unique. 47 Inflection points In this section, f : I R is continuous on n intervl I. An inflection point is point on grph of function t which the function chnges from being concve to convex, or vice vers. The x-coordinte of n inflection point must be n interior point of I. If f is differentible on the interior of I then (x, f(x)) is n inflection point if x is n interior locl extreme point of the derivtive f. From the first derivtive test, if f is differentible on the interior of I (f is twice differentible) nd (x, f(x)) is n inflection point then x must be sttionry point of f or f (x) = 0. From the second derivtive test, If f is twice differentible on the interior of I (f is three times differentible) nd x is n interior sttionry point of f (f (x) = 0) then x is n infection point if f (x) 0. If f (x) = 0 no conclusion cn be mde nd we hve to investigte if there is sign-chnge in f (x). 48 Definite integrls Throughout the section, I = [, b] is given closed nd bounded intervl nd f : [, b] R is continuous function (it is continuous t every point in (, b) nd f(+) = f() nd f(b ) = f(b)). If f(x) 0 for ll x I then we define b f(x)dx s the re of the shpe bounded by the grph of the function, the x-xis nd the verticl lines through x = nd x = b. Nottion: The symbol is clled the integrl sign. The function inside the integrl between the integrl sign nd the symbol dx is clled the integrnd., b re clled the (lower nd upper) its of the integrl. dx indictes tht x is the vrible of integrtion. b f(x)dx is lwys rel number. x is dummy-vrible nd b f(x)dx = b f(u)du. 31