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# Integrated Calculus IB (CHEM) MATH 1116

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This 95 page Class Notes was uploaded by Mrs. Kara Jacobs on Monday October 12, 2015. The Class Notes belongs to MATH 1116 at Georgia College & State University taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/221932/math-1116-georgia-college-state-university in Mathematics (M) at Georgia College & State University.

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Chapter 4 Power Functions 41 The Algebra of Power Functions Note This section corresponds to section 41 in your text pages 325 337 De nition 41 power function A power function is afunetion that can be written in the form f AM for some nonzero real number A and some rational number k Example 41 Which of the following are power functions Explain 1 2 f 96 x2 fa 77956 fx 109 fa 226 f 96 96 1 f 96 MO De nition 42 zero and negative exponents Ifz E R and z 31 0 and h is a positive integer then Dr Allen7s Math 1116 Class Notes 01 74 De nition 43 kth root Ifz E R and h is a positive integer then xi Furthermore7 o If h is odldl7 then ki is the unique number whose hth power is x o If h is even and z is nonnegative7 then kE is the unique nonnegative number whose hth power is x o If h is even and z is negative7 then kE is not a real number De nition 44 rational exponent Ifx E R and g is a positive rational number in 1 lowest terms then x WE Ifx lt 0 and q is even then xi is not a real number Example 42 Evaluate the following 1 1115 0 39 1116 639 071116 Theorem 24 Algebra Rules for Exponents For any numbers a y a and b such that each erpression is de ned Dr Allen7s Math 1116 Class Notes 75 Example 43 Which of the following are power functions Explain 1 fz1116 2 New 3 fx2m 5 Page 3367 Problem 48 f Dr Allen7s Math 1116 Class Notes 76 Example 44 Simplify and rewrite each expression until you can cancel a common factor from the numerator and denominator 1 3 h1 7 3 1 39 h V1h71 Example 45 Find the domain of the following functions Explain 1 we lt3 we 2 fx 4 x2 5 1 3 Page 3367 Problem 43 f 7 V5 7 x2 Exercise Set 41 Assignment Pages 335 3377 Problems 1 7 all7 16 28 all7 32 39 all7 447 467 477 507 517 687 737 75 Dr Allen7s Math 1116 Class Notes 77 42 Limits of Power Functions Note This section corresponds to section 42 in your text Recall the following terms 0 fx is continuous at z a 0 continuous function Example 46 For the following power functions a give the domain b sketch the graph and c give the intervals where the function is continuous What conclusion can you make about the continuity of power functions 1 m 95 2 M i 3 fzx 4 M g 5 fzx Dr Allen7s Math 1116 Class Notes 78 Theorem 25 Continuity of Power Functions Every power function fx AM is continuous on its domain Write the above theorem in terms of limits Hint What does it mean for a power function to be continuous on its domain Example 47 Evaluate the following limits7 if possible 1 lim 3x3 za72 1 lim 7 maizl x2 3 9 lim 41100 4 i j lim 2x 2 1419 Cf lim 1116z 417100 Dr Allen7s Math 1116 Class Notes 79 2 03 lim 7 40 5 1 l1m z 2 410 Example 48 Suppose k is a positive integer Evaluate the following limits7 if possible 1 lim M 14100 2 lim 711 14100 Example 49 Jim is asked to evaluate the following limit lim 7 7 10000x2 Jim does the fol lowing calculations lim 957 7 10000952 lim 957 7 lim 10000952 oo 7 oo 0 ls Jim7s work correct Justify Dr Allen7s Math 1116 Class Notes 80 De nition 45 function composition Let f andg be two functions The compo sition of the functions f and g denoted f o g is de ned by f 0 9M f9 Comments 0 To take the composition of two functions7 substitute g the value of the function g at s in place of z in o The domain of f o g is the set of all z in the domain of 9 such that g is in the domain of f Example 410 Let f 2 7 2 and let g 1 Find f o 2 Find 9 o Example 411 Let h 2x 58 Express the function h as a composition of two simpler functions f and 9 Example 412 Evaluate linx2 12 Dr Allen7s Math 1116 Class Notes 81 Theorem 26 Composite Limit Theorem ff is continuous at L and lim g L then 3331mm imam fL Example 413 Evaluate linx3 11395 Exercise Set 42 Problem 41 Let f g and g 1 7 4 Compute the following 1 f 0995 2 Find 9 o Problem 42 Let f 2 1 and gm 1 71 1 Compute f o 2 What is the domain of f o g 3 Graph the function f o g Problem 43 Let g f7 3 and f z 71 Sketch the graph of g Specify the domain and range DH Sketch the graph of f Specify the domain and range 9 Compute f o Graph the function f o g and specify its domain and range 7 Find a formula for What is this function7s domain Problem 44 Express each function as a composition of two simpler functions 1 Cs 3x 714 2 m 3 G 2x 7 3 Assignment Pages 345 3467 Problems 3a7 4a7 8a7 9 14 all7 23 28 all7 317 327 347 357 377 387 417 437 477 487 507 527 557 597 61 Dr Allen7s Math 1116 Class Notes 82 43 Derivatives of Power Functions Note This section corresponds to section 43 in your text Recall the following from chapter 3 o the de nition of the derivative 0 the interpretations of the derivative 0 differentiation rules In particular recall the following differentiation rule Power Rule for positive integer powers lf f x where n is a positive integer then f 96 d l L b t tt i n n e1 niznoaiond Example 414 Find f where f 3x9 7 2 We will generalize the power rule In particular we will investigate if the power rule77 holds when n 071 71 and n Does the power rule hold when n 0 Dr Allen7s Math 1116 Class Notes 83 Example 415 1 Let f 71 Find f Give the domain of f and f 2 Let f z Find f Give the domain off and f Theorem 27 Generalized Power Rule fr is my real number d 7 Example 416 Differentiate the function with respect to x 1 fx x wxw 2 fx 73 Dr Allen7s Math 1116 Class Notes 84 Example 417 Evaluate i dx x Example 418 E 1 t d z zxg va uaei dx x54x 32 Example 419 2 7 4 3 8 Hint This limit represents the derivative of what function at Evaluate linEig ma 7 what number Then use a differentiation rule Dr Allen7s Math 1116 Class Notes 85 Exercise Set 43 Problem 45 From Stewart7s Single Variable Calculus Early Transcendentals77 Section 31 o Differentiate the function 4 f V30 7 f z374z6 1 9 7 t4 8 4lt gt 11 y z2 4 13 Vr gwrg 15 As 7 18 y 53 20 N e i V 22 y 71 x2 4x 3 23 y 25 y 471392 0 Find the equation of the tangent line to the curve at the given point 33 y V5 11 Find the rst and second derivatives of the function 46 cm f7 377 Assignment Pages 358 3597 Problems 28 33 all7 617 637 65 Dr Allen7s Math 1116 Class Notes 86 Chapter 5 More Differentiation Rules and Applications of Differentiation 51 The Product and Quotient Rules Note This section corresponds to sections 63 and 72 in your text Example 51 Let gd d h 12x7 and u Find g xh x7 and f d Does f W 91901195 Theorem 28 Product Rule ff and g are di erentidble functions then fch WWW WWW The derivative of a product of 2 functions is the rst times the derivative of the second plus the second times the derivative of the rst Example 52 d xE 2x3z 4 g 795 5 Find Dr Allen7s Math 1116 Class Notes 87 Example 53 i d 54 Flnd 6 7 Theorem 29 Quotient Rule Let f and g be di erentiable with 9a 31 0 Then 1 996f 96 f969 96 g 906 The derivative of a quotient is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the square of the denominator Example 54 2 7 1 Let f 4 1 Flnd f Example 55 d 2 3 Edi 7 m ddltd41d Example 56 Let fx Find f z Dr Allen7s Math 1116 Class Notes 88 Exercise Set 51 Be prepared to discuss or present your solutions in Class Frorn Stewart7s Single Variable Calculus Early Transcendentals77 Section 32 o Differentiate 7 3x 7 1 7 2x 1 7 996 9 Vz 2x3 3 4 7 2x 1 3 11 Fy 777 y5y3 y2 y4 3 x 13 7 y 1 7 x2 t2 2 15 y t4 7 31 1 19 y 113 7 211 1 2t 21 t 25 fx c where 0 represents a constant Hint Simplify the expression x and then differentiate 0 Find the equation of the tangent line to the curve at the given point 2x z1 31 y 11 2 d 1 1951mm 41 um o 43 a Very good problem Suppose that f5 1 f 5 6 95 73 and g 5 2 Find the following values a f9 5 b fg 5 C 9f 5 Dr Allen7s Math 1116 Class Notes 89 52 The Chain Rule Note This section corresponds to section 72 in your text Example 57 Let hx z 22x 200591116 How would you nd the derivative of hz7 Using the Chain Rule we obtain h yc1116yc2 22x 2005911151295 22 Similarly suppose xx 2x 3 Using the chain rule we obtain that the derivative is given by 1 hx W4 2954953 e 6a 4 What do the functions h and have in common Based on these answers what would you guess is the chain rule If p 2x2 7 4x 1 nd p Theorem 30 Chain Rule Let y fu and u Ifg is di erentiable at z and f is di erentiable at u 9z then the composite function f o 9 de ned by f o fgz is di erentiable at z and f o 9 96 that is dy dy du 7 39 E39 The derivative of a composite function is the derivative of the outer function evalu ated at the inner function times the derivative of the inner function Dr Allen7s Math 1116 Class Notes 90 Example 58 Find y if y 5x3 W Example 59 Suppose fx 4 7 13x3 14 What differentiation rules will you use in nding the derivative of f Find f showing all steps Example 510 1 5 Suppose y 7 3 x2 22x 2008 What differentiation rules will you use in s nding the derivative of y Find y showing all steps Example 511 avery good problem Suppose that w um and u0 1v0 27 u 0 37 u 2 4 v 0 57 and v 2 6 Find w Dr Allen7s Math 1116 Class Notes 91 Exercise Set 52 Be prepared to discuss or present your solutions in class Problem 51 From Stewart7s Single Variable Calculus Early Transcendentals77 Section 34 0 Find the derivative of the function 7 4 3x2 7 25 8 4x 7 2100 9 71 2x 2 10 fz 1 95 1 11 gt W 17 g 1 4x53 z 7 x28 20 z 1 z2 2 x21 3 21 y 24 271 25 Fz121 Find the rst and second derivatives of the function 47 h xz2 1 Find the equation of the tangent line to the curve at the given point 51 y 1 2x10 01 fgz where f72 8 f 72 4 f 5 3 95 72 and 61 111 6 nd F 5 9 5 Assignment Pages 499 501 Problems 15 17 20 33 34 35 36 42 all 51 53 55 58 59 63 70 Dr Allen7s Math 1116 Class Notes 92 53 Some Major Theorems Revisited The Extreme Value Theorem If f is continuous on a closed interval 17 then f attains both an absolute maximum and an absolute minimum on 17 Intermediate Value Theorem Suppose f is continuous on the closed interval cab If N is a real number satisfying fa lt N lt fb or fb lt N lt fa7 then there exists a number 0 between a and b a lt c lt b such that fc N In other words7 if f is continuous on the closed interval 17 then f takes on every value between fa and fb Rolle7s Theorem Suppose f is continuous on the closed interval 17 f is differ entiable on the open interval ab7 and fa fb Then there is a number 0 in a7b such that f c Mean Value Theorem If f is continuous on the closed interval cab and differen tiable on the open interval 17 b7 then there exists a number 0 in 17 b such that f 0 Test for Increasing and Decreasing Functions Let I be an interval 1 If f gt 0 for all z in I7 then f is increasing on I 2 If f lt 0 for all z in I7 then f is decreasing on I Concavity Test Let I be an interval 1 If f z gt 0 for all z in I7 then the graph is concave upward on I 2 If f z lt 0 for all m in I7 then the graph is concave downward on I Dr Allen7s Math 1116 Class Notes 93 The Extreme Value Theorem 0 Where do extrema values occur 0 What are critical numbers Example 512 Find the critical numbers of the function 71 1 h 24 2 9xx 7x3 Example 513 Find the absolute maximum and absolute minimum values of f W8 7 s on the interval 08 Dr Allen7s Math 1116 Class Notes 94 Exercise Set 53 Be prepared to discuss or present your solutions in class Problem 52 From Stewart7s Single Variable Calculus Early Transcendentals77 section 41 0 Find the critical numbers of the function 31 fx x3 3x2 7 24x y71 35 gy yz iy 37 ht t1 i 25 0 Find the absolute maximum and absolute minimum values of f on the given interval 48 fx 33 7 3x1 on 03 52 fx 2 713 on 712 53 fx 962 1 on 02 55 ft 1N4 7 t2 on 712 Dr Allen7s Math 1116 Class Notes 95 54 More on How Derivatives Affect the Shape of a Graph Test for Increasing Decreasing Functions Suppose c is a critical number of a continuous function f Further suppose that f gt 0 for all z lt c and f lt 0 for all z gt c What can you deduce about a local extreme value at z 0 Example 514 Find where f 3x4 74x3 7 12x2 5 is increasing and where it is decreasing Also nd the local minimum and maximum values of f Concavity Test Dr Allen7s Math 1116 Class Notes 96 Example 513 Sketch a possible graph of a function that satis es the following conditions fz gt 0 on eoog and f z lt 0 on 100 f z gt 0 on eoo 72 and 200 and f z lt 0 on 722 0 lim fz 72 and lim fz0 Example mar Consider an 09 7 4x3 Find the domain the intercepts the intervals on which f is increasing or decreasing the local extrema tne luterva s of concavity and the in ection points Use this information to sketch the graph of m Drl Allen s Math 1116 Class Notes 97 Tips on Curve Sketching When sketching a graph nd the following domain intercepts asyrnptotes intervals of increase or decrease local rninirna and rnaxirna intervals of concavity points of in ection Example 517 Consider f 3mg 7 x Find the domain the intercepts the intervals on which f is increasing or decreasing the local extrerna the intervals of concavity and the in ection points Use this information to sketch the graph of Dr Allen7s Math 1116 Class Notes 98 Exercise Set 54 Be prepared to discuss or present your solutions in class Problem 53 For the given function nd the intervals on which f is increasing or decreasing the local extrema the intervals of concavity and the in ection points 10 f 4z33z2 7 6x1 2 12 f m Problem 54 Sketch the graph of a function f that satis es all of the given condi tions 24 f gt 0 for all z 31 1 vertical asymptote z 1 f x gt 0 if z lt 1 or z gt 3 f zlt01f1ltzlt3 25 f 0f 2f 40f zgt0iflt00r2ltzlt4f lt0if0ltzlt2 orxgt4f zgt0if1ltzlt3andf xlt0ifxlt1orzgt3 29 f lt 0 and f x lt 0 for all z Problem 55 For the given function nd the domain the intercepts the intervals on which the function is increasing or decreasing the local extrema the intervals of concavity and the in ection points Use this information to graph the given function 34 f 2 3 if 39 AW xxx 3 41 Cz 9595 4 Problem 56 Consider f azzi 1 Find the domain the intercepts the intervals on which f is increasing or decreasing the local extrema the intervals of concavity and the in ection points Use this information to graph Dr Allen7s Math 1116 Class Notes 99 55 Optimization Problems Example 518 A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river If the farmer does not fence the side along the river7 what is the largest area that can be enclosed Theorem 31 First Derivative Test for Absolute Extreme Values Suppose that c is a critical number of a continuous function f de ned on an interval 1 If f z gt 0 for all z lt c and f z lt 0 for all z gt c then fc is the absolute maximum value off 2 If f z lt 0 for all z lt c and f z gt 0 for all z gt c then fc is the absolute minimum value off Guidelines for Solving Optimization Problems Step 1 If possible7 draw an appropriate gure and label the quantities relevant to the problem Step 2 Find a formula for the quantity to be maximized or minimized Step 3 Express the quantity to be maximized or minimized as a function of one variable This may involve using conditions constraints stated in the problem to eliminate variables Step 4 Find the interval of possible values for this variable from the physical re strictions in the problem that is7 determine the values for which the stated problem makes sense Step 5 Determine the desired maximum or minimum value by using calculus tech niques This may involve using the Extreme Value Theorem or the First Deriva tive Test for absolute extreme values Dr Allen7s Math 1116 Class Notes 100 Example 519 Find a positive number such that the sum of the number and its reciprocal is as small as possible Example 520 A manufacturer wants to design an open box having a square base and a surface area of 108 square inches What dimensions will produce a box with maximum volume Dr Allen7s Math 1116 Class Notes 101 Example 521 A small island is 2 miles from the nearest point P on the straight shoreline of a large lake Nikki7 who has been camping on the island for three days7 is in a hurry to get more supplies The nearest town is 10 miles down the shore from P If Nikki can row a boat 3 miles per hour and can walk 4 miles per hour7 where should she land the boat in order to arrive at the town in the least time How long will it take her to get to the town if she takes the route that minimizes her time Dr Allen7s Math 1116 Class Notes 102 Exercise Set 55 Be prepared to discuss or present your solutions in class Use the following guidelines in solving the following optimization problems Step 1 If possible7 draw an appropriate gure and label the quantities relevant to the problem Step 2 Find a formula for the quantity to be maximized or minimized Step 3 Express the quantity to be maximized or minimized as a function of one variable This may involve using conditions stated in the problem to eliminate variables Step 4 Find the interval of possible values for this variable from the physical re strictions in the problem that is7 determine the values for which the stated problem makes sense Step 5 Determine the desired maximum or minimum value by using calculus tech niques This may involve using the Extreme Value Theorem or the First Deriva tive Test for absolute extreme values Some functions of importance to an economist or a manufacturer are 0 cost function Cz is the total cost of producing x units of a product during some time period 0 marginal cost If Cz is the cost function7 then the marginal cost is the rate of change of C with respect to x that is7 the marginal cost function is the derivative7 Cz CT of the cost function z demand function or price function pz is the price per unit that the company can charge if it sells x units revenue function Rz is the total revenue from selling x units of the product during the time period Note that if pz is the demand function and if x units are sold7 then the total revenue is given by Rz dR marginal revenue The derivative7 Rz 7 of the revenue function is called the marginal revenue function and is the rate of change of revenue with respect to the number of units sold pro t function Pz is total pro t obtained by selling x units of the product during the time period If x units are sold7 then the total pro t is Pz Rz 7 o marginal pro t function If Pz is the pro t function7 then the marginal pro t function is given by Pz CT the derivative of the pro t function z Dr Allen7s Math 1116 Class Notes 103 Problem 57 A price function7 p7 is de ned by pz 20 4x 7 z237 where z 2 0 is the number of units 1 Find the revenue function 2 Find the marginal revenue function 3 On what interval is the total revenue increasing 4 For what number x is the marginal revenue a maximum Problem 58 A liquid form of penicillin manufactured by a pharmaceutical rm is sold in bulk at a price of 200 per unit If the total production cost in dollars for x units is Cz 5007 000 80x 0003z2 and if the production capacity of the rm is at most 30000 units in a speci ed time7 how many units of penicillin must be manufactured and sold in that time to maximize the pro t Problem 59 At which points on the curve y 1 40z3 7 3x5 does the tangent line have the largest slope Problem 510 Stewart7 Find two numbers whose sum is 23 and whose prod uct is a maximum Problem 511 Stewart7 Find two positive numbers whose product is 100 and whose sum is a minimum Problem 512 Stewart7 Find the dimension of a rectangle with perimeter 100 m whose area is as large as possible Problem 513 Stewart7 A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil measured in appropriate units is kN YN 1 2 where k is a positive constant What nitrogen level gives the best 71 yield Problem 514 Stewart7 11 A farmer wants to fence an area of 1500000 square feet in a rectangular eld and then divide it in half with a fence parallel to one of the sides of the rectangle How can he do this so as to minimize the cost of the fence Problem 515 Stewart7 55 A baseball team plays in a stadium that holds 557 000 spectators With ticket prices at 107 the average attendance had been 277 000 When ticket prices were lowered to 87 the average attendance rose to 337 000 1 Find the demand function7 assuming that it is linear 2 How should ticket prices be set to maximize revenue Dr Allen7s Math 1116 Class Notes 104 56 Implicit Differentiation Note This section corresponds to section 73 in your text Consider the equations7 y 73x 5 and z yz Example 522 Identify whether the equation is in implicit or eaplieit form 1 xy1 2 aziy6 3 yz73 4 y37yx3 CI Note 7y means that we are differentiating with respect to x Differentiate the a following expressions with respect to x d 1 7 3 MW d 2 7 3 61969 d Example 523 For the equations in Example 522 hd 1 xy1 2 aziy6 3 yz73 4 y37yx3 105 Dr Allen7s Math 1116 Class Notes Use implioit differentiation to nd 31 of my 1 Back to nding the derivative ofy with respect to a of y3 7y Procedure for Implicit Differentiation 1 Dl erentlate both sides of the equation with respect to a 2 Solve for 21 Example 5241 Give the equation of the tangent line to the curve ya 7y x3 at the point 21 Assignment Pages 5087509 Problems 77 12 all 21 23 24 26 Drl Allen s Math 1116 Class Notes 106 Chapter 6 Exponential and Logarithmic Functions 61 The Algebra of Exponential Functions De nition 61 exponential function Ah exponential function is a mction that can be written in the form Ab for some real numbers A and b such that A 74 0 b gt 0 and b 74 1 Example 61 Consider the function 2quot 1 Sketch the graph of a 7 5 5 3 2 1 2 Give the domain of f 3 Give the range of f 4 Evaluate lim 2 5 Evaluate ljrn 2 Dr Allen s Math 1116 Class Notes 107 Example 52 On the same set o1 axes graph the touowxng exponentan tunctxons M 3 1a 5 MPG M G Obsexvatxons about gaphs o1 exponentan tunctxons ot the tonn m If 1 All gaphs pass through the poxnt z 15 a honzontal asymptote a Fox bases gxeatex than 1 the tunctxons axe Wham A Fox bases between 0 and 1 the tunctxons axe mum 5 Fox bases geatex than1 the graphs otthe tunctxons axe concave 6 Fox bases between 0 and 1 the gaphs ot the tunctxons axe concave 7 All unctxons have domaxn a All tunctxons have range Dr Allen s Math 1115 Chess Notes 108 Example 63 Determine which of the following are exponential functions Write each exponential function in the form fx Abw for some real numbers A and b with b gt 0 and b 7 0 21 1 n m 2 fx 23H4 3 fa 3m52 m Note If x y then bm by Also7 if bm by7 then x y Example 64 Find the domain of the following functions 1 L mpil 1 Wm Example 65 Solve the following equation 3311 x233 Exercise Set 61 Be prepared to discuss or present your solutions in class Assignment Taalman7 Pages 537 5387 Problems 107 117 127 147 157 187 197 207 227 247 257 267 287 297 317 347 36 Problem 61 Evaluate the following two limits by using a table of values i l 1 11351hh 1 m 2 lim 1 7 4100 z Dr Allen7s Math 1116 Class Notes 109 62 The Natural Exponential Function De nition 62 the number e The numbere is de ned as the value ofthe following limit 1 e lim1 h mo e number e is an irrational number meaning that its decimal expansion will never terminate or repeat A commonly used approximation for e is 2718 Example 66 1 it Use the de nition of e to evaluate lim 1 Example 67 eh 7 1 h Evaluate lirn mo Example 68 Consider the natural exponential function de ned by e 1 Sketch the graph of a 7 5 5 3 2 1 2 Give the domain of e 3 Give the range of e 4 Evaluate lirn e 5 Evaluate lirn e Dr Allen s Math 1116 Class Notes 110 Example 69 Let f em Use the de nition of the derivative to nd f Theorem 32 The Derivative of the Natural Exponential Function d Cl 5 7 Example 610 Find the equation of the tangent line to the graph of f em at the point 01 Example 611 Find each of the following derivatives Justify your answer d 1 7100 dzlt 6 d 2 7100 5 dz 96 d 339 i 100 61956 Dr Allen7s Math 1116 Class Notes 111 Exercise Set 62 Be prepared to discuss or present your solutions in class Problem 62 Find the derivative of each of the following functions d 1 7 2 M d em 2 i 7 dxltz2 3 d 2 39 dx 6 d m 5 EQEe d 6 7 dz6 Problem 63 Find the equation of the tangent line to the curve at the given point 1 y 4 2em at 02 2 y 2xew at 00 Problem 64 Let f em Find f1116z Problem 65 Let f 62m Find the 1116th derivative of Problem 66 Find dydx by implicit differentiation 6 z 7 y Dr Allen7s Math 1116 Class Notes 112 63 Inverse Functions Note This section corresponds to section 17 in your text Example 612 Consider the function f with domain a7b70 The function f is de ned as follows fa E N C7 and f0 A 1 Represent f with an arrow diagram What is the range of f D Construct the inverse function which we will call 9 of f Represent the func tion with an arrow diagram What is the domain of 9 What is the range of 9 9 Calculate f o g for all z in the domain of g 7 Based on your calculation above7 ll in the blank to the following statement Since 9 is the inverse of f7 fg for all z in the domain of CT Calculate g o f for all z in the domain of f 03 Based on your calculation above7 ll in the blank to the following statement Since 9 is the inverse of f7 gfz for all z in the domain of Dr Allen7s Math 1116 Class Notes 113 De nition 63 inverse of a function A function g is the inverse of a function f if and only if 1 x for allx in the domain off and 2 x for all x in the domain ofg We denote that g is the inverse off with the notation f 1x Example 613 Let 2x 1 What is f 1x7 2 Use the de nition of inverse function to show that 2x and f 1x are inVerses of each other 3 Graph 2x and f 1x on the same set of axes Also graph the identify function7 y x The graph of f 1x is the re ection of the graph of 230 about the graph of Dr Allen s Math 1116 Class Notes 114 Example 614 Does every function have an inverse function Explain De nition 64 one to one function A function f is oneto one iffor each a and b in the domain off if whenevera 31 b then fa 31 fb or equivalently if whenever fa fb then a b In other words a one to one function is afunction in which no range value repeats Theorem 33 A function f has an inverse function if and only iff is one to one Theorem 34 horizontal line test A function f is one to one if and only if every horizontal line passes through the graph off in at most one point Example 615 Do the following functions have inverses Use the horizontal line test to determine if the functions are one to one7 and hence7 invertible fx 2 7 3 g x3 h x r em Example 616 Find the inverse of f 7z3 27 if possible Dr Allenls Math 1116 Class Notes 115 Exercise Set 63 Be prepared to discuss or present your solutions in Class Problem 67 Use the de nition of inverse function to show that f 3 and g 5 are inverses of each other Problem 68 Let f xz 7 2 Pr Pr H 3 9 7 Cf 03 5 00 p Graph f Give the domain of f Give the range of f Give the range of f Why does f l exists Find f 1z Graph f 1z Give the domain of f l Give the range of f l oblem 69 Suppose f172 74 and f 174 75 Solve the equation for x oblem 610 Let f 2m H 3 9 7 CT CT 5 00 Graph f Give the domain of f Give the range of f Give the range of f Does f l exist Explain If f 1 exists graph f 1z If f 1 exists give the domain of f l If f 1 exists give the range of f l Problem 611 Taalman Pages 131 132 Problems 7 8 9 10 21 23 28 31 33 Dr Allen7s Math 1116 Class Notes 116 64 The Algebra of Logarithmic Functions Note This section corresponds to section 91 and pages 540 547 in section 82 in your text De nition 65 logarithmic function For each base b gt 0 with b 31 1 the loga rithmic function With base b denoted by y logb x is de ned by y logbd if and only if by x In other words the logarithm of a number x to a given base b is the exponent y that the base b must be raised to equal the number a The logarithmic function u logbd is the inverse of the eccponential function g bm Example 617 Evaluate the following using the de nition of logarithm 1 log5 25 2 log4 2 3 10g11161 1 4 log3 Example 618 Express 53 125 in logarithmic form Example 619 Express log464 3 in exponential form De nition 66 common logarithm A logarithm with base 10 denoted logz is called a common logarithm that is logl0 z log x Example 620 1 Evaluate log 1000 and log 7 100 Dr Allenls Math 1116 Class Notes 117 De nition 67 natural logarithm A logarithm with base 6 denoted ln x is called a natural logarithm that is loge x lnx De nition 68 natural logarithmic function The inverse of the function 9a ex is called the natural logarithmic function and is denoted by lnx Theorem 35 Some Properties of the Natural Logarithm 1 ln1 2 ln 6 5 ln 61 439 elnz Example 621 Evaluate the following 1 ln 7 61116 239 In 6111mm Example 622 Consider the natural logarithmic function de ned by lnx 1 Sketch the graph of 2 Give the domain of lnx 3 Give the range of ln x 4 Evaluate lim lnx ma0 5 Evaluate lim lnx Dr Allen s Math 1116 Class Notes 118 Theorem 36 Properties of Logarithms Suppose that b gt 0 with b 31 1 Then for all values of x y and a for which the ewpressioas below are de ned we have 1 logbb 2 logb1 Fe logb bm 439 blogb m F 10gbzy 10gb z logw 63 10gb E 10gb 10gb y gt2 lo 1 gbxii 00 logb x n logb z Example 623 Use the properties of logarithms to write each expression as a sum or difference of multiples of logarithms The expanded forms may be easier to differentiate 1 ln 43 z1 21 7 Og2271 Dr Allenls Math 1116 Class Notes 119 Example 624 Use the properties of logarithms to write each expression as a single logarithm7 if possible In solving equations involving logarithms7 it may be necessary to write the logarithm as a single quantity 1 logsz 7 2 logs 1 3 2 2ln ln2x71i 5ln2x1 Example 625 Give the domain and range of the function f loglM6 x Example 626 Find the domain of the following functions 1 log3x 1 2 logz 39 37 Exercise Set 64 Be prepared to discuss or present your solutions in class Problem 612 Taalman7 Page 6087 Problems 25 44 all Problem 613 Taalman7 Page 6087 Problems 497 53 Problem 614 Taalman7 Page 5487 Problems 15 23 all Dr Allen7s Math 1116 Class Notes 120 Problem 615 Write the expression as a single logarithm with a coef cient of 1 1 lnxlnx4 2 ln372ln4ln32 3 ln372ln4ln32 Problem 616 Use the properties of logarithms to write each expression as a sum or difference of multiples of logarithms The expanded forms may be easier to differentiate 2 ll 7 Og12 2 ln9 7 x2 Problem 617 Suppose that ln 2 a and ln3 b Use the properties of logarithms to write ln 56 in terms of a and b Problem 618 Suppose that lnx t and lny u Write each expression in terms of t and u 1 lnez 2ln1yln e Dr Allen7s Math 1116 Class Notes 121 65 Solving Exponential and Logarithmic Equa tions Example 627 Solve the following exponential equations 1 5M 20 2 7143em 1 3 62mem760 Example 628 V2 1 Find the domain of f 73 5 6 1 7 Example 629 Solve the following logarithmic equations 1 ln4x 1 Dr Allen7s Math 1116 Class Notes 122 2 logx log 9 1 3 ln 2 ln12 7 s 4 2 ln3z2 i 1 4 Exercise Set 65 Be prepared to discuss or present your solutions in Class Problem 619 Taalman Page 548 Problems 30 31 32 33 36 37 38 Problem 620 Taalman Page 608 Problems 58 60 61 62 68 Problem 621 Solve the following equation lnlnz 1116 Dr Allen7s Math 1116 Class Notes 123 66 Limits of Exponential and Logarithmic Func tions Note Exponential and logarithmic functions are continuous on their domains Example 630 Consider the exponential function fx 2m 1 What is the domain of f D Suppose a is in the domain of f What can you say about the value of the following limit and why lim 2w 141a 9 What is the inverse function of x 4 What is the domain of f 1x7 Cf Suppose c is in the domain of f l What can you say about the value of the following limit and why lim log2 z mac Theorem 37 Continuity of Exponential and Logarithmic Functions 1 IfAbc E R with b gt 0 but b 31 0 then lim Abw 2 Ifbc E R with b gt 0 but b 31 0 then lim logl7 z Example 631 Calculate the following limits H lim log2 x 418 3 lim 1116cm 410 3m 7 1 lim 412 2m 9 Dr Allen7s Math 1116 Class Notes 124 67 Derivatives of Exponential and Logarithmic Func tions We will rst explore how to calculate the derivative of an exponential function in the form f am where a gt 0 and a 31 0 If a 6 then we know that the derivative of the natural exponential function f cm is given by f Recall that we showed this using the de nition of the derivative What is the chain rule Now lets take a 2 that is we are considering the speci c exponential function f 2w Our goal is to determine If our base was 6 then we would know s how to differentiate f Recall that with logarithms we can do a base conversion Exponential functions are inverses of logarithmic functions so it seems reasonable that we could do a change of base for the exponential function f 2w Recall that the inverse of g cm is the function h In particular using the inverse properties eh for each x in the domain of lnx Using this property what is 611127 Use this information to write f 2m with base 6 f 95 2m Using the above differentiate f 2m d d 7 2m 7 d d Now we will generalize and we will nd the derivative of fx am First write the exponential function with base 6 Then differentiate using the chain rule and the rule for differentiating the natural exponential function maxi d m d a Theorem 38 Let a gt 0 Then d gm Dr Allen7s Math 1116 Class Notes 125 Example 632 d 1 7 3w dzlt d H 210 Next7 we will explore how to differentiate logarithmic functions Recall the technique of implicit differentiation Example 633 d Suppose 3 y3 6mg Find s One very useful application of implicit differentiation is to help us nd the derivative of logarithmic functions and of inverse trigonometric functions d Suppose that y lnx We would like to nd 1 First rewrite y lnx in exponential form using the de nition of logarithm 2 Note that in step 1 the equation in exponential form is in implicit form Use implicit differentiation on the equation in exponential form to nd Be sure s to express the derivative in explicit form Theorem 39 For x gt 0 d nx Dr Allen7s Math 1116 Class Notes 126 d Now suppose y loga x Find m Theorem 40 For x gt 0 d log 95 Example 634 Find the following d 1 og7 m 2 d 1 4 13 7 o x dx g 3 011 Theorem 41 For x 31 0 ion 1x1 Dr Allen7s Math 1116 Class Notes 127 Sometimes7 you may nd it helpful to use the properties of logarithms to write each expression as a sum or difference of multiples of logarithms The expanded forms may be easier to differentiate Example 635 Find In Exercise Set 66 Be prepared to discuss or present your solutions in class Problem 622 Taalman7 Page 5657 Problems 15 24 all Problem 623 Very nice problems Taalman7 Page 5657 Problems 297 307 317 32 Sim plify your results Problem 624 Differentiate the following functions 1 fx lnz2 10 2 fx log217 3x 4 ln 2 7 Hint Use properties of logarithms before differentiating 5 y lne w xe w ln z Problem 625 If f 7 nd f 1 Problem 626 Find the equation of the tangent line to the curve y lnxe 2 at the point 11 Problem 627 Taalman7 Page 6197 Problems 407 427 437 45 Dr Allen7s Math 1116 Class Notes 128 68 Logarithmic Differentiation Example 636 On a calculus test students are asked to nd the derivative of f Below are two student solutions Assign a grade of A correct C partially correct or F failure to the solutions Justify assignments of grades other than A and tell what the student is thinking 1 Jim says that the derivative of fx is given by f zm 1 2 Janet says that the derivative of f is given by f ln Jessica says that the derivative of f xx can be found by rst taking the natural logarithm of both sides of y She then says that we can use the power rule for logarithms to simplify the right hand side She notes that this equation is in implicit form and suggests that we use implicit differentiation to nd dydm She concludes that we can put the derivative in explicit form since the original function is in explicit form Using Jessica7s suggestion nd f Logarithmic differentiation is a differentiation technique used in calculating deriva tives of complicated functions involving products quotients or powers Example 637 Suppose that y e wz 1 Find y Exercise Set 67 Be prepared to discuss or present your solutions in class Problem 628 Taalman Page 628 Problems 32 36 37 41 42 43 Dr Allen7s Math 1116 Class Notes 129 Chapter 7 Trigonometric Functions 71 Radian Measure nition 71 angle An angle is the mm of mo mys hmng o camman 212 mm callei the 2mm mined by iolaiing a iay about iis endpoinl The initial side is 39 s39d ilie position oi ilie i Liv A angle is in Stan c e n air gulai oooiolinale plane w en the veiiex oi the angle is at the g p iecian oiigin and ilie mlhal side oi ilie an le is on ilie posiiive wraxls congue aios in the aio contained in the angle nL arcs then the degiee measuie oi the angle is the numbei oi the oongiuenl and quot5 interior Dr Allen s Math 1116 Class Nata o der why would someone choose to use degees to measure angles that rs w e rnto S60 preoas7 TJ sugg ts that there ls nothrng o 360 preoes and he suggests that they rnvent therr ol the mtegated oaloulus students thmk that thrs ls alantastro rdea Nrkkrsuggests that they call therr new unrt olmeasurement lor angles would someone dwlde a orrol speoral about dmdrng a orrole rnt own way ol measurrng angles All 1 Express 360 m Cechoy measure 2 Express mm m Cechoy measure 3 Express 1 m Cechoy measure A Express mm In terms ol Cechoys 5 Express 1 Cechoy In terms ol degees 6 Express 93 Caloloys m degee measure Dr Allen s Math 1115 3355 Notes 131 Jess tells the integrated calculus students that their way of measuring angles is quite festive7 but that it is not anymore natural than degree measures She tells them to consider the unit circle circle with radius one and center at the origin 1 What is the circumference of the unit circle 2 What is the distance halfway around the unit circle 3 What is the distance one quarter of the way around the unit circle 4 What is the distance one eighth of the way around the unit circle 5 How many arcs7 each of length 17 can the unit circle be divided into De nition 72 radian fa circle of radius 1 is drawn with the vertem of an angle at its center then the measure of this angle in radians abbreviated rad is the length of the are that subtends intersects the angle Example 71 1 Express 1 in radian measure 2 Express 1 radian in terms of degrees Example 72 1 Express 60 in radians 7r 2 Express 6 radians in degrees Dr Allenls Math 1116 Class Notes 132 Exercise Set 71 Be pzepaxed Lo ollseuss oz pxesent you soluuons m class Problem 71 Taalman Page 648 onblems 33737 all Problem 72 Taalman Page 649 onblems 3842 all Problem 73 Fox che unlL clzcle above label che zadlans couespondmg to me fol lowmg degzees 030456090120135150180210225240270300315330 360 Dr Allen s Math 1116 Class Nata 133 72 De nition of the Trigonometric Functions Recall 1 What is the unit circle 2 What is the circumference of the unit circle 3 For 1 revolution around the unit circle what is the length of the arc 4 What is a radian Consider the unit circle Let t E R On the unit circle above position the t aXis so that it is vertical with the positive direction up and so that t 0 is located at the point 10 in the rectangular coordinate system On your t aXis choose a t gt 0 Let s be the distance from the origin t 0 to your selected point t Mark this distance on the number line On the unit circle beginning at the point 10 travel 5 t units in the counterclockwise direction along the circle to arrive at the point Pc y Mark this distance note that is is an arc length What is the length of the are that you drew in the previous step What is the measure of the angle with initial side on the positive axis and terminal side intersecting the point Pc y Note If t lt 0 then you begin at the point 10 on the unit circle and travel 5 t units in the clockwise direction to arrive at the point 1333 y Dr Allen s Math 1116 Class Notes 134 Observation For any real number t we can locate a unique point Py on the unit circle In other words no matter what real number t you select for example you could select t 1116 then there exists a unique point P on the unit circle corresponding to it De nition 73 trigonometric functions Lett be a real number and let Py be the point on the unit circle that corresponds to t Z The sine function is de ned as sint y 2 The cosine function is de ned as cost 93 Ifx 31 0 the tangent function is de ned as tant 4x Ify 31 0 the cosecant function is de ned as csct E 96 1 y F 1 Ifx 31 0 the secant function is de ned as sect 7 63 Ify 31 0 the cotangent function is de ned as cott i 9 Using the de nition of the trigonometric functions write the tangent cosecant se cant and cotangent functions in terms of sine or cosine 1 tantg 1 2 csct7 y 1 3 sect7 4 cott 9 Example 73 1 3 Let t be a real number and let P lt75 be the point on the unit circle that corresponds to t Find the values of sin t cos t and tan t Dr Allen7s Math 1116 Class Notes 135 Example 74 Find the exact values of sin 0 cos 0 and tan6 for the following values of 0 1 6 0 26 ma Example 75 Evaluate the following 1 cos77r D sin77r was sinlt777rgt cos 37139 9 7 Cf 03 sin 37139 57139 cos 7 2 5 57139 sm 7 2 00 3 cos 11167139 H O sin 11167139 Dr Allen7s Math 1116 Class Notes 136 Example 76 For 6 2 nd cos 6 sin 6 and tan 6 in in II39IIIIIIIIIIIFIIIIII39III Ms m Example 77 For 6 g nd cos 6 sin 6 and tan 6 in in II39IIIIIIIIIIIFIIIIII39III Ms m Example 78 For 6 nd cos 6 sin 6 and tan 6 II39IIIIIIIIIIIFIIIIII39III Ms m Dr Allen s Math 1116 Class Notes 137 Quadrant signs of the trig functions the signs of the functions depend on the signs of x and y where Pxy is the point on the unit circle that corresponds to t a 3 Example 79 Evaluate the following Exercise Set 72 Be prepared to discuss or present your solutions in class Problem 74 For the unit circle above7 nd cos9 and sin9 for each radian labeled on the circle Dr Allen s Math 1116 Class Notes 138 73 Graphs 0f the Trigonometric Functions De nition 74 periodic function A functton f had pertod of tf ts the smallest postttoe real number such that fc fa for all cc tn the domatn of De nition 75 even function A functton fc ts even tf f a fa for each cc tn the domatn of Note An even function is symmetric with respect to the y axis De nition 76 odd function A functton fa ts odd tf f c fa for each cc tn the domatn of Note An odd function is symmetric with respect to the origin Add the vertical t aXis Where t 0 is positioned at the point 1 0 to the unit circle above Recall that each point on the t aXis gets mapped to a point cc y on the unit circle Recall the de nition of cos t and sin t t cost t sint V D OI39 Dr Allen s Math 1116 Class Notes 139 Example 710 Graph fx sin x Give the domain and range Example 711 Graph fx cos x Give the domain and range Example 712 Graph fx tan x Give the domain and range Dr Allen7s Math 1116 Class Notes 140 Example 713 Graph fx cs0 x Give the domain and range Example 714 Graph fx sec x Give the domain and range Example 715 Graph fx cot x Give the domain and range Dr Allen7s Math 1116 Class Notes 141 74 Trigonometric Identities 1 Reciprocal Identities 6 2 Quotient Identities 2 3 Pythagorean Identities 3 a Basic Identity b 1 tan2z sec2z c 1 cot2 csc2 4 EvenOdd Identities 6 5 Sum and Difference Identities a cosz y cosx cosy 7 sinz siny sinx y sinz cosy cosx siny tan tany V c tanz y W d cosz 7 y cosx cosy sinz siny m inz 7 y sinx cosy 7 cosx siny 7 tan 7 tany f Juana i y i 1 tan tany V 6 Double Angle Identities a cos2z cos2 7 sin2z b cos2z 1 7 2sin2x c cos2z 2 cos2 7 1 d sin2 2 sin z cos x 7 2tanz e tan2x 7 m 7 Half Angle Identities a cos2 w 7 1 7 cos2x b sin2z 7 2 c cosltggt iHHOSWgt d sing MW Dr Allen7s Math 1116 Class Notes 142 Example 716 Verify the following trigonometric identities 1 sin zcscx 7 sin s cos2 x Exercise Set 73 Be prepared to discuss or present your solutions in class Problem 75 Simplify the following trigonometric expressions t 0 1 CO Rewrite in terms of the sine and cosine functions csc0 COW Mlt lth t dd t b 1 79 1 Smw u 1p y e nurnera or an enornina or y sin 3 1 smw M Find a common denominator s1n0 cost9 sin20 7 1 439 tan0 sin0 7 tan0 FaCtor Problem 76 Establish the following trigonometric identities cscx tan secz DH sin27x cos27 1 339 Sm70 goose mm 7mm 1 tanx i 4 1 COtW i tanz 5 1C 0Sw 2csc6 39 1 cos0 sin0 tan cotx secx cscx ln sec6 7 ln cost9 03 1 5 Dr Allen7s Math 1116 Class Notes 143 75 Limits of Trigonometric Functions Exercise Set 74 Be prepared to discuss or present your solutions in class Problem 77 Investigate with a table of values or with a graph the following limits For each problem7 what is your conclusion Justify sin z 1 lim 410 D i i 1 lim sin 7 410 z 9 lim cos x 14100 7 lim tan z magl Problem 78 1 Sketch a graph of f tanxcotm What is the domain of x Use your graph to nd liH Dr Allen7s Math 1116 Class Notes 144 2 Some rst year calculus students claim that they have a proof that zero equals one Their argument follows below Do you believe the students in other words do you accept that zero equals one If you doubt the students7 claim7 there must be an error in their reasoning Find the error The students Claim that by looking at the graph of y tanmcotm it is easily observed that 1 lim tan 1 not 1 cc gtO and using limit laws and properties of the real number 0 we have lim tan 1 not 1 lim tan 1 lim not 1 Cc gt0 cc gt0 cc gt0 0 lim not 1 cc gtO 0 Therefore 1 0 What is wrong with the students7 argument Theorem 42 Continuity of the Trigonometric Functions The sicc trigonometric functions are all continuous on their domains Problem 79 Suppose c is a number in the natural domain of the stated trigono metric functions Use continuity to evaluate the following limits H 3 9 7 Cf 03 lim sin z mac lim cos x mac lim tan z mac lim csc z mac lim sec z mac lim cot z mac Problem 710 Evaluate the following limits7 if possible Recall the Composite Limit Theorem 1 2 lirn cos2z ma 2 71 limlt gt zal 71 Dr Allen7s Math 1116 Class Notes 145 76 Derivatives of Trigonometric Functions Note This section corresponds to section 105 in your text Example 717 Let f sin x 1 Graph 2 Use the graph of f to estimate where f 0 3 Use the graph of f to estimate f 07 f 7r7 f 27rf 77r7 f 727r 4 Graph f This is the graph of what trig function So f Recall On the previous homework assignment7 we used a table of values and a graph s1n m to estimate lim 7 maO Dr Allen7s Math 1116 Class Notes 146 Theorem 43 sint 39 Ho t 739 N w 5 Theorem 44 Derivative of Sine Function The function fx sinx is dz eren tiable with f Proof Similarly7 you can ShOW Dmcos m Dr Allen7s Math 1116 Class Notes 147 Example 718 d Find tan z Theorem 45 Derivatives of the Trigonometric Functions 1 6sinz 2 00S 3 diz anz 4 digescm 5 6sec m d 6 Cot m Example 719 Find an equation of the tangent line to the curve y em cosm at the point 01 Dr Allen7s Math 1116 Class Notes 148 Exercise Set 75 Be prepared to discuss or present your solutions in Class Problem 711 Use the quotient rule to prove the differentiation rule of cscz Problem 712 Use the quotient rule to prove the differentiation rule of secz Problem 713 Use the quotient rule to prove the differentiation rule of cot x Problem 714 Find an equation of the tangent line to the curve y 2x sinx at the point 772777 Problem 715 Taalman7 Page 6777 Problems 7 20 all7 227 237 287 30 Dr Allen7s Math 1116 Class Notes 149 77 The Inverse Trigonometric Functions The Inverse Cosine Function y arccos z or y cos 1 x Example 720 1 Graph y cos x 2 Does y cosa have an inverse on 700 00 on 027T Why or why not 3 Does y cosa have an inverse on Om Why or why not 4 Restrict the domain of y cosa to 07T Give the domain and range of 71 y cos a 5 Graph y cos 1 x arccos x De nition 77 inverse cosine function The inverse cosine function denoted y arccosz or y cos la is the inverse of the portion of the cosine function with the domain restricted to the interval 07T The domain ofy arccosz is 711 and its range is 07T Furthermore y arccosx if and only if cosy a that is y arccosz or y cos 1 x means that y is the angle between 07T whose cosine is a Dr Allenls Math 1116 Class Notes 150 plg cang m g39 Example 721 Evaluate the following 1 arccos 7 2 1 BICCOS 2 U a a O m A l H V Example 722 ls arccos x differentiable Why or Why not Example 723 Find 030871 Dr Allen s Math 1116 Class Notes 151 Example 724 z e 1 e e2 Flnd o Slmpllfy you answez Letoeeos The Inverse Sine Function o alesine 01 o Smquot z Example 725 Giapno 77r27r2 N v w intewal and hence has an inveise whlch we denote by o axcslnz 01 o sin z Giapn o axcslnz lee the domain and iange ottloe mVelse sine function De nition 78 inveise sine function The inverse sine function denoted o aicsinz W y sinquotz is the tnoemse of the pomon of the stne fzmmun mth the umm mastmztad to the mtamul r7r27r 2 The dumum ufy aiesins ZS 711 ond tts mnge ZS r7r2 7r2 Mtlmmooe o aiesins 2f ond unly 2f sino 2 thot ZS y amine W y sin 1 2 meons tlmt y is the ongle between i Z rZ whose Sma is 2 Example 726 Evaluate the followlng alesinu alcsmz Smquot0 Smquot Dr Allen s Math 1116 Class Notes 152 The Inverse Tangent Function y arctana or y tanquot 0 De nition 79 inverse tangent function The inverse tangent function denoted y emtene 0r tan 1x ts the tntetse 0f the nottton 0f the tangent functwn ntth the demotn testtttted to the tntettot en2m 2 The damam Ufy 7 70000 and tts Tangtz ts en2m 2 Fttthettnote y are aria if and nntn 2f tany st thot ts y are was M y tan 1x tneons that y ts the angle between en2mm nhose tangent ts st Example 727 Evaluate the following enema meten tan3917 3 Theorem 46 Derivatives of the Inverse Trigonometric thctions 1 sin 1x 2 0105quot e e 3 tan 1x 5 c0t 1x if Dre Allen s Math 1116 Class Notes 153 Example 728 Find the derivative of each of the following 1 y tan ls 1 2 y 7 arcs1nx Exercise Set 76 Be prepared to discuss or present your solutions in class Problem 716 Find the exact value of each of the following 1 sin 1 2 arccos71 1 tan 1 arctan1 sin 11 arccos705 3 9 7 CT 5103 arcsinsin77r3 Problem 717 Find the derivative of each of the following 1 fx z arctan 2 y sin 12x 1 3 fx arcsinm 4 y arccos62w Dr Allen7s Math 1116 Class Notes 154 Chapter 8 Integration 81 Antiderivatives Inde nite Integrals Example 81 Find a function whose derivative is given by u 4f one function ls there more than De nition 81 antiderivative A function F is called an antiderivative off on an interval I ifF x f for all z in I De nition 82 Inde nite Integral The eppression fzd is called an inde nite integral and it indicates the antideriuatiue of u with respect to x Notes In the equation fxd C o the symbol7 is called the integral sign 0 fx is called the integrand o C is called the constant of integration 0 C is the most general antiderivative of fx d Ewe o M o the equation is read The integral of u with respect to z is equal to plus a constant 0 to antidifferentiate is also to integrate Dr Allenls Math 1116 Class Notes 155 Example 82 Evaluate the following 1 z4dz 2 4dx 3 xrdx for r 31 71 l 5 dz 6 Edz Some Inde nite Integrals 1 2dz 2 r7 71xrdx Dr Allen7s Math 1116 Class Notes 156 4 ewd 5 amdz 6 sinxd 7 Cosxdx 8 seczxdx 9 Csczxdx 10 secxtanzd 11 Csczcotxdx 12 d W 13 dz x21 14 cfxdzcfxd 15 favgzdzfzdavgzdx Dr Allen7s Math 1116 Class Notes 157 Example 83 Evaluate the following integrals 1 3z4752dz z1 2 d sinx 3 2 dz cos m 4 3ew 7se02 m d 5 ves smz Example 84 Let f x Cosx Find Example 85 Suppose f z 4 7 6x 7 40x37 f0 27 and f 0 1 Find Dr Allen7s Math 1116 Class Notes 158 Exercise Set 81 Be prepared to discuss or present your solutions in class Problem 81 Find the most general antiderivative of the function Check your answer by differentiation 1 3 4 1 7 7 2 7 7 3 f96 2 496 596 2 i 7 Multiply rst Then antidj erentiate 3 we 7 6f 7 92 4 3 z 4 95 ff ate First separate the fraction into two simpler fractions simplify then antidjfferenti7 5 ft9 cos0 7 5sin0 Problem 82 Find the antiderivative of fx 5x4 7 2x5 that satis es the condition F0 4 Problem 83 Find 1 fm 6m Be careful 2 f 0 sin0 cost97 f0 37 and f 0 4 Problem 84 A particle is moving such that at t 7 27 50 17 and 00 3 Find the position of the particle Recall the relationship between the position func tion 5t7 the velocity function 11t and the acceleration function at 11t s t7 and at v t s t Problem 85 Evaluate the following integrals 1 x2 72 dx 2 17 z2 x2d 3 3 7 Q dx x 4 0 7 csc6cot 9 d0 Dr Allen7s Math 1116 Class Notes 159 82 Areas and Distances Area Problem Given a function f that is continuous and nonnegatiVe on an inter Val 11 nd the area between the graph of f and the interval 11 on the xeaxis Example 86 Estimate the area under the graph of 25 7 x2 from x 0 to x 5 I using 5 approximating rectangles and right endpoints I using 5 approximating rectangles and left endpoints gt 2 Dr Allen s Math 1116 Class Notes 160 Summation Notation De nition 83 Let n be a counting number The symbol 2 is a shorthand notation meaning to add 07quot sum V L 20739 a10203 an71an j1 Example 87 4 Find 2 1 11 Special Sum Formulas 1 1 j123nm 7391 2 12n1 239 212 22 32 2nn 27 n 6 71 3 Zcnc j1 4 anjcZaj j1 j1 5 BaaW j1 j1 i1 6 ZWrWFZaeri j1 j1 i1 Example 88 Find the area under the graph of f x2 from x 0 to z 1 Dr Allen7s Math 1116 Class Notes 161 I Partition the interval 07 1 into 2 subinterVals of equal length7 and use 2 approx imating rectangles and right endpoints to estimate the area under the curve I Partition the interval 07 1 into 4 subinterVals of equal length7 and use 4 approx imating rectangles and right endpoints to estimate the area under the curve I Partition the interVal 07 1 into 8 subinterVals of equal length7 and use 8 approx imating rectangles and right endpoints to estimate the area under the curve A722 333mm I Partition the interVal 01 into 16 subinterVals of equal length7 and use 16 approximating rectangles and right endpoints to estimate the area under the curve Am 3252mm Allen s Math 1116 Class Notes 162 I Partition the interval 01 into 32 subinterVals of equal length7 and use 32 approximating rectangles and right endpoints to estimate the area under the curve Ami ammw I Partition the interval 01 into 64 subinterVals of equal length7 and use 64 approximating rectangles and right endpoints to estimate the area under the curve Ame 3M 5233 I Partition the interVal 071 into 128 subinterVals of equal length7 and use 128 approximating rectangles and right endpoints to estimate the area under the curve Na 33721197553 Dr Allen s Math 1116 Class Notes 163 Exercise Set 82 Be prepared to discuss or present your solutions in class Problem 86 1 Estimate the area under the graph of f coss from x 0 to z 7T2 using four approximating rectangles and right endpoints Include a sketch of the graph and the rectangles ls your estimate an underestimate or an overes timate E0 Estimate the area under the graph of f coss from x 0 to z 7T2 using four approximating rectangles and left endpoints Include a sketch of the graph and the rectangles ls your estimate an underestimate or an overestimate Problem 87 1 Estimate the area under the graph of f from x 0 to z 4 using four approximating rectangles and right endpoints Include a sketch of the graph and the rectangles ls your estimate an underestimate or an overestimate D Estimate the area under the graph of f from x 0 to z 4 using four approximating rectangles and left endpoints Include a sketch of the graph and the rectangles ls your estimate an underestimate or an overestimate Problem 88 1 Estimate the area under the graph of f 1 x2 from x 71 to z 2 using three rectangles and right endpoints Include a sketch of the graph and the rectangles D Improve your estimate ofthe area under the graph of f 1z2 from x 71 to z 2 by using six rectangles and right endpoints Include a sketch of the graph and the rectangles Problem 89 Using your calculator7 graph the function f 67 on the interval 727 2 Using right endpoints7 estimate the area under the curve of f by using four approximating rectangles Include a sketch of the graph and the rectangles Dr Allen7s Math 1116 Class Notes 164 83 The De nite Integral De nition 84 de nite integral Let f be a continuous function de ned for a S x S b Diuide a7b into n subintepuals of equal width Ax b 7 an Let 0 a 1 2 wwl and xn b be the endpoints of these subintepuals Choose sample points xi z 271 and x in each subintepual so denote the sample point in the ith interval pl1 fopi 17 27 7n Then the de nite integral off from a b to b denoted fxdp is de ned by bfltzgtdz gin imam 0 i1 o is called an integral sign 0 fx is called the integrand o a and b are called the limits of integration 0 a is called the lower limit 0 b is called the upper limit 0 Z fpAz is called a Riemann sum i1 0 Integration is the process of calculating an integral Geometric interpretation of the de nite integral 7 fzdp gives the net area of the region trapped between the curve y u and I the p axis on the interval a7 b Example 89 1 Evaluate 2 d Evaluate the integral by interpreting it in terms of area 72 Dr Allenls Math 1116 Class Notes 165 Example 810 2 Evaluate 4 7 2 Evaluate the integral by interpreting it in terms of area 72 Properties of the De nite Integral 1 bafxd7abfdx 2 afzd0 b 3 a cdzcb7a 4 bltfltzgtzltzgtgtdz bfddd bzltzgtdz 5 abcfxdcabfzdz b 6 ltfltzgt 7 dz dz bfd dz 7 bzltzgt dz 7 fzdz 0b fzdz 1b fz dz 7 8 lff 20fora b7then fdx20 b b 9 If f 2 g for a S x 3 b7 then fzd 2 g dx Dr Allen7s Math 1116 Class Notes 166 Example 811 5 7 Suppose0 fxdz10 and5 fdz3 nd 7 1 d 0 M x 2 fzd 3 fd 4 053fxd Example 812 1 1 Evaluate fees 2 dz 1 3 3 2 Evaluate 26m 7 1 dx given that em dx 63 7 e 1 1 4 5 3 Evaluate f dx given that ff fx dx 12 and f dx 36 1 4 Dr Allen7s Math 1116 Class Notes 167 Exercise Set 83 Be pzepaxed Lo dlscuss oz pzesenc you soluclons m class blem 810 Evaluate che Rlemann sum fox f z 2 7 e2 on 02 wlch foul submlemls Lakmg lhe sample polnts to be lhe ughl endpomls Include a skeleh Explam usmg you skeleh what lhe Rlemann sum zepzesenls roblem 511 The gzaph of m mcegzals by lnLezpzeng IL m Lelme 1Ozmm 2 Aswan a A mm 4 Aamm IS glven below Evaluate eael of lhe followmg of 3163 Problem 512 Evaluate lhe lntegzal by lnLezpzeLlng ll m lem of axes 1 lrzrldz 02 2 10sz 2 a Mix 1 Dr Allen s Math 1116 Class Nata

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