This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples by yourself, you my sk me if you hve ny question, you cn either sk me directly fter clss, or come to my office, it s C527 WH, during office hours, you my emil me before you come. * Function Nottion nd Intercepts We write y = f (x) to express the fct tht y is function of x. The independent vrible is x, the dependent vrible is y, nd f is the nme of the function. The grph of function hs n intercept where it crosses the horizontl or verticl xis. The horizontl intercept is the vlue of x such tht f (x) = 0. The verticl intercept is the vlue of y when x = 0, which is given by f (0). Exmple 1 The following figure shows the mount of nicotine, N = f (t), in mg, in person s bloodstrem s function of the time, t, in hours, since the person finished smoking cigrette. N (mg) 0.5 0.4 0.3 0.2 0.1 0 t (hours) 1 2 3 4 5 6 () Estimte f (3) nd interpret it in terms of nicotine. (b) About how mny hours hve pssed before the nicotine level is down to 0.1 mg? (c) Wht is the verticl intercept? Wht does it represent in terms of nicotine? (d) If this function hd horizontl intercept, wht would it represent? * Incresing nd Decresing Functions Pge 1 of 15
A function f is incresing if the vlues of f (x) increses s x increses. A function f is decresing if the vlues of f (x) decreses s x increses. The grph of n incresing function climbs s we move from left to right. The grph of n decresing function descends s we move from left to right. * Slope nd Rte of Chnge Given two points (x 1, y 1 ) nd (x 2, y 2 ) on the grph of liner function y = f (x). The slope is given by Slope = Rise Run = y x = y 2 y 1. x 2 x 1 * Liner Functions in Generl A liner function hs the form y = f (x) = b + mx. Its grph is line such tht () m is the slope, or the rte of chnge of y with respect to x. (b) b is the verticl intercept or vlues of y when x is zero. If m is positive, then f is incresing. If m is negtive, then f is decresing. The eqution of line of slope m through the point (x 0, y 0 ) is (.k. point slope form) y y 0 = m(x x 0 ). Exmple 2 World grin production ws 1241 million tons in 1975 nd 2048 million tons in 2005, nd hs been incresing t n pproximtely constnt rte. () Find liner function for world grin production, P, in million tons, s function of t, then number of yers since 1975. (b) Interpret the slope in terms of grin production. (c) Interpret the verticl intercept in terms of grin production. * Recognizing Dt from Liner Function Vlues of x nd y in tble could come from liner function y = b + mx if the difference in y-vlues re constnt for equl differences in x. Pge 2 of 15
Exmple 3 Which of the following tbles of vlues could represent liner function? For ech tble tht could represent liner function, find formul for tht function. () (b) (c) x 0 1 2 3 f (x) 2.5 3.0 3.5 4.0 x 0 2 4 6 g(x) 100 90 70 40 t 20 30 40 50 h(t) 2.4 2.2 2.0 1.8 * Averge Rte of Chnge For function y = f (x), the Averge Rte of Chnge (AROC) of f between x = nd x = b is given by AROC = y x The units of AROC re units of y per unit of x. f (b) f () =. b Exmple 4 Using n intervl of width 0.01, wht is the formul for the verge rte of chnge(aroc) of p(x) = 0.85 x t x = 8? * Concvity The grph of function is concve up if it bends upwrd s we move left to right, or the AROC is incresing from left to right. The grph of function is concve down if it bends downwrd s we move left to right, or the AROC is decresing from left to right. A line is neither concve up nor concve down. Exmple 5 The following tble gives vlues of g(t). Is g incresing or decresing? Is the grph of g concve up or concve down? t 0 2 4 6 8 10 g(t) 40 37 33 28 22 15 Pge 3 of 15
* The Cost Function The cost function, C(q), gives the totl cost of producing quntity q of some good. The totl costs = Fixed Costs + Vrible Costs, where Fixed Costs re incurred even if nothing is produced nd Vrible Costs depend on how mny units re produced. If C(q) is liner cost function, Fixed costs re represented by the verticl intercept. Mrginl cost is represented by the slope. * The Revenue Function The revenue function, R(q), gives the totl revenue received from firm from selling quntity, q, of some good. If the good sells for price of p per unit, then which is exctly the sme s Revenue = Price Quntity, R = pq. If the price does not depend on the quntity sold, so p is constnt, the grph of revenue s function of q is line through the origin, with slope equl to the price p. The mrginl revenue is lso represented by the slope. * The Profit Function Let π denote the profit, then Profit = Revenue Cost. π = R C. The brek-even point is the point where the profit is zero, or equivlently, revenue equls cost. If the profit function is liner function, then the mrginl profit is represented by the slope. Exmple 6 A compny tht mkes Adirondck chirs hs fixed costs of $5000 nd vrible costs of $30 per chir. The compny sells the chirs for $50 ech. () Find formul for the cost function. (b) Find formul for the revenue function. (c) Find the brek-even point. * The Generl Exponentil Function Pge 4 of 15
We sy tht P is n exponentil function of t with bse if P = P 0 t, where P 0 is the initil quntity (when t = 0) nd is the fctor by which P chnges when t increses by 1. If > 1, we hve exponentil growth; if 0 < < 1, we hve exponentil decy. The fctor is given by = 1 + r, where r is the deciml representtion of the percent rte of chnge; r my be positive (for growth) or negtive (for decy). * Comprison Between Liner nd Exponentil Functions A liner function hs constnt rte of chnge. An exponentil function hs constnt percent, or reltive, rte of chnge. Exmple 7 The nnul net sles for chocolte compny in 2008 ws 5.1 billion dollrs. In ech of the following cses, write formul for the nnul net sles, S, of this compny s function of t, where t represents the number of yers fter 2008. () The nnul net sles increses by 1.2 billion dollrs per yer. (b) The nnul net sles decreses by 0.4 billion dollrs per yer. (c) The nnul net sles increses by 4.3% per yer. (d) The nnul net sles decreses by 1% per yer. * Recognizing Dt from n Exponentil Function Vlues of t nd P in tble could come from n exponentil function P = P 0 t if rtios of P vlues re constnt for eqully spced t vlues. Exmple 8 Which of the following functions in the following tble could be liner, exponentil, or neither? Find formuls for those functions. x 2 1 0 1 2 f (x) 500 600 700 800 900 g(x) 14 20 24 29 35 h(x) 16 24 36 54 81 Pge 5 of 15
* The Fmilies of Exponentil Functions nd Number e The formul P = P 0 t gives fmily of exponentil functions with prmeters P 0 (the initil quntity) nd (the bse). If > 1, then the function is incresing. If 0 < < 1, then the functions is decresing. The lrger is, the fster the function grows; the closer is to 0, the fster the functions decys. The most commonly used bse is the number e = 2.71828..., which is clled the nturl bse. Properties of the Nturl logrithm ln (AB) = ln A + ln B (Product Rule) ( ) A ln = ln A ln B (Quotient Rule) B ln ( A p ) = p ln A (Power Rule) ln e x = x e ln x = x In ddition, ln 1 = 0 nd ln e = 1. Exmple 9 Solve 7 3 t = 5 2 t for t using nturl logrithms. * Exponentil Functions with Bse e Writing = e k, so k = ln, ny exponentil function cn be written in two forms P = P 0 t or P = P 0 e kt. If > 1, we hve exponentil growth; if 0 < < 1, we hve exponentil decy. If k > 0, we hve exponentil growth; if k < 0, we hve exponentil decy. k is clled the continuous growth or decy rte. * Doubling Time nd Hlf-Life The doubling time of n exponentilly incresing quntity is the time required for the quntity to double. The hlf-life of n exponentilly decying quntity is the time required for the quntity to be reduced by fctor of one hlf. Pge 6 of 15
Exmple 10 $5, 000 is deposited into n ccount tht doubles in vlue every 2.5 yers. () Determine the continuous growth rte of the ccount. (b) Use your work from prt () to determine how long it will tke for the ccount to rech vlue of $49, 000. Exmple 11 $5, 000 is deposited into n ccount pying 2.15% interest compunded nnully. () Determine formul for the blnce, P fter t yers. (b) How long will it tke the blnce in the ccount to triple? The derivtive of function t the point A is equl to the slope of the line tngent to the curve t A. Exmple 12 Estimte the derivtive of f (x) t x = 0 grphiclly. y 5 4 3 2 1-2 -1 0 1 2 x Exmple 13 Given the grph of f (x) below. Sketch the grph of f (x). y 3 y 2 1-3 -2-1 -1 x 1 2 3-3 -2-1 x 1 2 3-2 -3 * Definition of the Derivtive Using Averge Rtes Pge 7 of 15
For ny function f, we define the derivtive function, f, by f f (x + h) f (x) (x) = lim, h 0 h provided the limit on the right hnd side exists. The function f is sid to be differentible t ny point x t which the derivtive function is defined. We write lim f (x) x c to represent the number pproched by f (x) s x pproches c. Exmple 14 () lim (b) lim x 1 0.5+0.9e 0.2x (x 2)(3x 2 ) = x 2 x 2 4 * Wht Does the Second Derivtive Tell Us? f > 0 on n intervl mens f is incresing, so the grph of f is concve up there. f < 0 on n intervl mens f is decresing, so the grph of f is concve down there. Exmple 15 For ech function given in the following tbles, do the signs of the first nd second derivtives of the function pper to be positive or negtive over the given intervl? x 1.0 1.1 1.2 1.3 1.4 1.5 () f (x) 10.1 11.2 13.7 16.8 21.2 27.7 (b) (c) (d) x 1.0 1.1 1.2 1.3 1.4 1.5 g(x) 10.1 9.9 8.1 6.0 3.5 0.1 x 1.0 1.1 1.2 1.3 1.4 1.5 h(x) 1000 1010 1015 1018 1020 1021 x 10 20 30 40 50 w(x) 10.7 6.3 4.2 3.5 3.3 * Mrginl Anlysis The cost function, C(q), gives the totl cost of producing quntity q of some good. Define Mrginl Cost = MC(q) = C (q), which gives us tht Mrginl Cost C(q + 1) C(q). Pge 8 of 15
The revenue function, R(q), gives the totl revenue received from firm from selling quntity, q, of some good. Define Mrginl Revenue = MR(q) = R (q), which gives us tht Mrginl Revenue R(q + 1) R(q). Exmple 16 A compny s cost of producing q liters of chemicl is C(q) dollrs; this quntity cn be sold for R(q) dollrs. Suppose C(2000) = 5930 nd R(2000) = 7780. () Wht is the profit t production level of 2000? (b) If MC(2000) = 2.1 nd MR(2000) = 2.5, wht is the pproximte chnge in profit if q is incresed from 2000 to 2001? Should the compny increse or decrese production from q = 2000? (c) If MC(2000) = 4.77 nd MR(2000) = 4.32, should the compny increse or decrese production from q = 2000? * For derivtives of functions, refer to the derivtive worksheet, mke sure you cn remember those useful derivtive formuls by the end of tht worksheet. Exmple 17 Determine the derivtive of P(x) = (3x + 9)7 x + 10e x +7 x 4 * Using the Derivtive Formuls Exmple 18 Find n eqution for the tngent line t x = 1 to the grph of y = x 3 + 2x 2 5x + 7. Sketch the grph of the curve nd its tngent line on the sme xes. Exmple 19 Let h(x) = f (g(x)) nd k(x) = g( f (x)). Use the following figure to estimte () h (1) nd (b) k (2). Pge 9 of 15
8 y f (x) 6 4 2 g(x) 0 2 4 6 8 x * Testing For Locl Mxim nd Minim First Derivtive Test for Locl Mxim nd Minim Suppose p is criticl point of continuous function f. Then, s we go from left to right: If f chnges from decresing to incresing t p, then f hs locl minimum t p. If f chnges from incresing to decresing t p, then f hs locl mximum t p. Second Derivtive Test for Locl Mxim nd Minim Suppose p is criticl point of continuous function f, nd f (p) = 0. If f is concve up t p ( f (p) > 0), then f hs locl minimum t p. If f is concve down t p ( f (p) < 0), then f hs locl mximum t p. If the second derivtive test filed, then you need to use first derivtive test to determine the mximum nd minimum. * Concvity nd Inflection Points A point t which the grph of function f chnges concvity is clled n inflection point of f. If p is n inflection point of f, then either f (p) = 0 or f is undefined t p. Exmple 20 Find the criticl points nd inflection points of f (x) = x 3 9x 2 48x + 52, nd then find those locl minimums nd mximums. Pge 10 of 15
Exmple 21 Mike is building fence for grden, three sides of the enclosure will be mde up beutiful lttice fencing mteril nd the fourth side will be the wll of his house.given tht he hs 260 feet of fencing mteril vilble wht is the lrgest re he cn enclose? * Left- nd Right-Hnd Sums nd Definite Integrls Let f (t) be function tht is continuous for t b. We divide the intervl [, b] into n equl subdivisions, ech of width t, so t = b n. Let t 0, t 1, t 2,, t n be endpoints of the subdivisions. For left-hnd sum, we use the vlues of the function from the left end of the intervl. For right-hnd sum, we use the vlues of the function from the right end of the intervl. Actully, we hve Left-hnd sum = Right-hnd sum = n 1 i=0 f (t i ) t = f (t 0 ) t + f (t 1 ) t + + f (t n 1 ) t n f (t i ) t = f (t 1 ) t + f (t 2 ) t + + f (t n ) t i=1 The definite integrl of f from to b, written f (t)dt, is the limit of the left-hnd or right-hnd sums with n subdivisions of [, b] s n gets rbitrrily lrge. In other words, f (t)dt = lim n (Left-hnd sum) = lim n ( ) n 1 f (t i ) t i=0 nd f (t)dt = lim n (Right-hnd sum) = lim n ( n i=1 f (t i ) t ). Ech of these sums is clled Riemnn sum, f is clled the integrnd, nd nd b re clled the limits of integrtion. Exmple 22 Vlues for function f (t) re in the following tble. Estimte 30 f (t)dt by constructing 20 left- nd right-hnd sums with n = 5. Pge 11 of 15
t 20 22 24 26 28 30 f (t) 5 7 11 18 29 45 Exmple 23 Given the grph of y = f (t) in the below. Estimte 6 f (t)dt by constructing left- nd 0 right-hnd sums with t = 2. Drw the corresponding rectngles for both left- nd right-hnd sums. y 32 24 16 8 1 2 3 4 5 6 7 8 t * The Definite Integrl s n Are: When f (x) is Positive When f (x) is positive nd < b: Are under grph of f between nd b = f (x)dx. When f (x) is positive for some x-vlues nd negtive for others, nd < b: f (x)dx is the sum of the res bove the x-xis, counted positively, nd the res below the x-xis, counted negtively. * Are Between Two Curves Given functions f (x) nd g(x) for x b, then Are between grphs of f (x)nd g(x)for x b = f (x) g(x) dx. Exmple 24 Use definite integrl to find the re enclosed by y = 2 + 8x 3x 2 nd y = 22 + 2x. * The Nottion nd Units for the Definite Integrl Pge 12 of 15
The unit of mesurement for f (x)dx is the product of the units for f (x) nd the units for x. If f (t) is rte of chnge of quntity, then the Totl chnge in quntity between t = nd t = b is given by f (t)dt. Exmple 25 A cup of coffee t 90 is put into 20 room when t = 0. The coffees s temperture is chnging t rte of r(t) = 7(0.9 t ) per minute, with t in minutes. Estimte the coffee s temperture when t = 10. * The Fundmentl Theorem of Clculus The Fundmentl Theorem of Clculus If F (t) is continuous for t b, then F (t)dt = F(b) F(). In words: The definite integrl of the derivtive of function gives the totl chnge in the function. By The Fundmentl Theorem of Clculus, if you re sked to use ntiderivtives(without using clcultor) to estimte n integrl f (t)dt, first find the ntiderivtive F(t) of f (t), nd then F(b) F() Exmple 26 The grph of derivtive f (x) is shown in the following figure. 1 y 0-1 1 2 3 4 5 6 x Fill in the tble of vlues for f (x) given tht f (3) = 2. x 0 1 2 3 4 5 6 f (x) 2 Pge 13 of 15
Exmple 27 Consider the grph of f (x), NOT f (x), given below: f (x) 6 5 4 3 2 1 0-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 11 x -1-2 () List ll vlues of x where the function f (x) hs n inflection point. (b) List ll vlues of x where the function f (x) hs locl minimum. (c) At x = 1 the function f (x) is incresing, decresing or neither? (d) At x = 2 the function f (x) is concve up, concve down, or neither? (e) Sort the vlues of f ( 2), f (7), nd f (9). Pge 14 of 15
* Mrginl Cost nd Chnge in Totl Cost If C (q) is mrginl cost function nd C(0) is the fixed cost, Cost to increse production from units to b units = C(b) C() = Totl vrible cost to produce b units = 0 C (q)dq C (q)dq Totl cost of producing b units = Fixed cost + Totl vrible cost = C(0) + 0 C (q)dq Exmple 28 A business hs mrginl cost given by C (x) = 5 ln(2x 2 + 3) + 10 nd mrginl revenue given by R (x) = 0.2x 2 + 50 where x is the number of goods, nd cost nd revenue re in dollrs. () Explin the mening of R (30), no clcultion is necessry, include units. (b) Determine the chnge in profit when production is chnged from 5 to 35 items. Pge 15 of 15