10-6 Functions as Infinite Series

Transcription

1 Use Use to find a power series representation of g(x) Indicate the interval on which the series converges Use a graphing calculator to graph g(x) and the sixth partial sum of its power series 1 g(x) to find a power series representation of g(x) Indicate the interval on which the series converges Use a graphing calculator to graph g(x) and the sixth partial sum of its power series 1 g(x) To find the transformation that relates f (x) to g(x), use u-substitution Substitute u for x in f (x), equate the two functions, and solve for u To find the transformation that relates f (x) to g(x), use u-substitution Substitute u for x in f (x), equate the two functions, and solve for u Replace x with Replace x with can be represented by the power series This series converges for can be represented by the equivalent to 7 < x < 1 the sixth partial sum S 6(x) of this series power series This series converges for < 1, which is < 1, which is Graph g(x) and S 6(x) equivalent to 7 < x < 1 the sixth partial sum S 6(x) of this series Graph g(x) and S 6(x) g(x) Page 1

2 The sixth partial sum S 6(x) of this series is Graph g(x) and S 6(x) g(x) To find the transformation that relates f (x) to g(x), use u-substitution Substitute u for x in f (x), equate the two functions, and solve for u 3 g(x) To find the transformation that relates f (x) to g(x), use u-substitution Substitute u for x in f (x), equate the two functions, and solve for u Replace x with f(x) for x < 1 for < 1 can be represented by the power series This series converges for Replace x with f(x) for x < 1 < 1, which is equivalent to 1 < < 1 or < 1 for can be represented by the The sixth partial sum S 6(x) of this series is power series Graph g(x) and S 6(x) This series converges for equivalent to 1 < < 1, which is < 1 or 1 < x < 1 The sixth partial sum S 6(x) of this series is Page Graph g(x) and S 6(x)

3 This series converges for equivalent to 1 < < 1 or 1 < x < 1 The equivalent to 1 < < 1, which is < 1 or 4 < x < sixth partial sum S 6(x) of this series is The sixth partial sum S 6(x) of this series is Graph g(x) and S 6(x) Graph g(x) and S 6(x) 4 g(x) 5 g(x) To find the transformation that relates f (x) to g(x), use u-substitution Substitute u for x in f (x), equate the two functions, and solve for u To find the transformation that relates f (x) to g(x), use u-substitution Substitute u for x in f (x), equate the two functions, and solve for u Replace x with f(x) for x < 1 Replace x with for < 1 can be represented by the power series for x < 1 This series converges for equivalent to 1 < f(x) < 1, which is for < 1 can be represented by the power series < 1 or 4 < x < This series converges for < 1, which is The sixth partial sum S 6(x) of this series is equivalent to 1 < Graph g(x) and S 6(x) < 1 or <x< The sixth partial sum S 6(x) of this series is Page 3

4 This series converges for < 1, which is equivalent to 1 < < 1 or <x< The sixth partial sum S 6(x) of this series is can be represented by the power series This series converges for equivalent to 1 < < 1, which is < 1 or <x< Graph g(x) and S 6(x) The sixth partial sum S 6(x) of this series is or Graph g(x) and S 6(x) 6 g(x) To find the transformation that relates f (x) to g(x), use u-substitution Substitute u for x in f (x), equate the two functions, and solve for u Use the fifth partial sum of the exponential series to approximate each value Round to three decimal places 7 e05 Replacing x with f(x) for x < 1 < 1 for 8 e 05 can be represented by the power series ThisManual series- Powered converges for esolutions by Cognero equivalent to 1 < < 1, which is < 1 or <x< Page 4 9 e 5

5 8 e 05 1 e35 9 e 5 13 ECOLOGY The population density P per square meter of zebra mussels in the Upper Mississippi 008t River can be modeled by P 35e, where t is measured in weeks Use the fifth partial sum of the exponential series to estimate the zebra mussel population density after 4 weeks, 1 weeks, and 1 year e 08 x Use the power series representation of e to find the 008t fifth partial sum of P 35e and evaluate the expression for t 4, t 1, and t 5 t4 11 e 03 t 1 t 5 1 e35 Therefore, the zebra mussel population density after 4 weeks, 1 weeks, and 1 year will be 48 mussels/m, 91 mussels/m, and 1340 mussels/m, respectively Use the fifth partial sum of the power series for cosine or sine to approximate each value Round to three decimal places 14 sin 13 ECOLOGY The population density P per square Page 5

6 Therefore, the zebra mussel population density after 4 weeks, 1 weeks, and 1 year will be 48, 91 mussels/m, and 1340 mussels/m, 10-6mussels/m Functions as Infinite Series respectively Use the fifth partial sum of the power series for cosine or sine to approximate each value Round to three decimal places 17 cos 14 sin Substitute Substitute ; 18 cos 15 cos Substitute Substitute 19 sin 16 sin Substitute Substitute 0 AMUSEMENT PARK A ride at an amusement park is in the shape of a giant pendulum that swings riders back and forth in a 40º arc to a maximum height of 137 feet The pendulum is supported by a tower that is 85 feet tall and dips below ground-level into a pit when swinging below the tower Use the fifth partial sum of the power series for cosine or sine to approximate the length of the pendulum Refer to the photo on Page cos esolutions Manual - Powered by Cognero Substitute ; Page 6 Draw a diagram in which the ride is swinging in a 40º arc to a maximum height of 137 feet Include a

7 into a pit when swinging below the tower Use the fifth partial sum of the power series for cosine or sine to approximate the length of the pendulum 10-6Refer Functions asoninfinite to the photo Page 64Series {Note: We should already know that this value is 05) r Draw a diagram in which the ride is swinging in a 40º arc to a maximum height of 137 feet Include a horizontal in the diagram, as shown Therefore, the length of the pendulum is 104 feet Write each complex number in exponential form 1 i Write the polar form of Because the tower is 85 feet tall, the distance from the maximum height of the ride to the horizontal is feet So, the length of the side opposit the 30º angle in the right triangle shown is 5 feet i In this expression, a, b 1, a > 0 r An acute angle measure and the opposite side length are known, so the sine function can be used to write an expression for the length of the pendulum r Use the fifth partial sum of the power series for sine to find sin 30º Converting to radians, 30º Now write i in exponential form i Write the polar form of {Note: We should already know that this value is 05) i In this expression, a, b 1, a > 0 r r Page 7

8 i in exponential form Now write i in exponential form Now write i 3 Write the polar form of i In this expression, a, b 1, a > 0 Write the polar form of expression, a i in exponential form Now write, a > 0 iθ Therefore, because a b i re, the exponential 4 i Write the polar form of expression, a i is i i In this,b Write the polar form of a i In this expression,, b 1, a < 0, a > 0 r r,b form of 3 i In this r r i iθ Now write i in exponential form Therefore, because a b i re, the exponential form of i is 4 i Page 8

9 iθ Therefore, because a b i re, the exponential form of 4 i is 5 1 i Write the polar form of a i In this expression,, b 1, a < 0, a > 0 r i in exponential form Now write Now write i 6 a 1, b i in exponential form i Write the polar form of 1 i In this expression,, a > 0 Write the polar form of expression, a 1, b r i In this expression, Write the polar form of 1 a 1, b r i i In this, a < 0 r i NowManual write- 1Powered exponential esolutions byin Cognero form Write i in exponential form Page 9

10 i in exponential form Now write 1 i in exponential form Write 6 i 7 i In this Write the polar form of expression, a 1, b, a < 0 i in exponential form i, a < 0 Write i in exponential form i Write the polar form of expression, a 8,b i In this, a < 0 r,b i In this r Write 7 Write the polar form of expression, a r i Write the polar form of expression, a 1, b i In this, a < 0 r Write i in exponential form Write i in exponential form Page 10

11 Write i in exponential form 31 ln ( 45) 8 3 ln ( 7) i Write the polar form of expression, a 1, b i In this, a < 0 r 33 ln ( 436) 34 ln ( 91) 35 POWER SERIES Use the power series Write i in exponential form the value of each natural logarithm in the complex number system 9 ln ( 6) representations of sin x and cos x to answer each of the following questions a Graph f (x) sin x and the third partial sum of the power series representing sin x Repeat for the fourth and fifth partial sums Describe the interval of convergence for each b Repeat part a for f (x) cos x and the third, fourth, and fifth partial sums of the power series representing cos x Describe the interval of convergence for each c Describe how the interval of convergence changes as n increases Then make a conjecture as to the relationship between each trigonometric function and its related power series as n a 30 ln ( 35) Graph y sin x and 31 ln ( 45) 3 ln ( 7) Sample answer: convergence, (15, 15) Graph y sin x and 33 ln ( 436) Page 11

12 Sample answer: convergence, (15, 15) Graph y sin x and Sample answer: convergence, (5, 5) Graph y cos x and Sample answer: convergence, (5, 5) Graph y sin x and Sample answer: convergence, (3, 3) c Sample answer: The interval of convergence widens as n increases As n approaches infinity, the power series equals the trigonometric function that it represents Solve for z over the complex numbers Round to three decimal places 36 ez 5 0 Sample answer: convergence, (35, 35) b Graph y cos x and 37 ez 1 0 Sample answer: convergence, (15, 15) Graph y cos x and Sample answer: convergence, (5, 5) 38 4ez 7 6 Graph y -cos x andby Cognero esolutions Manual Powered Page 1

13 38 4ez 7 6 Therefore, the solutions are iπ or about ez 17ez 3 4 ECONOMICS The total value of an investment of z 39 3(e 1) 5 P dollars compounded continuously at an interest rt rate of r over t years is Pe Use the first five terms of the exponential series to approximate the value of an investment of $10,000 compounded continuously at 55% for 5 years 43 RELATIVE ERROR Relative error is the z z 40 e e absolute error in estimating a quantity divided by its true value The relative error of an approximation a of a quantity b is given by the relative 1 error in approximating e using two, three, and six terms of the exponential series two terms: Therefore, the solutions are iπ or about ez 17ez 3 three terms: Page 13

14 % 43 RELATIVE ERROR Relative error is the absolute error in estimating a quantity divided by its true value The relative error of an approximation a of a quantity b is given by the relative 1 error in approximating e using two, three, and six terms of the exponential series Approximate the value of each expression using the first four terms of the power series for sine and cosine Then find the expected value of each 44 sin cos two terms: sin 1 cos 1 or or three terms: The expected value is exactly 1 because sin x cos x 1 45 sec 1 tan 1 35% sin 1 1 six terms: cos 1 1 tan 1 sec % Approximate the value of each expression using the first four terms of the power series for sine and cosine Then find the expected value of each 44 sin cos 1 cos The expected value is exactly 1 because sec x tan x 1 46 RAINBOWS Airy's equation, which is used in physics to model the diffraction of light, can also be used to explain how a light wave front is converted into a curved wave front in forming rainbows sin or or esolutions Manual - Powered by1cognero Page 14 The expected value is exactly 1 because sin x cos x 1 This equation can be represented by the power

15 a power series representing E(x) expected value is exactly 1 because sec x 10-6The Functions as Infinite Series tan x 1 if k is a constant and d 1 First, rewrite E(x) in terms of one fraction 46 RAINBOWS Airy's equation, which is used in physics to model the diffraction of light, can also be used to explain how a light wave front is converted into a curved wave front in forming rainbows To find the transformation that relates E This equation can be represented by the power series shown below f (x) 1 (x) to f (x), use u-substitution Substitute u for x in f (x), equate the two functions, and solve for u Use the fifth partial sum of the series to find f (3) Round to the nearest hundredth 47 ELECTRICITY When an electric charge is accompanied by an equal and opposite charge nearby, such an object is called an electric dipole It consists of charge q at the point x d and charge q at x d, as shown below Replace x with f(x) Along the x-axis, the electric field strength at x is the sum of the electric fields from each of the two charges This is given by E(x) for x < 1 for a power series representing E(x) < 1 if k is a constant and d 1 can be First, rewrite E(x) in terms of one fraction represented by the power series Page 15

16 < 1 g(x) as Infinite Series can be 10-6Therefore, Functions represented by the power series 50 cos (x) cos x 48 SOUND The Fourier Series represents a periodic function of time f (t) as a summation of sine waves and cosine waves with frequencies that start at 0 and increase by integer multiples The series below represents a sound wave from the digital data fed from a CD into a CD player f (t) APPROXIMATIONS The infinite series for the 1 inverse tangent function f (x) tan Graph the series for n 4 Then analyze the graph Graph S 4(t) x is given by However, this series is only valid for values of x on the interval (1, 1) a Write the first five terms of the infinite series 1 representation for f (x) tan x b Use the first five terms of the series to 1 approximate tan 01 1 c On the same coordinate plane, graph f (x) tan x and the third partial sum of the power series 1 representing f (x) tan x On another coordinate plane, graph f (x) and the fourth partial sum Then graph f (x) and the fifth partial sum d Describe what happens on the interval (1, 1) and in the regions x 1 or x 1 a tan x x Sample answer: The graph oscillates in a pattern that repeats about every 003 second IDENTITIES Use power series representations from this lesson to verify each trigonometric identity 49 sin (x) sin x b c Graph tan x and S 3(x) 50 cos (x) cos x Page 16

17 more closely resemble the graph of f (x) tan x on the interval (1, 1) Outside of the interval (1, 1), the end behavior of the polynomial approximations causes the graphs of the partial sums to diverge from the graph of f (x) tan x c Graph tan x and S 3(x) 5 WRITING IN MATH Describe how using additional terms in the approximating series for e affects the outcome x Sample answer: In general, using additional terms provides an approximation that is closer to the actual x value of e Consider the fourth, fifth, and sixth partial sums of e Graph tan x and S 4(x) Using a calculator to evaluate e results in e 739 Therefore, as the number of terms increases, the x approximation approaches the actual value of e Graph tan 53 REASONING Use the power series for sine to x and S 5(x) explain why, for x-values on the interval [01, 01], a close approximation of sin x is x Sample answer: The power series representation for sine is given by sin x x For x- values on the interval [ 01, 01], the cubic term and those of higher degree have values less than or equal to d As n increases, the graphs of the partial sums more closely resemble the graph of f (x) tan x on the interval (1, 1) Outside of the interval (1, 1), the end behavior of the polynomial approximations causes the graphs of the partial sums to diverge from the graph of f (x) tan x This represents less than one-tenth of 1 percent of the value Therefore, for the values of x on the interval [01, 01], the first term of the series, x, is a close approximation of sin x 54 CHALLENGE Prove that 5 WRITING IN MATH Describe how using additional terms in the approximating series for e affects the outcome x Sample answer: In general, using esolutions Manual - Powered by Cognero additional terms provides an approximation that is closer to the actual x value of e Consider the fourth, fifth, and sixth Page 17

18 iα found by adding integer multiples of to θ For e iβ and e, α and β represent θ Thus, if α and β differ by an integer multiple of, then the complex one-tenth of 1 percent of the value Therefore, for the values of x on the interval [01, 01], the first of the series, is a close approximation of sin 10-6term Functions asx,infinite Series x iα iβ numbers represented by e and e will be the same PROOF Show that for all real numbers x, the following are true 54 CHALLENGE Prove that 56 First, find an expression for e ix ix Since e cos x i sin x, 55 REASONING For what values of α and β does eiα iβ e? Explain The exponential form of a complex number a b i is iθ given by a b i re, where θ is the measure of an angle in radians Since θ is the measure of an angle, θ has infinitely many coterminal angles that can be Thus, e ix cos x i sin x iα found by adding integer multiples of to θ For e iβ and e, α and β represent θ Thus, if α and β differ by an integer multiple of, then the complex iα iβ numbers represented by e and e will be the same PROOF Show that for all real numbers x, the following are true First, find an expression for e i sin x, First, find an expression for e ix ix Since e cos x i sin x, ix Since e cos x Thus, e Thus, e ix ix ix cos x i sin x cos x i sin x Page 18

19 58 CHALLENGE The hyperbolic sine and hyperbolic 57 First, find an expression for e i sin x, ix ix Since e cos x cosine functions are analogs of the trigonometric functions that you studied in Chapters 4 and 5 Just as the points (cos x, sin x) form a unit circle, the points (cosh t, sinh t) form the right half of an equilateral hyperbola An equilateral hyperbola has perpendicular asymptotes The hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions are defined below the power series representations for these functions cosh x sinh x Begin with the right side of sinh x Thus, e ix cos x i sin x The expanded form of the series, x 58 CHALLENGE The hyperbolic sine and hyperbolic cosine functions are analogs of the trigonometric functions that you studied in Chapters 4 and 5 Just as the points (cos x, sin x) form a unit circle, the points (cosh t, sinh t) form the right half of an equilateral hyperbola An equilateral hyperbola has perpendicular asymptotes The hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions are defined below the power series representations for these functions sinh x, resembles the power series representation for sin x except that the terms are all positive Therefore, the power series representation for sinh x is sinh x x Begin with the right side of cosh x cosh x Begin with the right side of sinh x The expanded form of the series, 1, resembles the power series representation for cos x except that the terms are all positive Therefore, the power series representation for cosh x is cosh x Use Pascal's triangle to expand each binomial The expanded form of the series, x, resembles the power series representation for sin x 59 (3m ) 4 Page 19

20 except that the terms are all positive Therefore, the power series representation for cosh x is cosh x 1, 8, 8, 56, 70, 56, 8, 8, and 1 Use these numbers as the coefficients of the terms in the series Then simplify Series 10-6 Functions as Infinite Use Pascal's triangle to expand each binomial 59 (3m ) 6 Prove that (3n 1) 4 for all positive integers n 4 Write a series for (3m coefficients ) omitting the Let Pn be the statement (3n 1) The numbers in the 4th row of Pascal s triangle are 1, 4, 6, 4, and 1 Use these numbers as the coefficients of the terms in the series Then simplify is a true Because 4 statement, Pn is true for n 1 Assume that Pk : (3k 1) is true for a positive integer k Show that Pk 1 must be 60 Write a series for omitting the true coefficients The numbers in the 5th row of Pascal s triangle are 1, 5, 10, 10, 5, and 1 Use these numbers as the coefficients of the terms in the series Then simplify This final statement is exactly Pk 1, so Pk 1 is true Because Pn is true for n 1 and Pk implies Pk 1, Pn is true for n, n 3, and so on That is, by the principle of mathematical induction, (3n 1) is true for all positive integers n 61 (p q)8 each power, and express it in rectangular form 63 ( i)3 8 Write a series for (p q) omitting the coefficients First, write i in polar form The numbers in the 8th row of Pascal s triangle are 1, 8, 8, 56, 70, 56, 8, 8, and 1 Use these numbers as the coefficients of the terms in the series Then simplify 6 Prove that (3n 1) The polar form of i is Now use De Moivre s Theorem to find the third power for all positive integers n Page 0 Let Pn be the statement (3n 1) Because 4 is a true Therefore,

21 1 n the principle of mathematical induction, (3n 1) is true for all positive integers n each power, and express it in rectangular form 63 ( i)3 Therefore, 65 ( i) First, write i in polar form First, write i in polar form The polar form of i is Now use De Moivre s Theorem to find the third power Therefore, Therefore, 66 Given t < 9, 3, c>, u <8, 4, 3>, v <, 5, i)4 64 (1 i The polar form of is Now use De Moivre s Theorem to find the negative second power 6>, and that the volume of the parallelepiped having adjacent edges t, u, and v is 93 cubic units, find c i in polar form First, write 1 The polar form of The volume is the absolute value of the triple scalar product, so t (u v) can equal 93 or 93 i is Now use De Moivre s Theorem to find the fourth power Therefore, 65 ( i) First, write i in polar form Use an inverse matrix to solve each system of 1 Page equations, if possible 67 x 8y 7

22 Use an inverse matrix to solve each system of equations, if possible 67 x 8y 7 x 5y 8 The solution is (9, ) 68 4x 7y 9x 11y 4 Write the system in matrix form AX B 4x 7y 9x 11y 4 Write the system in matrix form AX B Use the formula for the inverse of a matrix to Use the formula for the inverse of a matrix to 1 find A Multiply A find A by B to solve the system The solution is (9, ) esolutions Manual - Powered by Cognero 68 4x 7y 9x 11y 4 Multiply A by B to solve the system The solution is (, ) 69 w x 3y 18 4w 8x 7y 41 w 9x y 4 Page

23 The solution is (, ) 69 w x 3y 18 Multiply A by B to solve the system 4w 8x 7y 41 w 9x y 4 w x 3y 18 4w 8x 7y 41 w 9x y 4 Write the system in matrix form AX B Use a graphing calculator to find A Enter the values for the matrix A Enter [A] Select to get the values in reduced fraction form The solution is (7, 1, 3) Determine whether A and B are inverse matrices 70 A,B A Multiply A by B to solve the system,b If A and B are inverse matrices, then AB BA I Page 3

24 Determine whether A and B are inverse matrices 70 A, B 7 A, B If A and B are inverse matrices, then AB BA I A, B If A and B are inverse matrices, then AB BA I AB I, so A and B are not inverses AB I, so A and B are not inverses 73 CONFERENCE A university sponsored a conference for 680 women The Venn diagram shows the number of participants in three of the activities offered Suppose women who attended the conference were randomly selected for a survey 71 A, B If A and B are inverse matrices, then AB BA I a What is the probability that a woman selected participated in hiking or sculpting? b Describe a set of women such that the probability of their being selected is about 039 Because AB BA I, B A and A B a From the diagram, 98 women participated only in hiking, 75 women participated in both hiking and sculpting, 13 women participated only in sculpting, 38 women participated in sculpting and quilting, 15 women participated in all three events, and 1 women participated in hiking and quilting Page 4

25 Therefore, the probability that a randomly selected woman participated in hiking or sculpting is about 5441% 74 SAT/ACT PQRS is a square What is the ratio of the length of diagonal to the length of side? b Sample answer: First, find the size of a set with a probability of 039 or 39% 39% of (680) or about 65 Look at the Venn diagram and find a set of numbers whose sum is close to These numbers would represent the set of women that participated in hiking or quilting but not sculpting Check: So, the probability of a selecting a woman who participated in hiking or quilting but not sculpting is about 039 A B C 1 D E Since PQRS is a square and all angles are right angles Let this side length be represented by x Consider triangle QRS This is a right triangle with side lengths labeled x Apply the Pythagorean Theorem to assign a value for the hypotenuse The ratio of The correct answer is B Page 5

26 75 REVIEW What is the sum of the infinite geometric series? F G 1 H 1 a J 1 The sum S of an infinite geometric series for which r < 1 is given by S So, first find the common ratio The common ratio r is < 1 This infinite geometric series has a sum Use the formula the sum of an infinite geometric series b There is 1 dot in the first figure, 4 dots in the second figure, 9 dots in the third figure and 16 dots in the fourth figure a 1 1 a 4 1 or 3 a or 5 a or 7 The correct answer is F 76 FREE RESPONSE Consider the pattern of dots shown a Draw the next figure in this sequence b Write the sequence, starting with 1, that represents the number of dots that must be added to each figure in the sequence to get the number of dots in the next figure c the expression for the nth term of the sequence found in part b d the expression for the number of dots in the nth figure in the original sequence e Prove, through mathematical induction, that the sum of the sequence found in part b is equal to the expression found in part d Therefore, the sequence is 1, 3, 5, 7, c Each term a n in this sequence can be found by subtracting 1 from n (1) 1 1 () 1 3 (3) 1 5 (4) 1 7 So, an explicit formula for this sequence is a n n 1 d The sequence of dots is 1, 4, 9, 16,, which can also be written as 1,, 3, 4, Therefore an expression for the number of dots in the nth figure in the original sequence is n e Let P n be the statement (n 1) n Because 1 1 is a true statement, P n is Page 6

27 true for n 1 Assume that (k 1) k is true for a positive integer k Show that P k 1 must be true This final statement is exactly P k 1, so P k 1 is true Because P n is true for n 1 and P k implies P k 1, P n is true for n, n 3, and so on That is, by the principle of mathematical induction, (n 1) n is true for all positive integers n Page 7