1.1 Functions. 0.1 Lines. 1.2 Linear Functions. 1.3 Rates of change. 0.2 Fractions. 0.3 Rules of exponents. 1.4 Applications of Functions to Economics

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1 0.1 Lines Definition. Here re two forms of the eqution of line: y = mx + b y = m(x x 0 ) + y 0 ( m = slope, b = y-intercept, (x 0, y 0 ) = some given point ) slope-intercept point-slope There re two importnt vritions: A verticl line hs n eqution of the form x = C, where C is constnt. A horizontl line hs n eqution of the form y = C, where C is constnt. 0.2 Frctions Here re the rules for ing, subtrcting, multiplying n iviing frctions 1 : b + c + bc = b b c bc = b b c = c b b c = b c 0.3 Rules of exponents Recll the following rules: b 1 mens b n m = n+m 1/b mens b n m = n m ( n ) m = nm (b) n = n b n 1 If you like, you cn tke these rules s the efinitions of ition, multipliction, etc. You coul even tke them s some me up formuls tht tell us how to ply gme. However, the best thing to o is go look up where they come from so you unerstn why they mke sense. 1.1 Functions Definition. Here s the most common function nottion: f(x). In this, f is the nme of function, x represents the input, n f(x) represents the output. Usully we re tol wht f(x) equls with formul, but sometimes we re tol wht it equls with tble of numbers or with grph. Sometimes we replce f with some fmilir function such s nturl log: ln(x), or sine: sin(x). 1.2 Liner Functions Definition. A liner function is function whose grph is stright line. 1.3 Rtes of chnge Definition. Given ny function f(x), we efine verge rte of chnge from x = to x = b = f(b) f() b We sometimes bbrevite this nottion s y. This number equls x the slope of the secnt line s picture (, f()) m = f(x) f() x (b, f(b)) f(x) Definition. We sy tht function is incresing if its grph goes upwrs s we move to the right. We sy tht function is ecresing if its grph goes ownwrs s we move to the right. Incresing n ecresing functions look like combintions of the shpes shown in Figure Applictions of Functions to Economics Definition. The cost function gives the totl cost of proucing quntity of some goo. The stnr nottion is: q = quntity, C(q) = cost. In this section, unless we explicitly sy otherwise, we ssume tht C(q) is liner function, i.e. C(q) = mq + b. Then the fixe cost is efine s b n the vrible cost is efine s m.

2 Figure 1: Shpes of incresing/ecresing grphs Incresing Decresing The revenue function gives the totl revenue receive for quntity of some goo. Typicl nottion: q = quntity, R(q) = revenue. Note tht R(q) = pq, where p is the price per item. The profit function gives the totl profit for quntity of some goo. Typicl nottion π(q) = profit. Note tht π(q) = revenue minus cost = R(q) C(q). A brek-even point is quntity tht prouces 0 profit, i.e. π(q) = 0. Definition. Let C(q), R(q) n π(q) be the cost, revenue n profit functions. We efine the functions MC(q), MR(q) n Mπ(q) s follows: M C(q) (nive version) = chnge in cost resulting from incresing q by 1 M R(q) (nive version) = chnge in revenue resulting from incresing q by 1 M π(q) (nive version) = chnge in profit resulting from incresing q by 1 We nme these mrginl cost, mrginl revenue, n mrginl profit respectively. Mrginl here refers to the fct tht q hs chnge by only smll mount. Nive version mens we will lter reefine these function using erivtives. If C is liner function, then the mrginl cost equls the slope; similr comments pply to R n π. Definition. The supply curve is the grph relting the quntity q tht mnufcturers re willing to supply, to the price p for which the item cn be sol. Note tht price is viewe s the input vrible (however economists usully put p on the verticl xis bse on historicl trition). As price increses, the quntity q lso increses. The emn curve is the grph relting the quntity q tht consumers re willing to buy, to the price p. Note tht price is viewe s the input vrible (however economists usully put p on the verticl xis bse on historicl trition). As price increses, the quntity q ecreses. The equilibrium point is the intersection of the supply n emn curves. The price vlue of this intersection is clle the equilibrium price n the quntity vlue is clle the equilibrium quntity. Typicl nottions for these vlues re p n q respectively. It is ssume tht mrkets ten to move towrs, n settle in, equilibrium points. Definition. A tx juste supply or emn curve tkes the originl curve, n substitutes ifferent vlue of p into the formul for q. Let q = D(p) be the emn curve, written s function of p. Here re two types of shifte emn curves: D(p + c) flt tx/rebte of c: consumer pys p + c on ech item D(cp) percentge tx/rebte of c: consumer pys cp on ech item Let q = S(p) be the emn curve, written s function of p. Here re two types of shifte supply curves: S(p + c) flt tx/rebte of c: seller gets p + c on ech item S(cp) percentge tx/rebte of c: seller gets cp on ech item 1.5 Exponentil Functions Definition. An exponentil function is one of the form f(x) = C t where C n re constnts, n > 0. Note tht C is the y-intercept of f(t). We cn lwys write s = 1 + r. In this cse, we cll r the percentge rte of chnge. Note tht r is the eciml representtion of the percentge. Note tht r cn be negtive, in which cse is less thn 1, n the function f(x) is ecresing. All exponentil functions hve bout the sme shpe of grph: growing if > 1 n shrinking if < 1. We picture some below:

3 1. ln(b) ( = ln() + ln(b) ) 2. ln = ln() ln(b) b 3. ln( b ) = b ln() 1.7 Exponentil Growth n Decy In this section we istinguish between two wys of interpreting percentge chnge, s shown 0.8 t, 0.5 t, 0.2 t n 10 t, e t n 1.5 t. 1.6 Nturl Logrithm Definition. Let e be number tht s pproximtely The nturl logrithm function ln(x) is the inverse function of e x. In other wors ln(e x ) = x n e ln(x) = x. Note: x represents nything in the bove formuls. So in fct we lso hve ln(e y ) = y, ln(e z ) = z, ln(e ) = n ln(e 2x2 5x ) = 2x 2 5x. Similr comments pply to e ln(x). To remember wht the grph of ln(x) looks like, tke the grph of e x n reflect it over the line y = x: If r is the iscrete nnul/monthly/ hourly/per-time-perio percentge chnge (written s eciml), then we use y = C t with = 1 + r If r is the continuous percentge chnge (written s eciml) then we use y = Ce rt 1.8 New Functions From Ol If f(x) n g(x) re ny two functions then we cn combine them s follows f(x) + g(x) functions f(x) g(x) subtrct functions f(x) g(x) or f(x)g(x) multiply functions f(x) g(x) or f(x) g(x) ivie functions f(g(x)) plug formul for g(x) into f(x) Note tht the grph is not efine for n x tht is negtive or 0. Also, the x intercept is x = 1. Fct. Properties of logrithms. 1 outputs is the sme s ing the inputs. For ln(x), it s the reverse. Property (2) comes from e e b = e b. In other wors, for e x, iviing the outputs is the sme s subtrcting the inputs. For ln(x), it s the reverse. Property (3) comes from (e ) b = e b. In other wors, for e x, rising the output to the power b is the sme s multiplying the input by b. For ln(x), it s the reverse. 1 Property (1) comes from e e b = e +b. In other wors, for e x, multiplying the

4 1.9 Power Functions Recll the following rules: b 1 mens b n m = n+m 1/b mens b n m = n m ( n ) m = nm (b) n = n b n Figure 2: Four Common Power Functions You shoul recll wht the grphs of some bsic power functions look like: x 2 1, x, x3 n x. You shoul be ble to figure out the shpe without clcultor, n lso just be ble to recognize them if the grphs re shown in Figure 2 Definition. A power function is one of the form f(x) = Cx n, where C is constnt, clle the proportionlity constnt, n n is rel number. We cll n the power n we sy tht x n is power of x. We i the worksheet on power functions x 2 Even if x is negtive, squring it mkes positive y-vlue. A little bit bigger x-vlue mke lot bigger y-vlue (e.g. 2 2 = 4 but 5 2 = 25). 1 x Bigger x-vlues mke y-vlues close to 0 (like 1/100 = 0.01). There s verticl symptote t x = 0 (this is wht hppens when you try to ivie 1 by 0). x 3 x It looks kin of like x 2 on the right, but it s negtive on the left. It hs exctly the sme shpe s left hlf of x 2, but turne on it s sie.

5 2.1 Tngent n Velocity Problems Definition. The instntneous velocity is the limit of verge velocities over shorter n shorter time intervls. Definition. Given function f, n number in the omin of f, we efine the number f () s follows: f () = the number tht the frction f(x) f() pproches s x x gets close to. We cll f () the erivtive of f t. ifference quotient (of f(x) t ). We cll f(x) f() x The erivtive, f () is interpret s the instntneous rte of chnge of f(x) t. If f(t) equls position s function of time, then f () is the instntneous velocity t t =. Fct. The number f () equls the slope of the line tht is tngent to the grph of f(x) t the point (, f()). More briefly: f () is the slope of the tngent line of f t. People frequently sy the slope of f t inste of the slope of the tngent line of f t n there is nothing wrong with this provie one remembers the role of the tngent line. 2.2 The erivtive s function Definition. Given ny function f, we efine the erivtive function f s follows: f (x) = the erivtive of f t x Now we escribe wht the erivtive tells us grphiclly: If f (x) > 0 then f is incresing roun x If f (x) < 0 then f is ecresing roun x If f (x) = 0 then the grph of f is horizontl t x the 3.1 Derivtives of power functions Leibniz nottion for erivtive: f (x) = f, V (t) = V t, C (q) = C q, etc. res s the erivtive of.... E.g. x2 = 2x res s the erivtive of x 2 equls 2x. Nottion for plugging in number: f (5) = f, V (15) = V x=5 t, C (1200) = C t=15 q, etc. q=1200 Constnt rule: Liner rule: C = 0 where C is constnt. (mx + b) = m. Constnt multiple rule: constnt. Sum n Difference rule: Power rule: (C f(x)) = C f (x) where C is [f(x) ± g(x)] = f (x) ± g (x) xn = nx n 1 where n is ny rel number. Definition. The tngent line to function f(x) t point x = is given by y = m(x x 0 ) + y 0 where x 0 =, y 0 = f(), m = f (). Definition. Recll tht we cn strt with one function, f(x), n efine secon function f (x), the erivtive function of f(x). Since f (x) is function, we cn repet this process n efine the secon erivtive f (x) = the erivtive of f (x) = 2 f 2 Now we escribe wht the secon erivtive tells us grphiclly:

6 If f (x) > 0 then f is concve up If f (x) < 0 then f is concve own There re four pictures tht relte concve up/own to incresing/ecresing. f (x) > 0, f (x) > 0 f (x) < 0, f (x) > 0 Usul version 1 x = 1 x 2 1 x = 2 x xn = nx n 1 ex = e x ln(x) = 1 x 3.4 Prouct n Quotient Rules Chin rule version 1 = = n = n n 1 e = e ln ( ) = 1 Prouct rule f (x) > 0, f (x) < 0 f (x) < 0, f (x) < Derivtives of exponentils n logrithms Exponentil rule: ex = e x Generl Exponentil rule: x = ln() x Logrithm rule: ln(x) = 1 x 3.3 The Chin Rule Rule (Chin Rule in Leibniz Nottion). If y is function of z n z is function of t then y t = y z z t. Rule. The following tble shows the erivtive of vriety of bsic functions, n then chin rule version for ech bsic function. Function Nottion: (f g) = f g + f g u Leibniz Nottion: (uv) = v + u v Wor Nottion : The erivtive of the first, times the secon, plus the first, times the erivtive of the secon. Function Nottion: Leibniz Nottion: Quotient Rule ( ) f = f g f g g g 2 ) ( u v = u v u u v 2 Wor Nottion : The erivtive of the top, times the bottom, minus the top times the erivtive of the bottom, everything over the bottom squre.

7 4.1 Locl Mx n Mins Definition. Let x = c be in the omin of f(x). x = c is locl mximum if f(x) f(c) for ll x ner c (we llow enpoints) x = c is locl minimum if f(x) f(c) for ll x ner c (we llow enpoints) Definition. If x = c is in the omin of f n f (c) = 0, then we cll x = c criticl point. We lso cll the (x, y)-point (c, f(c)) criticl point. We cll f(c) the criticl vlue. Theorem 1 (First erivtive test). To fin the locl mx/mins of function f(x) o the following. 1. First fin the criticl points. 2. Figure out whether f (x) is + or on ech sie of ech criticl point (four cses, lots of pictures): f (x) = +, left of c right of c outcome + x = c locl mx + x = c locl min + + x = c neither x = c neither Definition. A 1D#tble (1st Derivtive Number Line Tble) shows the following: 1. A number line, with ech criticl point mrke on the number line. The criticl points ivie the line into regions. Any points where the f(x) is iscontinuous lso ivie the line into regions. 2. The following informtion bout the erivitve shoul be recore below the line. Ech region of the line, n ech criticl point, shoul be lbele with f > 0, f < 0 or f = 0 (or something equivlent). 3. The following informtion bout the function shoul be recore bove the line. Ech region of the line shoul be lbele with f or f (or something equivlent). Ech criticl point shoul be lbele with l.mx, l.min, or neither (or something equivlent). 1. First fin the criticl points. 2. Figure out whether f (c) is + or (three cses): f (c) outcome + locl min locl mx 0 or DNE test sys nothing 4.2 Intervls of Increse, Decrese, Concvity, Inflection points Definition. For function f(x), n inflection point is number x = c such tht f(x) chnges concvity t x = c. We fin inflection points the sme we wy fin locl mx/mins: (1) tke the secon erivtive, (2) set it equl to 0, (3) solve this eqution, (4) confirm your nswers by looking t the grph. 4.3 Globl mx n min Definition. Let x = c be in the omin of f(x). x = c is globl mximum if f(x) f(c) for ll x in the omin of f(x) (we llow enpoints) x = c is globl minimum if f(x) f(c) for ll x in the omin of f(x) (we llow enpoints) Globl mx/mins re lso sometimes clle bsolute mx/mins. Theorem 1 (Globl mx/min test (k close intervl metho )). To fin the bsolute mx/min of function f(x) on n intervl [, b], o the following. 1. Fin the criticl points of f(x) in the intervl [, b]. 2. Clculte the y-vlue of f(x) t ech criticl point. 3. Clculte the y-vlue of f(x) t ech enpoint. 4. The bsolute mx vlue is the biggest y-vlue from steps 2 n 3. The bsolute min vlue is the smllest y-vlue from steps 2 n 3. Theorem 2 (Secon erivtive test). To fin the locl mx/mins of function f(x) try the following.

8 4.4 Optimizing Cost n Revenue Recll tht mrginl cost, mrginl revenue n mrginl profit hve been efine before, but in nive fshion. Now tht we know wht the erivtive is, we cn efine these ifferently. Definition. Let C(q), R(q) n π(q) be the cost, revenue, n profit functions. We efine the functions MC(q), MR(q), Mπ(q) s follows: MC(q) (clculus version) = C (q) MR(q) (clculus version) = R (q) Mπ(q) (clculus version) = π (q) Let R(q), C(q) n π(q) = R(q) C(q) be the cost, revenue n profit functions. If MC = MR then π (q) = 0 n so π is t criticl point (mybe mximum) If MR > MC then π (q) > 0 n so incresing q will increse π. If MR < MC then π (q) < 0 n so incresing q will ecrese π. 4.5 Averge Cost Definition. The verge cost is the totl cost ivie by the number of items, in other wors (q) = C(q) q where q is the number of items, C(q) is the totl cost, n (q) is the verge cost. If (q) = MC then (q) is t criticl point (mybe minimum) If (q) < MC then incresing q will increse (q) If (q) > MC then incresing q will ecrese (q) 4.6 Elsticity of Demn Definition. Let q be the quntity of some prouct emne (bought) when the price is p (so q is function of p). elsticity E efine s E = p q q p E pproximte by E q/q p/p E interprete s: percentge chnge in emn, compre to percentge chnge in price. preicting percentge chnge in emn: q q E > 1 mens elstic emn E < 1 mens inelstic emn E p p Rule. In generl, the elsticity etermines whether R is n incresing function of p or not: If E < 1 then incresing p will increse R If E > 1 then incresing p will ecrese R If E = 1 then R is t criticl point. 5.2 The Definite Integrl Definition. If f(t) is ny function, n [, b] ny intervl tht f is efine on, we efine number f(t) t = the number tht the sum f(t) t s follows: f(t 1 ) t + f(t 2 ) t + + f(t n ) t pproches s n gets lrger n lrger (we ssume tht t 1,..., t n, re n eqully spce points in the intervl [, b], n t = b n ). We cll the sum f(t 1 ) t + f(t 2 ) t + + f(t n ) t Riemnn Sum, n we cll f(t) t the efinite integrl. If t is time, n f is velocity, then f(t) t = vel time

9 so = istnce f(t) t = istnce The units re foun s follows: units of the totl chnge = units of rte of chnge units of time If f(x) is ny positive function, then so f(x) x = height with = re f(t) t = re 5.3 The Definite Integrl s Are Fct. f(x) = Are uner f(x) from x = to x = b (when f(x) > 0 n < b) Fct. If f(x) is sometimes positive, n sometimes negtive, then f(x) = Are bove the x-xis minus re below x-xis (between grph of f(x) n the x-xis n from to b) ( < b) 5.4 The Interprettions of the Definite Integrl Fct. If we pply n integrl to rte of chnge of ny type of quntity we get the following: f(t) t = sum of terms like f(t) t so rte of chnge t = sum of rte of chnge t These will be the units of whtever quntity we were mesuring the rte of chnge of. Totl chnge mens net chnge. In other wors, it s the increse minus the ecrese of whtever quntity we were mesuring the rte of chnge of. 5.5 The Funmentl Theorem of Clculus Theorem 1 (The Funmentl Theorem of Clculus). F (b) F () = Fct. If C(q) is cost function then Totl chnge in cost from q = to q = b F (t) t = C(b) C() = One specil cse of the preceing formul is when = 0: C(b) = C(0) + 0 C (q) q C (q) q (C(0) = fixe cost) Note tht C(b) is the totl cost of mking b units. This is ifferent thn 0 b units. C (q) q which is clle the totl vrible cost of mking In other wors Note the following: = sum of chnges (becuse rte of chnge t = chnge) = totl chnge rte of chnge t = totl chnge from t = to t = b