Section 8.2: Integration by Parts When you finish your homework, you should be able to

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1 Section 8.2: Integration by Parts When you finish your homework, you should be able to π Use the integration by parts technique to find indefinite integral and evaluate definite integrals π Use the tabular method to organize an integral requiring integration by parts π Recognize trends and establish guidelines for integrals requiring integration by parts Warm-up: 1. Differentiate with respect to the independent variable. f x = x a. ( ) arcsin 5 b. y = ln ( 5x+ 1) c. r ( θ) = tan θ CREATED BY SHANNON MYERS (FORMERLY GRACEY) 1

2 2. Find the indefinite integral. arctan x a. 2 dx x + 1 b. ( ln x) 3 dx x c. x 5 xdx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 2

3 3. Evaluate the definite integral. π tan 6θsec 6 d θ θ by is based on the formula for the of a and is useful for involving products of algebraic and functions. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 3

4 Consider the following product of two functions of x that have continuous. THEOREM: INTEGRATION BY PARTS If u and v are functions of x and have derivatives, then This technique turns a super complicated integral into ones. The trick is to choose your function so that is than. Oh yeah and PRACTICE A OF PROBLEMS!!! Okay let s look at the types of Integration by Parts (IBP) problems. Which types of expressions do not have integration formulas? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 4

5 EXAMPLE 1: Find the following indefinite integrals. a. 2 4x ln xdx b. arcsin xdx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 5

6 When you don t have a factor, you need to play around with the. Oftentimes it works out to let be the factor whose is a simpler than u. Then would be the more remaining factor. Use!!! There is a lot of and --especially at first **Remember: ALWAYS includes! EXAMPLE 2: Find the indefinite integral. 6x a. 7 x dx e CREATED BY SHANNON MYERS (FORMERLY GRACEY) 6

7 b. x 5 xdx c. 2 x sec xdx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 7

8 Sometimes, you need to use IBP times. You may even need to like (yes, you can do that)! EXAMPLE 3: Find the indefinite integral. x a. e cos3xdx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 8

9 b. 2 x x e dx THE TANZALIN (AKA TABULAR) METHOD is a way of organizing an integration by parts problem. Let s rework the last example using this method. DERIVATIVES INTEGRALS ALTERNATE SAME-COLOR u SIGNS PRODUCTS dv Let s try to bring this all together CREATED BY SHANNON MYERS (FORMERLY GRACEY) 9

10 In general, use the following choice for u, in order **When you have e ax ax cosbxdx or e sin bxdx, let and let or let. **To evaluate a definite integral, first find the integral and then back substitute. EXAMPLE 5: Find the indefinite integral or evaluate the definite integral. a. 1 0 x arcsin 2 x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 10

11 b. 3 x xe 2 2 ( x + 1) 2 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 11

12 Section 8.3: Trigonometric Integrals When you finish your homework, you should be able to π Find indefinite integrals and evaluate definite integrals involving the sine and cosine functions which are raised to positive powers π Find indefinite integrals and evaluate definite integrals involving the secant and tangent functions which are raised to positive powers π Use trigonometric identities to find indefinite integral and evaluate definite integrals involving the sine and cosine functions Warm-up 1: Simplify. a. b. 2 1 sin x 2 1+ tan x c. 1 cos 2 x 2 d. 1 + cos 2 x 2 Warm-up 2: Complete the statement. a. If u = sin 2x, then du =. b. If u = cos 4x, then du =. c. If u = tan x, then du =. d. If u = sec6x, then du =. EXAMPLE 1: Find the indefinite integral. a. cos x dx sin x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 12

13 b. sin x cos 3 2 xdx So we discovered that if the sine portion of the integrand has an, positive integer as a power and the cosine portion has any other power, then we save sine factor, and the others to factors. Then and. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 13

14 c. 4 3 sin 2x cos 2 xdx So we discovered that if the portion of the integrand has an odd, positive integer as a power and the sine portion has any other power, then we save cosine factor, and the others to factors. Then and. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 14

15 d. 4 sin 5xdx So we discovered that if only one sine or cosine factor is in the integrand and has an, positive integer as a power, you use the formula or until you can use basic integration formulas. What should we do if both the sine and cosine are raised to even, positive powers? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 15

16 e. sec x tan 4 3 xdx So we discovered that if the portion of the integrand has an even, positive integer as a power and the tangent portion has any other exponent, then we save factors, and convert the rest to factors. Then and. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 16

17 f. 3 3 sec 5xtan 5 xdx So we discovered that if the portion of the integrand has an odd, positive integer as a power and the secant portion has any other exponent, then we save a - factor, and convert the rest to factors. Then and. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 17

18 g. 4 tan xdx So we discovered that if there is only a factor raised to a positive, power, rewrite as two factors, one of which is, convert the other to minus, and then and. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 18

19 h. 3 sec xdx So we discovered that if there is only a factor raised to a positive, power, we need to use by. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 19

20 If none of these techniques work, try converting all factors to and factors. Then play around with identities. EXAMPLE 2: Find the indefinite integral. a. tan sec 2 5 x dx x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 20

21 b. sin x cos cos x 2 2 x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 21

22 PRODUCT TO SUM IDENTITIES If occur in the integral, use the following identities. sin mxsin nx = sin mx cos nx = cos mx cos nx = You do not need to memorize these identities. EXAMPLE 3: Find the indefinite integral. sin 7x cos 4xdx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 22

23 EXAMPLE 4: Find the area of the region bounded by the graphs of π π y = sin xcos x, x =, and x =. 2 4 y cos 2 = x, CREATED BY SHANNON MYERS (FORMERLY GRACEY) 23

24 Section 8.4: Trigonometric Substitution When you finish your homework, you should be able to π Find indefinite integrals using trigonmometric substitution π Evaluate definite integrals using trigonmometric substitution Warm-up 1: Consider the definite integral x dx. Do you have the skills to evaluate this definite integral?! What tool did we use in Calculus I?! Warm-up 2: Complete the figures. a. u = asinθ So, a u = 2 2 for π π θ. 2 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 24

25 b. u = atanθ So, a + u = 2 2 for π π < θ <. 2 2 c. u = asecθ So, u a = 2 2 u π for u > a, where 0 θ < a 2 = π for u< a, where < θ π CREATED BY SHANNON MYERS (FORMERLY GRACEY) 25

26 NOTE: These are the same intervals over which the,, and are defined. The restrictions on ensure that the function used for the substitution is -to-. EXAMPLE 1: Evaluate the definite integral x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 26

27 So we discovered that if the integrand has a and no basic integration rules,, or regular integrals work, we use the substitution. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 27

28 EXAMPLE 2: Find the indefinite integral. a. 2 x 16 dx x So we discovered that if the integrand has a and no basic integration rules,, or regular integrals work, we use the substitution. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 28

29 1 x 9x + 1 dx b. 2 So we discovered that if the integrand has a and no basic integration rules,, or regular integrals work, we use the substitution. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 29

30 c. x 2 2x x 2 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 30

31 EXAMPLE 3: Evaluate the definite integral ( 1 t ) dt CREATED BY SHANNON MYERS (FORMERLY GRACEY) 31

32 Section 8.5: Partial Fractions When you finish your homework you should be able to π Review how to decompose rational expressions into partial fractions π Utilize partial fractions to find indefinite integrals π Utilize partial fractions to evaluate definite integrals Warm-up: Find the indefinite integral. 2 x x 1 dx x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 32

33 (CASE 1) Q HAS ONLY NONREAPEATED LINEAR FACTORS Under the assumption that Q has only linear factors, the polynomial Q has the form where no two of the numbers are equal. In this case, the partial fraction decomposition of is of the form where the numbers are to be determined. Example 1: Write the partial fraction decomposition of the rational expression in the integrand, and find the indefinite integral x 1 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 33

34 (CASE 2) Q HAS REAPEATED LINEAR FACTORS If the polynomial Q has a linear factor, say,, n is an, then, in the partial fraction decomposition of, we allow for the terms where the numbers are to be determined. Example 2: Write the partial fraction decomposition of the rational expression in the integrand, and find the indefinite integral. 5x 2 ( x 2) 2 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 34

35 (CASE 3) Q CONTAINS A NONREAPEATED IRREDUCIBLE QUADRATIC FACTOR If Q contains a irreducible quadratic factor of the form, then, in the partial fraction decomposition of, allow for the term where the numbers are to be determined. Example 3: Write the partial fraction decomposition of the rational expression in the integrand, and find the indefinite integral. 6x dx 8 3 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 35

36 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 36

37 (CASE 4) Q CONTAINS A REAPEATED IRREDUCIBLE QUADRATIC FACTOR If the polynomial Q contains a irreducible quadratic factor of the form,, n is an, then, in the partial fraction decomposition of, allow for the terms where the numbers are to be determined. Example 4: Write the partial fraction decomposition of the rational expression in the integrand, and evaluate the definite integral. 1 x ( x + 16) 2 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 37

38 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 38

39 Example 5: Find the indefinite integral. 5cos x 2 sin x+ 3sin x 4 dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 39

40 Section 8.7: Indeterminate Forms and L Hôpital s Rule When you finish your homework you should be able to π Recognize all indeterminate forms π Apply L Hôpital s Rule to evaluate limits π Manipulate expressions so that L Hôpital s Rule may be applied to evaluate limits WARM-UP: Find the limit. It is okay to write ± as your answer. lim 1. 9 x x 3 x 9 2. lim x x x x x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 40

41 lim x+ x x x 0 x 4. lim csc x x π + CREATED BY SHANNON MYERS (FORMERLY GRACEY) 41

42 5. lim ln x x lim arctan x x 7. sin 4x lim x 0 x 8. lim x cos x x+ xcos x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 42

43 What indeterminate form did you encounter in some of these problems? What skills did you use to get these expressions into a determinate form?: L Hôpital s Rule Suppose f and g are differentiable and g ( x) 0 Suppose that or that ( ) = and ( ) lim f x 0 x a near a (except possible at a ). lim g x = 0 x a lim x a f ( x) = ± and lim g( x) x a = ± meaning that we have an form of or. Then If the limit on the right side or is or. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 43

44 *What should we check before applying L Hôpital s Rule? 1. and are near and 2. near. **L Hôpital s Rule is also valid for limits and for limits at or. ***Let s look at the special case when f ( a) g( a) 0. continuous, and g ( x) 0 = =, f and g are CREATED BY SHANNON MYERS (FORMERLY GRACEY) 44

45 Example 1: Determine if L Hôpital s Rule can be used to evaluate the limit. If so, apply L Hôpital s Rule to evaluate the limit. lim a. x 9 x 3 x 9 b. lim x x x x x 1 c. sin 4x lim x 0 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 45

46 d. lim x cos x x+ xcos x e. ln x lim x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 46

47 Indeterminate Forms We already know that and represent 2 types of forms. There are also indeterminate,, and. Indeterminate Products occur when the limit of 1 approaches and the other factor approaches or. Suppose and. If prevails, the result of the limit of the will be. If is the victor, the of the product will be. If they decide to sign a, the answer will be some, number. To find out, see if you can the difference into a. Example 2: Evaluate the limit. a. lim x xe x 2 b. lim sin xln x + x 0 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 47

48 Indeterminate Differences occur when both limits approach. Suppose and. If prevails, the result of the limit of the will be. If is the victor, the of the product will be. If they decide to sign a, the answer will be some number. To find out, see if you can the difference into a by using a,, or out a. Example 3: Evaluate the limit. a. lim ( csc x cot x) x 0 b. lim ln ( x 7 1) ln ( x 5 1) + x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 48

49 Indeterminate Powers occur when. There are 3 indeterminate forms that arise from this type of limit. 1. and. This yields the inderminate form. 2. and. This yields the inderminate form. 3. and. This yields the inderminate form. To find these types of limits, see if you can take the : or write the function as an : Example 4: Evaluate the limit. bx a. lim 1 a + x x b. lim x 0 + x x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 49

50 Section 8.8: Improper Integrals When you finish your homework you should be able to π Recognize when a definite integral is improper π Use your integration and limit techniques to evaluate improper integrals 2 =. x WARM-UP: Consider the function ( ) 3 1. Graph the function. f x 2. Find the limits. It is okay to write ± as your answer. 2 a. lim+ 3 x 0 x 2 b. lim x 3 x 3. Evaluate the definite integral. 1 b 2 3 x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 50

51 Let s put some stuff together Recall the if a function is on the interval, the integral is equal to the under the and bounded by the. Also remember that a function is said to have an infinite at when, from the or the left, TYPE 1: INFINITE INTERVALS Now consider the following definite integral dx x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 51

52 Definition: Improper Integrals With Infinite Integration Limits 1. Suppose f is continuous on the interval from, then a f ( ) x dx = 2. Suppose f is continuous on the interval from, then b f ( ) x dx = 3. Suppose f is continuous on the interval from, then f ( ) x dx = where is any real number. In the first two cases, the improper integral when the exists; otherwise the improper integral. In the third case, the improper integral on the left when either of the improper integrals on the diverge. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 52

53 Example 1: If possible, evaluate the following definite integrals and ascertain if they are convergent or divergent. a. 1 dx x dx 4 x b. 2 + CREATED BY SHANNON MYERS (FORMERLY GRACEY) 53

54 TYPE 2: DISCONTINUOUS INTEGRANDS Now consider the following definite integral. 1 0 ln x dx x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 54

55 Definition: Improper Integrals With Infinite Discontinuities 1. Suppose f is continuous on the interval from, and has an infinite at, then b ( ) a f x dx = 2. Suppose f is continuous on the interval from, and has an infinite at, then b ( ) a f x dx = 3. Suppose f is continuous on the interval from, except for some in at which f has an infinite at, then b ( ) a f x dx = In the first two cases, the improper integral when the exists; otherwise the improper integral. In the third case, the improper integral on the left when either of the improper integrals on the diverge. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 55

56 Example 2: If possible, evaluate the following definite integrals and ascertain if they are convergent or divergent. a. 3 6 dx 36 x 2 dx x b. 1 xln CREATED BY SHANNON MYERS (FORMERLY GRACEY) 56

57 Consider dx p x, where p is a real number. Let s find the indefinite integral on 1. p = 0 2. p 1 3. p = 1 Example 3: Determine all values of p for which the improper integral converges. dx x 1 p CREATED BY SHANNON MYERS (FORMERLY GRACEY) 57

58 THEOREM: A SPECIAL TYPE OF IMPROPER INTEGRAL 1 dx p x 1, p > 1 = p 1 diverges, p < 1 Example 4: If possible, evaluate the following definite integrals and ascertain if they are convergent or divergent. dx x a b. x 3 dx 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 58

59 9.1: Sequences When you finish your homework you should be able to π Identify the terms of a sequence, write a formula for the nth term of a sequence, and ascertain whether a sequence converges or diverges. π Use properties of monotonic sequences and bounded sequences. WARM-UP: Consider the function f ( x) = x. 1. Sketch the graph of the function. 2. Find the following: f 0 = a. ( ) b. f ( 1 ) = c. f ( 2 ) = d. f ( 3 ) = e. f ( 4 ) = f. f ( 5 ) = g. h. lim x 0 lim x 1 3. Now consider an x = x = 4. Find the following: a. a 1 = b. a 2 = = n. c. a 3 = d. a 4 = i. lim x 4 x = j. lim x = x e. a 5 = Hmmm so it looks like equals at all of the. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 59

60 5. Sketch the graph of the sequence. Definition of the Limit of a Sequence Let L be a real number. The of a sequence is, written as lim a n = n L if for ε > 0, there exists M > 0 such that an L < ε whenever n> M. If the limit L exists, then the sequence. If the limit does not exist, then the sequence. Looking at the two graphs we sketched, as it looks like. So, we would say the lim a n. and. n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 60

61 EXAMPLE 1: Write the first five terms of the sequence. a. a n 3n = b. n a1 = 6, ak+ 1 = a 3 2 k FACTORIALS are factors which decrease by one. So 5!, read as five factorial is 5! = =. We will be working with unknown factorials. In general, n! =, and 0! =. EXAMPLE 2: Simplify the ratio of factorials. n! n + 2! b. a. ( ) ( n + ) ( 2 n)! 2 2! CREATED BY SHANNON MYERS (FORMERLY GRACEY) 61

62 EXAMPLE 3: Find the nth term of the sequence. a ,,,,, b ,,,, Theorem: Limit of a Sequence Let L be a real number. Let f be a function of a real variable such that If { a n } is a sequence such that f ( n) an = for every positive integer n, then EXAMPLE 4: Find the limit of the sequence, if it exists. a n 2 = 6 + b. n a. 2 a n 2 = cos n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 62

63 Theorem: Properties of Limits of Sequences Let lim a n n = L and lim b n n = K. 1. lim ( a b ) n n ± = n 2. lim can =, c is any number. n 3. lim ( ab) n n n = a n 4. lim =, n and. bn EXAMPLE 5: Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. a. a n ( ) 1+ 1 n = 2 n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 63

64 b. a n = 3 3 n n + 1 c. a n = ( n ) n! 2! CREATED BY SHANNON MYERS (FORMERLY GRACEY) 64

65 Absolute Value Theorem For the sequence { a n }, if lim a = 0 then. n n Squeeze Theorem for Sequences If lim an = L= lim bnand there exists an integer N such that, n n then. EXAMPLE 6: Show that the sequence converges and find its limit. c n ( 1) = n 1 n! CREATED BY SHANNON MYERS (FORMERLY GRACEY) 65

66 Definition: Monotonic Sequence A sequence { a n } is when its terms are, a a a a or when its terms are nonincreasing n. Definition: Bounded Sequence 1. A sequence { a n } is above when there is a real number such that an M for all. The number is called an of the sequence. 2. A sequence { a n } is bounded when there is a real number such that N a n for all. The number is called a bound of the sequence. 3. A sequence is when it is bounded and below. Theorem: Bounded Monotonic Sequences If a sequence { a n } is and, it. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 66

67 EXAMPLE 7: Determine whether the sequence with the given nth term is monotonic and whether it is bounded. a n = cos n n EXAMPLE 8: Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 67

68 9.2: Series and Convergence When you finish your homework you should be able to π Understand and represent a convergent infinite series. π Use properties of infinite geometric series. π Use the nth term test for divergence. We spent the last section checking out, and ascertaining whether a given sequence, a, or n as. Remember, the of a sequence are represented as a or, which need not be ordered. There are finite and sequences. What if we were interested in a sequence? If we are interested in summing a finite number, say n, of the of a sequence, we would be finding the. If we are interesting in finding the sum of an infinite sequence, if it exists, we would be finding an sum, called an infinite, or just a. Our main interest will be to ascertain whether a series or. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 68

69 EXAMPLE 1: Consider the sequence we found above. a. Write the first five terms, and the nth term of the sequence. b. Sum the first five terms. c. Represent this 5 th partial sum as a summation. d. Find the limit of the sequence. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 69

70 e. Find an expression for the nth partial sum. f. What must the limit of this expression equal? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 70

71 Definition: Convergent and Divergent Series For the infinite series n= 1 a n, The sum is If the sequence of partial sums { S n } n= 1 a n to S, then the series converges. The limit S is called the of the. If { S n } diverges, then the series. So from our first example, S =, and this series since the sum. GEOMETRIC SERIES: Theorem: Convergence of a Geometric Series A geometric series with converges to the sum when. Otherwise, for, the series diverges. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 71

72 EXAMPLE 2: Express the number as a ratio of integers CREATED BY SHANNON MYERS (FORMERLY GRACEY) 72

73 EXAMPLE 3: Show that the series n= 1 n( n+ 2) 1 is convergent and find its sum. NOTE: The series in example 3 is called a series. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 73

74 EXAMPLE 4: Show that the series n= 1 1 n diverges. NOTE: The series in example 4 is called a series. Theorem: Properties of Infinite Series Let If sums. an and n= 1 an = A and n= 1 bn be convergent series, and let A, B, and c be real numbers. n= 1 n= 1 b n = B, then the following series converge to the indicate 1. can = 2. ( an + bn) = n= 1 3. ( an bn) n= 1 n= 1 = CREATED BY SHANNON MYERS (FORMERLY GRACEY) 74

75 EXAMPLE 5: Determine the convergence or divergence of the series. If the series converges, find its sum. n= n 3 n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 75

76 Theorem: Limit of the nth Term of a Convergent Series If n= 1 a n converges, then. Theorem: nth Term Test for Divergence If lim an 0 n, then an. = 1 n EXAMPLE 6: Determine the convergence or divergence of the series. Explain. a. n= 1 arctan n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 76

77 b. n= 1 n ( 3) 1 4 n c. n= n n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 77

78 EXAMPLE 7: Find all values of x for which the series converges. For these values of x, write the sum as a function of x. n= 0 x n EXAMPLE 8: A ball is dropped from a height of 16 feet. Each time it drops h feet, it rebounds 0.81h feet. Find the total distance traveled by the ball. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 78

79 9.3: The Integral Test, P-Series, and Harmonic Series When you finish your homework you should be able to π Use the Integral Test to ascertain whether an infinite series converges or diverges. π Determine whether a p-series converges or diverges. π Use properties of harmonic series. WARM-UP: Determine whether the improper integral converges or diverges ln x dx x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 79

80 2. 1 l 3 dx x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 80

81 3. 1 l x dx CREATED BY SHANNON MYERS (FORMERLY GRACEY) 81

82 Theorem: The Integral Test If f is,, and for 1 a = f n, then x and ( ) n Either both or both. ***NOTE: Our interest is whether the series converges or diverges as, so the index of the summation can start at some integer as opposed to a when we apply the integral test. EXAMPLE 1: Determine the convergence or divergence of the series. Explain. a. 3 n= 1 ln n n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 82

83 n b. 4 2 n n n = CREATED BY SHANNON MYERS (FORMERLY GRACEY) 83

84 P-Series and Harmonic Series A harmonic series is the of sounds represented by waves in which the of each sound is an multiple of the frequency. Pythagoras and his students discovered this relationship between the and the of the vibrating string. The most beautiful harmonies seemed to correspond with the simplest of numbers. Later mathematicians developed this idea into the series, where the in the harmonic series correspond to the node on a string that produce of the fundamental frequency. So, is the fundamental frequency, is times the fundamental frequency, and so on. In music, strings of the same,, and, and whose form a harmonic series, produce tones. A general harmonic series is of the form. The harmonic series is a special case of the, where. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 84

85 Theorem: Convergence of p-series The p-series for, and for. EXAMPLE 2: Determine the convergence or divergence of the series. Explain. a. n= 1 l n b CREATED BY SHANNON MYERS (FORMERLY GRACEY) 85

86 9.4: Series Comparison Tests When you finish your homework you should be able to π Use the Direct Comparison Test to ascertain whether an infinite series converges or diverges. π Use the Limit Comparison Test to ascertain whether an infinite series converges or diverges. WARM-UP: Determine whether the series converges or diverges. 1. n 1 n= CREATED BY SHANNON MYERS (FORMERLY GRACEY) 86

87 2. 2 n 1 n= CREATED BY SHANNON MYERS (FORMERLY GRACEY) 87

88 Our Tests So Far nth Term Test for. If, the series. If, we need to further!!! Geometric Series is of the form. If, the series and its is. Otherwise, the series diverges. Telescoping Series. Requires decomposition. The is the sum of the terms which do not out plus. p-series is of the form. If, the series. If, the series. The Integral Test requires that is, continuous, and for 1 f n x, and ( ) n = a for all n. If converges, converges. Otherwise, diverges. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 88

89 Theorem: The Direct Comparison Test Let for all n. 1. If 2. If bn, converges. i= 1 an, bn. i= 1 i= 1 EXAMPLE 1: Determine the convergence or divergence of the series. Explain. a. 1 n= 1 n + 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 89

90 b. n n n= 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 90

91 Theorem: The Limit Comparison Test If,, and where L is and, then an and i= 1 bn either both or both. i= 1 NOTE: When choosing your comparison, you can disregard all but the n 1 5n 2 powers of. So, if we are testing 2 n= +, our comparison series would be =. Proof: CREATED BY SHANNON MYERS (FORMERLY GRACEY) 91

92 EXAMPLE 2: Determine the convergence or divergence of the series. Explain. n a. 4 2 n n n = CREATED BY SHANNON MYERS (FORMERLY GRACEY) 92

93 b. n= 0 1+ sin n 10 n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 93

94 9.5: Alternating Series When you finish your homework you should be able to π Use the Alternating Series Test to ascertain whether an infinite series converges or diverges. π Use the Alternating Series Remainder to approximate the sum of an alternating series. π Classify a convergent series as conditionally convergent or absolutely convergent. WARM-UP: Determine whether the series converges or diverges. n= 0 n ( 1) n+ ( 2) 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 94

95 Theorem: Alternating Series Test Let. The alternating series ( 1) n + an and ( ) 1 converge when both conditions below are met. n= 1 n= 1 a 1 n n 1. 2., for all n. NOTE: The second condition can be modified to require that for all greater than some integer. EXAMPLE 1: Determine the convergence or divergence of the series. Explain. a. n= 0 n ( 1) n+ ( 2) 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 95

96 b. n= 1 n ( 1) ( n + ) ln 1 c. n= 1 n+ ( ) n n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 96

97 d. n= 1 1 cos nπ n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 97

98 e. ( 1) n= 1 n+ 1 ( n ) ( n ) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 98

99 Theorem: Alternating Series Remainder If a convergent alternating series satisfies the condition a n+ 1 an, then the value of the involved in approximating the sum by is less than or equal to the first term. EXAMPLE 2: Approximate the sum of the series by using the first six terms. n= 1 n+ ( 1) 1 3 n n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 99

100 EXAMPLE 3: Determine the number of terms required to approximate the sum of the series with an error of less than n= 1 ( 1) ( n) n 2! Theorem: Absolute Convergence If the series converges, then the series also converges. Which of our examples would be an example of this theorem? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 100

101 Definition of Absolute and Conditional Convergence 1. The series an converges. 2. The series an is convergent when is convergent when converges but diverges. EXAMPLE 4: Determine whether the series converges absolutely or conditionally, or diverges. a. n= 0 ( ) e 1 n n 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 101

102 n+ b. ( ) 1 n= 1 1 arctan n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 102

103 c. n= 1 π sin ( 2n + 1) 2 n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 103

104 9.6: The Ratio and Root Tests When you finish your homework you should be able to π Use the Ratio Test to ascertain whether an infinite series converges or diverges. π Use the Root Test to ascertain whether an infinite series converges or diverges. π Review Tests for convergence and divergence of an infinite series. Theorem: The Ratio Test Let an be a series with terms. 1. The series an converges when. 2. The series an diverges when or. 3. The Ratio Test is when. EXAMPLE 1: Determine the convergence or divergence of the series using the Ratio Test. a. n= 0 2 n e CREATED BY SHANNON MYERS (FORMERLY GRACEY) 104

105 b. n= 0 n 2 n! CREATED BY SHANNON MYERS (FORMERLY GRACEY) 105

106 c. n= 1 n+ 1 ( 1) ( n + 2) n( n+ 1) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 106

107 d. n= 0 ( n! ) ( n) 2 3! CREATED BY SHANNON MYERS (FORMERLY GRACEY) 107

108 Theorem: The Root Test 1. The series an 2. The series an converges when. diverges when or. 3. The Root Test is when. EXAMPLE 2: Determine the convergence or divergence of the series using the Root Test. a. n= 1 1 n n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 108

109 b. n= 1 n 2 5n + 1 n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 109

110 c. n= 1 ln n n n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 110

111 d. ( n )! n n ( ) 2 n= 1 n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 111

112 NOW IT S UP TO YOU!!! DETERMINE WHETHER THE FOLLOWING INFINITE SERIES CONVERGE OR DIVERGE 1. n= 1 ( n ) ( n ) Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 112

113 2. n= 1 ( 1) n n + 1 n Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 113

114 3. 3 n n n = n Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 114

115 4. n= 1 ( n ) ( n ) n 18 n! 2 1 Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 115

116 5. n= n n Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 116

117 6. n= 1 e 4n Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 117

118 7. n= 1 n 5 1 n 6 1 Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 118

119 8. n= 1 arctan n Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 119

120 9. 2 n= 1 ln n n Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 120

121 n n 1 n= 1 Step 1: Identify the test(s) and conditions (if applicable). Step 2: Run the test. Step 3: Conclusion. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 121

122 9.7: Taylor Polynomials When you finish your homework you should be able to π Find Taylor and Maclaurin polynomial approximations of elementary functions. π Use the remainder of a Taylor polynomial. Some uses of the Taylor series for analytic functions include: The of the series can be used as of the entire function. Keep in mind that you need a sufficient amount of. and of power series is since it can be done by term. operations can be done on the series. For example, formula follows from Taylor series for and functions. This result is important in the field of analysis. using the first few terms of a Taylor series can make otherwise problems possible for a restricted CREATED BY SHANNON MYERS (FORMERLY GRACEY) 122

123 domain. This is often used in. To find a function that another function, we choose a number in the of at which. This approximating is said to be about or at. The evil plan is to find a polynomial whose looks like the graph of this point. If we require that the of the polynomial function is the as the slope of the at, then we also have. Using these two requirements we can get a approximation of. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 123

124 EXAMPLE 1: Consider f ( x) x =. x + 1 a. Find a first-degree polynomial function ( ) P x = a + ax whose value and x =. slope agree with the value and slope of f at 0 x x x + 1 ( ) P x 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 124

125 P2 x = a0 + ax 1 + ax 2 whose value and slope agree with the value and slope of f at x = 0. b. Now find a second-degree polynomial function ( ) 2 P2 x x x ( x ) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 125

126 P x = a + ax+ ax + ax c. Let s go for a third-degree polynomial function ( ) x =. whose value and slope agree with the value and slope of f at 0 x x x P3 ( x ) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 126

127 Definition of nth Taylor and nth Maclaurin Polynomial If f has n derivatives at c, then the polynomial is called the polynomial for at. If, then is also called the polynomial for. Remainder of a Taylor Polynomial To the of approximating a function value by the Taylor polynomial, we use the concept of a. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 127

128 EXAMPLE 2: Consider the function f ( x) = x 2 cos x. a. Find the second Taylor polynomial for the function f ( x) = x 2 cos x centered at π. b. Approximate the function at x 7π 8 = using the polynomial found in part a. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 128

129 Taylor s Theorem If a function f is differentiable through order n + 1 in an interval I containing c, then, for each x in I, there exists z between x and c such that where A of this theorem is that where is the value of between and. For we have Does this look familiar? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 129

130 EXAMPLE 3: Use Taylor s Theorem to obtain an upper bound for error of the approximation. Then calculate the exact value of the error e ! 3! 4! 5! CREATED BY SHANNON MYERS (FORMERLY GRACEY) 130

131 EXAMPLE 4: Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than cos( 0.1 ) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 131

132 9.8: Power Series When you finish your homework you should be able to π Find the radius and interval of convergence of a power series. π Determine the endpoint convergence of a power series. π Differentiate and integrate a power series. x WARM-UP: Find the sixth-degree Maclaurin polynomial for f ( x) = e. This enables us to be able to the function near. We found out that the higher the of the approximating, the better the approximation becomes. In this section, you ll see that several important can be represented by series. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 132

133 Definition of Power Series If x is a variable, then an infinite series of the form is called a series at, where is a constant. If a power series is at, the power series will be of the form x EXAMPLE 1: Find the power series for f ( x) = e, centered at x = 0. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 133

134 Radius and Interval of Convergence A power series in can be thought of as a of. The of is the of all for which the power series. Every power series converges at its. Therefore, is always in the of. The domain of a power series can take on any one of the following forms: a an the of numbers CREATED BY SHANNON MYERS (FORMERLY GRACEY) 134

135 Theorem: Convergence of a Power Series For a power series centered at c, precisely one of the following is true: 1. The series converges only at. 2. There exists a number such that the series converges for and diverges for. 3. The series converges absolutely for. Endpoint Convergence Each must be for or. This results in possible forms an of can take on. 0 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 135

136 Example 2: Find the radius and interval of convergence (including a check for convergence at the endpoints) of the following power series. a. ( x) n= 0 2 n CREATED BY SHANNON MYERS (FORMERLY GRACEY) 136

137 b. n= 0 ( 3x) ( n) n 2! CREATED BY SHANNON MYERS (FORMERLY GRACEY) 137

138 c. n= 0 ( x 3) ( n + ) 14 n+ 1 n+ 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 138

139 Theorem: Properties of Functions Defined by Power Series If the function has a radius of convergence of, then, on the interval f is and thus. The derivative and antiderivative are given below: The radius of convergence of the series obtained by or a power series is the as that of the power series. What may change is the of convergence. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 139

140 Example 3: Let f ( x) = n 2n+ 1 ( 1) x and g( x) n= 0 ( 2n + 1! ) a. Find the interval of convergence of f. n= 0 n ( 1) ( n) 2n x =. 2! b. Find the interval of convergence of g. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 140

141 c. Show that f ( x) = g( x). d. Show that g ( x) = f ( x). CREATED BY SHANNON MYERS (FORMERLY GRACEY) 141

142 e. Identify the function f. f. Identify the function g. Example 4: Write an equivalent series with the index of summation beginning at 1 n =. n+ a. ( 1) 1 ( n+ 1) n= 0 x n b. n= 0 n 2n+ 1 ( 1) x 2n + 1 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 142

143 9.9: Representing Functions as Power Series When you finish your homework you should be able to π Manipulate a geometric series to represent a function as a power series π Differentiate or integrate a geometric series to represent a function as a power series. WARM-UP: Find the infinite sum of the convergent series n= n. Now consider the function f ( x) 1 =. 1 x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 143

144 This represents f ( x) = 1 1 x only on the interval from. What is the domain of f?. How would we represent f on another interval? We must develop a which is at a different value. Example 1: Find the power series for f ( x) = 1 1 x centered at c = 2. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 144

145 Example 2: Find a geometric power series for the function f ( x) 2 = 5 x centered at 0, (a) by manipulating the function into the format of a geometric power series and (b) by using long division. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 145

146 Example 3: Find a power series for the function, centered at c, and determine the interval of convergence. 3 f x =, c= 2 2x 1 a. ( ) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 146

147 4 f x =, c= 3 3x 2 b. ( ) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 147

148 Operations with Power Series f x ax n Let ( ) = n and g( x) 1. f ( kx) N 2. ( ) n= 0 n= 0 n = bx n be power series centered at 0. n= 0 n n = ank x, where is a. nn f x = ax n, where is a. n= 0 3. f ( x) ± g( x) = ( an ± bn) n= 0 Note: These operations can change the of for the resulting series. Example 4: Find a power series for the function, centered at c, and determine the interval of convergence. 5 f x =, c= x a. ( ) 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 148

149 3x 8 f x =, c= 0 3x + 5x 2 b. ( ) 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 149

150 1 1 x Example 5: Consider the functions f ( x) = + and g ( x ) a. Find a power series for f, centered at 0. 1 n = = x. 1 x n= 0 b. Use your result from part a to determine a power series, centered at 0, for the function h( x) convergence. x 1 1 = = ( ) ( ) 2 x x 21 x. Identify the interval of CREATED BY SHANNON MYERS (FORMERLY GRACEY) 150

151 c. Use your result from part a to determine a power series, centered at 0, for the function r( x) = 2 ( x + 1) 3. Identify the interval of convergence. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 151

152 d. Use your result from part a to determine a power series, centered at 0, for 2 the function s( x) ln ( 1 x ) =. Identify the interval of convergence. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 152

153 9.10: Taylor and Maclaurin Series When you finish your homework you should be able to π Find a Taylor series or a Maclaurin series for a function. π Find a binomial series. π Use a basic list of Taylor series to derive other power series. WARM-UP: Find the 8 th degree Maclaurin polynomial for the function f x = x. ( ) cos Now let s see if we can form a power series! What about that interval of convergence? CREATED BY SHANNON MYERS (FORMERLY GRACEY) 153

154 Theorem: The Form of a Convergent Power Series If f is represented by a power series f ( x) a ( x c) n interval I containing c, then = for all x in an open n and CREATED BY SHANNON MYERS (FORMERLY GRACEY) 154

155 Definition of Taylor and Maclaurin Series If a function f has derivatives of all orders at x = c, then the series is called the series for at. If, then the series is the series for. Example 1: Find the Taylor series, centered at c, for the function. a. ( ) 4 x f x = e, c= 0 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 155

156 1 f x =, c= 2 1 x b. ( ) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 156

157 Theorem: Convergence of Taylor Series If lim Rn = 0 for all x in the interval I, then the Taylor series for f converges n and equals f ( x ). Example 2: Prove that the Maclaurin series for f ( x) = cos xconverges to f ( x ) for all x. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 157

158 Binomial Series Let s check out the function f ( x) ( 1 x) k = +, where k is a rational number. What do you think the Maclaurin series is for this function? Guess what YOU KNOW HOW TO FIND IT!!! So, on your mark, get set, GO! f x a bunch of times and evaluate each 1. ( ) at. Evil plan: a. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 158

159 2. Determine the of...don t forget to test the! CREATED BY SHANNON MYERS (FORMERLY GRACEY) 159

160 Guidelines for Finding a Power Series 1. f ( x ) and each at until you find a. 2. Form the coefficient, and determine the of convergence for the series. 3. Determine whether the series to within the interval of convergence. Example 3: Find the Maclaurin series for the function using the binomial series. a. f ( x) = 1 ( 1+ x) 4 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 160

161 f x = 1+ x b. ( ) 3 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 161

162 A Basic List of Power Series for Elementary Functions FUNCTION INTERVAL OF CONVERGENCE 1 x = 0< x < n n 1 x x x x x ( 1) x 1+ x = < x < 1 ln x = 0< x 2 e n x x x x x x = 1+ x ! 3! 4! 5! n! < x < n n ( ) x ( n + ) x x x sin x= x ! 5! 7! 2 1! n ( ) ( n) 2n x x x x cos x = ! 4! 6! 2! < x < < x < arctan x = 1 x 1 arcsin x = 1 x 1 ( 1) ( 1)( 2) 2 3 k k k x k k k x ( 1+ x) = 1+ kx ! 3! *convergence at endpoints depends on k 1< x < 1* CREATED BY SHANNON MYERS (FORMERLY GRACEY) 162

163 Example 4: Find the Maclaurin series for the function using the basic list of power series for elementary functions. 2 a. f ( x) = ln ( 1+ x ) x x b. f ( x) = e + e CREATED BY SHANNON MYERS (FORMERLY GRACEY) 163

164 f x = cos x c. ( ) 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 164

165 d. f ( x) = xcos x CREATED BY SHANNON MYERS (FORMERLY GRACEY) 165

166 Example 5: Find the first four nonzero terms of the Maclaurin series for the = +. x function f ( x) e ln ( 1 x) CREATED BY SHANNON MYERS (FORMERLY GRACEY) 166

167 Example 6: Use a power series to approximate the value of the integral with an error less than arctan x dx 0 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 167

168 7.4: Arc Length and Surfaces of Revolution When you finish your homework you should be able to π Find the arc length of a smooth curve. π Find the area of a surface of revolution Arc length is approximated by infinitely many. A curve is one which has a arc length. A sufficient condition for the graph of a function to be rectifiable between and is that be continuous on. A function of this type is considered to be differentiable on and its graph on the interval is a. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 168

169 Definition of Arc Length Let the function represent a smooth curve on the interval. The arc length of between and is For a smooth curve on the interval the arc length of between and is CREATED BY SHANNON MYERS (FORMERLY GRACEY) 169

170 EXAMPLE 1: Find the arc length from ( 3, 4) clockwise to ( 4,3 ) along the circle x + y = 25. Show that the result is one-fourth the circumference of a circle. 2 2 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 170

171 Definition of Surface of Revolution When the graph of a continuous function is about a,the resulting surface is a of. Definition of the Area of a Surface of Revolution Let the function have a continuous derivative on the interval. The area of the surface of revolution formed by revolving the graph of about a horizontal or vertical axis is where is the distance between the graph of and the axis of revolution. If on the interval then the surface area is where is the distance between the graph of and the axis of revolution. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 171

172 EXAMPLE 2: Find the area of the surface generated by revolving the curve y 2 = 9 x about the y-axis. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 172

173 10.1: Conics and Calculus When you finish your homework you should be able to π Use properties of conic sections to analyze and write equations of parabolas, ellipses, and hyperbolas. π Classify the graph of an equation of a conic section as a circle, parabola, ellipse, or hyperbola. π Find the equations of lines tangent and normal to conic sections The graph of each type of section can be described as the intersection of a plane and two identical which are connected at their vertices. A parabola is the set of all that are from a fixed line called the and a fixed point called the. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 173

174 Theorem: Standard Equation of a Parabola The standard form of a parabola with vertex and directrix is Vertical axis The standard form of a parabola with vertex and directrix is Horizontal Axis The focus lies on the axis units from the vertex. The coordinates of the focus are Vertical axis Horizontal Axis CREATED BY SHANNON MYERS (FORMERLY GRACEY) 174

175 EXAMPLE 1: Consider 2 y y x = 0. a. Find the vertex, focus, and the directrix of the parabola and sketch its graph. y x -5 b. Find the equation of the line tangent to the graph at x = 4. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 175

176 An ellipse is the set of all the sum of whose distances from two distinct fixed points called is constant. Theorem: Standard Equation of an Ellipse The standard form of the equation of an ellipse with center and major and minor axes of lengths and, where, is Major Axis is Horizontal or Major Axis is Vertical The foci lie on the major axis, units from the center, with CREATED BY SHANNON MYERS (FORMERLY GRACEY) 176

177 Theorem: Reflective Property of an Ellipse Let be a point on an ellipse. The tangent line to the ellipse at point makes angles with the lines through and the. Definition of Eccentricity of an Ellipse The of an ellipse is given by the ratio For an ellipse that is close to being a, the foci are close to the and the is close to.an ellipse has foci which are close to the and the is close to. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 177

178 EXAMPLE 2: Consider x 25y 64x 150y =. Find the center, foci, vertices, and eccentricity of the ellipse and sketch its graph. y x -5 EXAMPLE 3: Find an equation of the ellipse with vertices ( 0,3 ) and ( 8,3) and eccentricity ¾. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 178

179 A hyperbola is the set of all for which the absolute value of the difference between the distances from two distinct fixed points called is constant. The line connecting the vertices is the, and the of the transverse axis is the of the hyperbola. Theorem: Standard Equation of a Hyperbola The standard form of the equation of a hyperbola with center is Transverse Axis is Horizontal or Transverse Axis is Vertical The vertices are units from the center, and the foci are units from the center with. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 179

180 Theorem: Asymptotes of a Hyperbola Transverse Axis is Horizontal or Transverse Axis is Vertical EXAMPLE 4: Consider 2 2 y x = a. Find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes. y x -5 CREATED BY SHANNON MYERS (FORMERLY GRACEY) 180

181 b. Find equations for the tangent lines to the hyperbola at x = 4. c. Find equations for the normal lines to the hyperbola at x = 4. CREATED BY SHANNON MYERS (FORMERLY GRACEY) 181