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# INTRODUCTIONTOCALCULUS MATH103

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This 27 page Class Notes was uploaded by Claudine Friesen on Monday September 28, 2015. The Class Notes belongs to MATH103 at University of Pennsylvania taught by N.Rimmer in Fall. Since its upload, it has received 42 views. For similar materials see /class/215391/math103-university-of-pennsylvania in Mathematics (M) at University of Pennsylvania.

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Date Created: 09/28/15

1282009 INVERSE FUNCTIONS INVERSE FUNCTIONS 73 Logarithmic Functions In this section we will learn about Logarithmic functions and natural logarithms LOGARITHMIC FUNCTIONS If a gt 0 and a 1 the exponential function fx ax is either increasing or decreasing so it is onetoone Thus it has an inverse function I which is called the logarithmic function with base a and is denoted by loga LOGARITHMIC FUNCTIONS Ifwe use the formulation of an inverse function given by 713 f1xy ltgt fyx then we have Definition 1 logaxy ltgt ayx LOGARITHMIC FUNCTIONS Thus if xgt 0 then logax is the exponent to which the base a must be raised to give x LOGARITHMIC FUNCTIONS Example 1 Evaluate a log381 b I09255 c log100001 1282009 LOGARITHMIC FUNCTIONS Example 1 LOGARITHMIC FUNCTIONS Definition 2 a Iog381 4 The cancellation equations Equations 4 since 34 81 in Section 71 when applied to the functions fx ax and f 1x logax become I09255 12 12 srnce 25 5 logamx x for every 966 R c log100001 3 a since 10393 0001 loga x x for every x gt 0 LOGARITHMIC FUNCTIONS The logarithmicfunction loga has domain 0 00 and range R LOGARITHMIC FUNCTIONS The figure shows the case where a gt 1 The most important It IS continuous srnce It IS the Inverse of logarithmic functions a continuous function namely the exponential have base a gt 1 function Its graph is the reflection of the graph of y ax about the line y X Flog agt1 LOGARITHMIC FUNCTIONS The fact that y ax is a very rapidly increasing function for Xgt 0 is reflected in the fact that y logax is a very slowly increasing function for Xgt 1 LOGARITHMIC FUNCTIONS The figure shows the graphs of y logax with various values of the base a gt 1 ISince loga1 0 the graphs of all quot Zloglx logarithmic functions F lg3 39 gt pass through the point 1 V 1 0 l quot log5 V logiuV yzlogux agtl 1282009 LOGARITHMIC FUNCTIONS The following theorem summarizes the properties of logarithmic functions PROPERTIES OF LOGARITHMS Theorem 3 If a gt 1 the function fx logaxis a onetoone continuous increasing function with domain 0 so and range R If x ygt 0 and ris any real number then 1 toga xy logax leggy 210ga i logax loga y 1 310gax rloga x PROPERTIES OF LOGARITHMS Properties 1 2 and 3 follow from the corresponding properties of exponential functions given in Section 72 PROPERTIES OF LOGARITHMS Example 2 Use the properties of logarithms in Theorem 3 to evaluate a log42 log432 b log280 log25 PROPERTIES OF LOGARITHMS Example2a Using Property 1 in Theorem 3 we have log42log4321og4232 log4 64 3 This is because 43 64 PROPERTIES OF LOGARITHMS Example 2 b Using Property 2 we have log2 80 log2 5 log2 log216 4 This is because 24 16 1282009 LIMITS OF LOGARITHMS The limits of exponential functions given in Section 72 are reflected in the following limits of logarithmic functions Compare these with this earlier figure y log x a gt1 LIMITS OF LOGARITHMS Equation 4 If a gt 1 then lim loga X 2 oo and lim loga X oo X gt X gtO In particular the yaxis is a vertical asymptote of the curve y logax LIMITS OF LOGARITHMS Example 3 Find lim lo tan2 X X gtO gm As x gt 0 we know that t tan2X gt tango 0 and the values of tare positive Hence by Equation 4 with a 10 gt 1 we have I I t 2 I t Im n X Im Xao ngo 3 H0 0910 2 00 NATURAL LOGARITHMS Of all possible bases a for logarithms we will see in Chapter 3 that the most convenient choice of a base is the number 9 which was defined in Section 72 NATURAL LOGARITHMS The logarithm with base 9 is called the natural logarithm and has a special notation loge x 111x NATURAL LOGARITHMS Definitions 5 and 6 If we put a e and replace loge with ln in 1 and 2 then the defining properties of the natural logarithm function become lnx y ltgt ey x 1nexx XE emzx xgt0 1282009 NATURAL LOGARITHMS In particular if we set X 1 we get NATURAL LOGARITHMS E g 4 Solution1 Find Xif In x 5 1n 6 1 From 5 we see that In x 5 means e5x Therefore x 95 NATURAL LOGARITHMS E g 4 Solution 1 NATURAL LOGARITHMS E g 4 Solution 2 If you have trouble working with the ln notation just replace it by loge Then the equation becomes logex 5 So by the definition of logarithm 95 X Start with the equation In X 5 Then apply the exponential function to both sides of the equation 939 e5 However the second cancellation equation in Equation 6 states that e39 X Therefore X 95 NATURAL LOGARITHMS Example 5 Solve the equation 95 39 3quot 10 We take natural logarithms of both sides of the equation and use Definition 9 h1e573X1n10 5 3x M110 3x 5 11110 1 7 5 1 10 x 3 n As the natural logarithm is found on scientific calculators we can approximate the solution tofour decimal places x 08991 NATURAL LOGARITHMS Example 6 Express In a n b as a single logarithm Using Properties 3 and 1 of logarithms we have lna 5111 1na1nb Z 1na1nZ 1nal NATURAL LOGARITHMS The following formula shows that logarithms with any base can be expressed in terms of the natural logarithm 1282009 CHANGE OF BASE FORMULA Formula 7 For any positive number a a at 1 we have lnx loga x Ina CHANGE OF BASE FORMULA Proof Let y logaX Then from 1 we have ay X Taking natural logarithms of both sides of this equation we get yln a 1n X NATURAL LOGARITHMS Scientific calculators have a key for natural logarithms 80 Formula 7 enables us to use a calculator to compute a logarithm with any base as shown in the following example Similarly Formula 7 allows us to graph any logarithmic function on a graphing calculator or computer Therefore y In x In a NATURAL LOGARITHMS Example 7 Evaluate log8 5 correct to six decimal places 5 Formula 7 gives 10g8 5 E z 0773976 NATURAL LOGARITHMS The graphs of the exponential function y ex and its inverse function the natural logarithm function are shown As the curve y 9 crosses the yaxis with a slope of 1 it follows that the reflected curve y 1n X crosses the Xaxis with a slope of 1 1282009 NATURAL LOGARITHMS In common with all other logarithmic functions with base greater than 1 the natural logarithm is a continuous increasing function defined on 0 co and the y axis is a vertical asymptote NATURAL LOGARITHMS Equation 8 If we put a e in Equation 4 then we have these limits NATURAL LOGARITHMS Example 8 Sketch the graph of the function y1nx 2 1 We start with the graph of y In x ln r limlnXoo lim lnX oo X gt X gtO NATURAL LOGARITHMS Example 8 Using the transformations of Section 13 we shift it 2 units to the right to get the graph of y lnx 2 NATURAL LOGARITHMS Example 8 Then we shift it 1 unit downward to get the graph of y 1nx 2 1 Notice that the line X 2 is a vertical asymptote since r 1x2 yLn39271 lirr21 lnx 2 1 OO 3 l l l l l l l l l l l I NATURAL LOGARITHMS We have seen that In x gt so as X gt so However this happens very slowly In fact In x grows more slowly than any positive power of X 1282009 NATURAL LOGARITHMS NATURAL LOGARITHMS To illustrate this fact we compare We graph the functions here approximate values of the functions y 1n X and y XV2 J in the table Initially the graphs grow at comparable rates Eventually though the root function far surpasses the logarithm l 3 5 Ill 50 IO Sill lllllll ll IUUIMX In 0 MN lb 23 R l 46 12 39 92 LS l lll 311 1 lo 707 00 224 RI 1 Hill llh Inf 0 L49 072 L73 055 046 02x 122 01 004 l 0 1000 X NATURAL LOGARITHMS In fact we Will be able to show In Section 78 that I lnX llm O x gtoo Xp for any positive power p So for large X the values of In x are very small compared with XP a il Math 103 Rimmer 37 Related Rates Goal Compute the rate of change of one quantity in terms of the rate of change of another quantity which may be more easily measured um Read the problem carefully Draw a diagram if possible Introduce notation Assign symbols to all quantities that are functions of time Express the given it ll tJt l l lilllJ39l l and the required rate in terms of derivatives Write an equation that relates the various quantities of the problem if necessary use the geometry of the situation to eliminate one of the variables by substitution as in Example 3 Use the Chain Rule to tlitTerentiate both sides of the equation with respeet to t Subntitute the given information into the resulting equation and solve for the unknown rate i1 w Math 103 Rimmer mg Maw 37 Related Rates A cylindrical tank with radius 5 In is bein filled with wateth at a rate of 3 In3 min OW fast is the height of the water increasing W dV 3 Find dt 2 chlinder r h V 257th since r 5 constant dV h dt dt dh 3 mmin dz 257 6X2 w Math103 Rimmer WM 37 Related Rates A plane ying horizontally at an altitude of 1 mi and a speed of 500 mihr passes directly over a radar station Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station dx o d 500 x F1nd l2 x2 z2 Zx 2 ZZ dt dt quotinstant snapshotquot 2 x x 2 dt z 1 z 2 M 250J mihr dt 2 6X3 w Ma13103aFFiimmer mwga 37 eate ates At noon ship A is 150 km west of ship B Ship A is sailing east at 35 kmhr and ship B is sailing north at 25 kmhr How fast is the distance between the ships changing at 400 pm Find t quotinstant snapshotquot Z dy x150 354 2 7 dz 5 x150 140 3x 10 10V101 100 y254 gty100 A X x is decreasing E Z V1002 102 7935 g0 10 z 10000100 dl 2 2 2 ZWm x y Z gtz1010 1 dx dy dz xdt y dt 22d dz dz 3502500 2152f 215 10 35 100 25 10 101 kmhr rgtdt 3dr mm mm tlol 6364 Mat 103Flimmer 53mg 37 e ate ates A street light is mounted at the top of a 15 ft tall pole A man 6 ft tall walks away from the pole with a speed of 5 fts along a straight path How fast is the tip of his shadow moving when he is 40 ft from the pole d x y Find dt Use similar triangles 15 6 15y6x6y xy y 9y6x 2x rx y 3 li i 2 5 dt gt x y 3x d x 15 Mzi z g fts dt 3 dt 3 X5 w Math 103 Rimmer 5 33ng 37 Related Rates Two sides of a triangle have lengths 12 m and 15 m The angle between them is increasing at a rate of 2quot n1in How fast is the length of the third side x increasing when the angle between sides of fixed length is 60 Fllld 2 2 l radians 90 Use Law of Cosines 62 at2 192 2abcos t9 49 is the A Opposite c x2 2152 122 21512Cos6 x2 369 36000s6 d6 360sin67 quotinstantsnapshotquot 2x360sin6d 6 2 d 2 2 x 6 7r f l 15 x x21369 36000s 236039739E 2366 z 96 E 2189 dt 23935 132547 3 3J dx 7Lquot 12 rnmin dt 3J7 216 1 anIDJ lemer i mm 37 Relaxed Rates Water is leaking out of an inveIted conical tank at a rate of 10000 cm3min at the same time that water is being pouted into the tank at a constant rate The La W raw water tank has height 6 m and the diameter at the top is 4 m Ifthe water level is is bang pumpec in Iising at a rate of 20 cmmin when the height of the wateris 2 m find the Find W rate at which the wateris being pumped into the tank 2 7139 7139 h 7139 Vmir2h Vi 7 h 2V7h3 3 3 3 27 d1 a M n dt 9 dt quotinstant snapshotquot Z 200 7 h 2 T W 10000 9200 20 h 200 800 000 7139 W 100007 cm3 lmmt dV dh iw 10000 720 dt W W I m out Math 103 Rimmer eamre 43 First and Second Derivative g Relation to the Graph of a function 43 Flew the derivative relates te the funetien M M graph has positive sloping f x is positive tangent lines I graph off x is above the x ax1s f is increasmg graph has negative sloping f x is negative tangent lines I f d graph off x is below the x ax1s 1s ecreasmg graph has a tangent line f x0 graph of f x comes into contact with the x axis with a slope of 0 horizontal tangent line it might go through the x axis it might not f is neither increasing or decreasing Math 103 Rimmer 43 First and Second Derivative Relation to the Graph of a function The First Derivative Test Purpose to find and classify local maximum and local minimum values Let f x be a continuous function at x c and let 6 be a critical number of f f39Xlt0 h f f c anges rom gt f hasalocal mp0 positive to negative at c max1mum at c 0i i a Local maximum ammth mm f 39 changes from gt f has a local negative to positive at 6 minimum at C 10302009 1 0302009 Math 103 Rimmer 43 First and Second Derivative Relationto the Graph of a function Math 103 Rimmer Wig 43 First and Second Derivative Example Use the first derivative test to find the local Relationmhemphmfunction maximum and local minimum values 5 f x x 5x 3 f x is a polynomial it exists 1 Find the critical numbers Off for all x gt it is never undefined f x 5x4 5 5x4 15x2 1x2 i5x ixix2 1 f x0gtxil 5 f39lx m m f 1 1 5 13 H U 2 15 3 7 the local maximum to to value of f U U f hasa fhasa f115 513 local local 15 3 1 the local minimum value off maximum minimum atx l atxl Math 103 Rimmer 43 First and Second Derivative Relation to the Graph of a function 13 How the second dervative relates to the function Concavity describes the shape of the function When the curve lies above the tangents When the curve lies belo the tangents of f on an interval 6119 we say that f of f on an interval 6119 we say that f is concave upward or concave up for short is concave downward or concave down for short 3 t B J A a Concave upward b Concave downward 0 Maxim mm an an f quotx is positive f quotx is negative Math 103 Rimmer 43 First and Second Derivative Relation to the Graph of a function When f changes concavity at a point we call that point an in ection point y A D i J l l 7 tr gt1quot C d p q a CD e CU CD cu l CU gt CD I Thomson Higher Education in ection points occur when x I C d and p 10302009 10302009 nun rRi u My tarimndsecnnd Deli The Second Derivative Test Purpose to find and classify local maximum and local minimum values Let f be continuous near x or Problem The second derivative f c O and 2 f has a local f C lt O mammum at C test does not catch local max and local min that occur whenf39 is unde ned f c O and 2 f has a local f C gt 0 minimum at c D Use the second derivative test when it isn39t that much trouble to take the second derivative Nhlhl ar immer V ng tarimndsecnnd Deliuali khlhnmlhe vhnh Example Use the second derivative test to find the local maximum and local minimum values fxx575x3 1 Find wheref c O f x 5x4 75 f x0gtxi1 Using the second derivative test local max and local min values can only come from these x 7 values 2 Evaluatef at these x 7 values f x20x3 f 120gt0 f 71720lt0 U U f has a local f has a local minimum at x 1 maximum at x 1 Math 103 Rimmer 3 15333515 43 First and Second Derivative fX x2 1 RelationtotheGraphofafunction Find the intervals where f x is increasing and decreasing Find all local maximum and local minimum values Find the intervals where f x is concave up and concave down Find all in ection points f x3x2 l22x 2 f x is a polynomial it exists f39x 6xc2 l 6cc l2 cl2 for all x gt it is never undefined 6xx 12 x12 0 x 101 These are the critical numbers off I ml fx I I I 1 W 1 to U f has a local minimum value of f 0 l at x 0 f is decreasing on oo lU l0 f is increasing on 01U1 Math 103 Rimmer swig a 43 First and Second Derivative Relation to the Graph of a function fx xZ l3 f x 6cc2 l2 m 6x2 122x2 1lt2xlt6x 6x2 1x2 14x2l 6x2 15x2 1 CCU CCD CCU CCD CCU f is concaveup on 00 1U 5 U1 in ection points 10 5 5 64 1 125 f is concave down on 1U l 10302009 Math 103 Rimmer 96x9 Hiya 3132 The Derivative Section 31 The limit of the slopes of the secant lines is the slope of the tangent line Math 103 Rimmer EM 3132 The Derivative Another expression for the slope of the tangent line Qa h fa h 242010 5152TheDlerivative If you zoom in on the point of tangency the function is quotlocally linearquot there Module 31 httpwwwstewartcacuuscomtec List the following numbers from smallest to largest vw mg i lve 0 939 2 9 0 9 2 9 4 y 1e21th steep y WI most steep N w 4 P a Thamwn mng Eduunnn g 0lt 0 lt 4 lt g 2lt g 2 242010 242010 Find the equation of the tangent line to a 1 a 3323 Smiths the graph of the function y 2x3 5x at 13 f 1 mlimmz mw haw H x 1 x 1 Hl x1 1 2 0 5 3 H4 J 2 2 3 W 2 2 3 x1im2x2 2x 3 gt 1 2 12 2 1 3 II E Equation of the tangent line m1 yzmxb gt31 1b 1 3 gtb4 H W Math 103 Rimmer 53 3132 The Derivative fx2fxl Thisiscalleda Ax x2 x1 difference quotient This is the average rate of change of y f x with respect to x over the interval x1 x2 average rate of change mPQ instantaneous rate of change slope of tangent at P a mums uuuuuuuuuuuu an 11mg 1irn M Ax O Ax xz xl x2 XI The derivative f a is the instantaneous rate of change of y f x with respect to x when x a lehl a 7 nimmev 3132 the Derivative Interpreting the derivative as a rate of change The cost of producing 5 ounces of gold from a new gold mine is Cx dollars What is the meaning of C39c 7 What are its units change in C 7 C39c measures the ratio 7 change in c A C15 is the rate of change of production cost with respect to the number of ounces produced this is called marginal cost The units for C15 are dollars per ounce What does C39800 17 mean C39800 is aratio so let39s turn 17 into a fraction C 800 When you are producing 800 ounces of gold and you increase production by l to 801 ounces cost will increase by 17 lehl a 7 nimmev Section 32 Let the number a vary fxh fx fx1h12f f39x can be thought of as a new function it is called the derivative off Iff a exists thenf is called differentiable at a f is called differentiable 0n ab if it is differentiable for all numbers in ab 242010 Find the derivative of the function using the definition of the derivative xh x fx4x7x2 fxh 4xh7xh2 4x4h 7x2 2xhh2 fxh 4364h 7JCZ 14xh 7h2 fx 4x 7362 fxh fx 4h 14xh 7h2 h4 14x 7h fxhfx r 14 ly 7h 1im4 14x 7h Mamma 7 nimmev 3132 m2 Dzvivzlivz W s Find the derivative of the function using the definition of the derivative 1 f x h f x x 7 f J f x h 1 1 1 fxhm fxh fxm V 1 1 f39x1imm hmW xh M h gm NAHUM 3 xh V Im xxh liw 4f 213 xxh xh 1H0 quotXXh xh 1 1 1 306 m Jx7 m 242010 5 Math 103 Rimmer a mge 3132 The Derivative 3 Ways for a function not to be differentiable at a a A corner Ownmm N39er Ewuwn RV y A y 0 a 3C 0 6 2 b A discontinuity c A vertical tangent em uuuuuuuuuu m gunman wa Edumhm g Math 103 Rimmer 3132 The Derivative Match the graph of each function in ad with the graph of its derivative in I IV u Ill 6 Thomson Higher Education 393 d V The main connection function sign of the slope of the tangent line derivative 2 above x axis 2 below x axis 0 2 quottouchesquot x axis a sign of the slope of the tangent line a 0 a a 0 a deriV belowthen 0 then above then 0 then below a 2 11 b sign of the slope of the tangent line adnea adnea deriV above then jump to below then jump to above b 2 IV 0 sign of the slope of the tangent line a 0 a deriV belowthen 0 then above cltgtI d sign of the slope of the tangent line a0a a0aa0a deriV above then 0 then below then 0 then above then 0 then below d 111 242010 Nhlhl ar immu w 3132 Deriwliv Animation of the graph of the derivative function httpwwwstewartcacuuscomtec Module 32 xx shim v 55 animals 242010

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