Worksheet 7, Math 10560

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Worksheet 7, Math 10560"

Transcription

1 Worksheet 7, Math 0560 You must show all of your work to receive credit!. Determine whether the following series and sequences converge or diverge, and evaluate if they converge. If they diverge, you must explain why to get credit. X 5 n + n (a) n X 5 n Solution: Because is a geometric series with ration 5 n X 5 n + n series diverges. Therefore, diverges. n >, this geometric (b) X ln( n + n + ) Solution: This is a typical telescoping series. More precisely, let s first consider a partial sum S N = NX ln( n + N n + )= X ln(n +) ln(n +) =ln ln {z } +ln ln 4 {z } =ln ln(n +). +ln4 {z ln 5 } ln(n +) ln(n +) {z } n= n=n Thus, X ln( n + )= lim n + S N =ln ln(n +)=. N! (c) a n =sin( n ) Solution: Looking at the graph of sin(x), we notice that for any even number a and for a < < (a +), thensin( ) > 0, and for an odd number b and b < < (b +) then sin( ) < 0. Now looking at the sequence {n/ } you can check that you can always find integers m and m such that a < m / < (a +) and b < m / < (b+) for a even and b odd. Hence the sequence sin(n/ ) is

2 constantly changing from positive numbers to negative numbers and vice-versa, which implies that if it converges, it has to converge to 0. However, sin(x) =0 only when x is an integer multiple of. Since {n/ } is never approaching multiples of, thenthesequencesin(n/ ) diverges. (d) a n =sin( n ) Solution: Because sin x is a continuous function for all x is a real number, we have lim n! sin( n )=sin(lim n! n )=sin0=0.

3 . (a) Show that f(x) = p x is a continuous, positive, decreasing function on [, ). Solution: f is continuous because it is a composition of continuous functions and the only points at which it is not defined are and -, which are not in our interval. f is clearly positive. f 0 x (x) = < 0on[, ), so f is decreasing on our interval. (x ) (b) Compute R p x dx. Solution: Observe that R p dx R t x t! p x dx. If you re good with hyperbolic trigonometry, you ll recognize straight away that this is the antiderivative of the inverse hyperbolic cosine. The rest of us, however, will have to use a trigonometric substitution. Let x =secu. Then dx =secutan udu, when x =,u =, and when x = t, u =arcsect. Z t lim p t! x Z arcsec t dx t! Z arcsec t t! Z arcsec t t! sec u tan u p sec u du sec u tan u p tan u du sec udu t! ln sec u +tanu t! t! arcsec t! ln sec (arcsec t)+tan(arcsect) ln +sin(arcsect) ln cos (arcsec t) t! ln + p t t t ln + p + p! ln t + p t ln + p t! =ln lim t + p t ln + p t! = ln + p = ln sec +tan Therefore, R p dx = x (c) Determine whether the series P p n is convergent or divergent.

4 Solution: By our work in parts (a) (checking the requirement) and (b) (taking the integral), the series diverges by the Integral Test.

5 . For each of the following series, determine whether they converge, and state why. (a) P sin(n) Solution: The sequence sin(n) does not converge, so the series is divergent by the Divergence Test. (b) P n (c) P Solution: This series converges by the p-test, with p =>. n cos (n) Solution: We can use the Comparison Test. 0 < cos (n) apple (whyiscos (n) never 0?), so n cos (n) apple n, and. P n cos (n) n diverges, and so the given n series diverges also. (d) P 000( 5 4 )n Solution: This is a geometric series with ratio 5 4. But 5 4 > sotheseriesis divergent. You can also see this by the Divergence Test.

6 4. The meaning of the decimal representation of a number 0.d d d... (where the digit d i is on of the numbers 0,,..., 9) is that Show that this series always converges. 0.d d d d 4... = d 0 + d 0 + d 0 + d Solution: We note that d 0 + d 0 + d 0 + d = X d n 4 0 n However, we know that each 0 apple d n apple 0, so 0 apple dn that Thus, by the Comparison Test, 0.d d d d 4... always con- so the series converges. verges. X 0 = n 0 = 0 9 apple 0 = 0 n 0 n 0 n. We now see Another way to see this is to consider the sequence of partial sums, s N = P N d n. 0 n This sequence is clearly increasing, since we are always adding on positive terms, and it is bounded above since 0.d d d d 4... apple, so it must converge, because it is a bounded monotone sequence. The limit is the sum of the series. 5. Use integration to calculate upper and lower bounds for the series P. (Hint: Draw n the graph of the function, and calculate two di erent Riemann sums.) Solution: On the interval (0, ), f(x) = is a continuous decreasing function. x Therefore R we could calculate upper and lower bounds for the series by the integrals f(x) dx and R f(x) dx. These evaluate as lim t! = and lim t t! t = respectively. 8 8

10.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1.

10.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1. 10.1 Sequences Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1 Examples: EX1: Find a formula for the general term a n of the sequence,

More information

8.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1.

8.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1. 8. Sequences Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = Examples: 6. Find a formula for the general term a n of the sequence, assuming

More information

MATH 1242 FINAL EXAM Spring,

MATH 1242 FINAL EXAM Spring, MATH 242 FINAL EXAM Spring, 200 Part I (MULTIPLE CHOICE, NO CALCULATORS).. Find 2 4x3 dx. (a) 28 (b) 5 (c) 0 (d) 36 (e) 7 2. Find 2 cos t dt. (a) 2 sin t + C (b) 2 sin t + C (c) 2 cos t + C (d) 2 cos t

More information

JUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson

JUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises

More information

18.01 EXERCISES. Unit 3. Integration. 3A. Differentials, indefinite integration. 3A-1 Compute the differentials df(x) of the following functions.

18.01 EXERCISES. Unit 3. Integration. 3A. Differentials, indefinite integration. 3A-1 Compute the differentials df(x) of the following functions. 8. EXERCISES Unit 3. Integration 3A. Differentials, indefinite integration 3A- Compute the differentials df(x) of the following functions. a) d(x 7 + sin ) b) d x c) d(x 8x + 6) d) d(e 3x sin x) e) Express

More information

MA 114 Worksheet # 1: Improper Integrals

MA 114 Worksheet # 1: Improper Integrals MA 4 Worksheet # : Improper Integrals. For each of the following, determine if the integral is proper or improper. If it is improper, explain why. Do not evaluate any of the integrals. (c) 2 0 2 2 x x

More information

Solutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) =

Solutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) = Solutions to Exam, Math 56 The function f(x) e x + x 3 + x is one-to-one (there is no need to check this) What is (f ) ( + e )? Solution Because f(x) is one-to-one, we know the inverse function exists

More information

Learning Objectives for Math 166

Learning Objectives for Math 166 Learning Objectives for Math 166 Chapter 6 Applications of Definite Integrals Section 6.1: Volumes Using Cross-Sections Draw and label both 2-dimensional perspectives and 3-dimensional sketches of the

More information

Multiple Choice. (c) 1 (d)

Multiple Choice. (c) 1 (d) Multiple Choice.(5 pts.) Find the sum of the geometric series n=0 ( ) n. (c) (d).(5 pts.) Find the 5 th Maclaurin polynomial for the function f(x) = sin x. (Recall that Maclaurin polynomial is another

More information

Integration by Parts

Integration by Parts Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u

More information

Section 11.1: Sequences

Section 11.1: Sequences Section 11.1: Sequences In this section, we shall study something of which is conceptually simple mathematically, but has far reaching results in so many different areas of mathematics - sequences. 1.

More information

Questions from Larson Chapter 4 Topics. 5. Evaluate

Questions from Larson Chapter 4 Topics. 5. Evaluate Math. Questions from Larson Chapter 4 Topics I. Antiderivatives. Evaluate the following integrals. (a) x dx (4x 7) dx (x )(x + x ) dx x. A projectile is launched vertically with an initial velocity of

More information

Arc Length and Surface Area in Parametric Equations

Arc Length and Surface Area in Parametric Equations Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length

More information

Solutionbank Edexcel AS and A Level Modular Mathematics

Solutionbank Edexcel AS and A Level Modular Mathematics Page of Exercise A, Question The curve C, with equation y = x ln x, x > 0, has a stationary point P. Find, in terms of e, the coordinates of P. (7) y = x ln x, x > 0 Differentiate as a product: = x + x

More information

Section 5.6. Integration By Parts. MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10

Section 5.6. Integration By Parts. MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10 Section 5.6 Integration By Parts MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10 Integration By Parts Manipulating the Product Rule d dx (f (x) g(x)) = f (x) g (x) + f (x) g(x)

More information

6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities

6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities Chapter 6: Trigonometric Identities 1 Chapter 6 Complete the following table: 6.1 Reciprocal, Quotient, and Pythagorean Identities Pages 290 298 6.3 Proving Identities Pages 309 315 Measure of

More information

5.5. The Substitution Rule

5.5. The Substitution Rule INTEGRALS 5 INTEGRALS 5.5 The Substitution Rule In this section, we will learn: To substitute a new variable in place of an existing expression in a function, making integration easier. INTRODUCTION Due

More information

Sequence. A list of numbers written in a definite order.

Sequence. A list of numbers written in a definite order. Sequence A list of numbers written in a definite order. Terms of a Sequence a n = 2 n 2 1, 2 2, 2 3, 2 4, 2 n, 2, 4, 8, 16, 2 n We are going to be mainly concerned with infinite sequences. This means we

More information

( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f

( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f Warm-Up: Solve sinx = 2 for x 0,2π 5 (a) graphically (approximate to three decimal places) y (b) algebraically BY HAND EXACTLY (do NOT approximate except to verify your solutions) x x 0,2π, xscl = π 6,y,,

More information

MATH : Calculus II (42809) SYLLABUS, Spring 2010 MW 4-5:50PM, JB- 138

MATH : Calculus II (42809) SYLLABUS, Spring 2010 MW 4-5:50PM, JB- 138 MATH -: Calculus II (489) SYLLABUS, Spring MW 4-5:5PM, JB- 38 John Sarli, JB-36 O ce Hours: MTW 3-4PM, and by appointment (99) 537-5374 jsarli@csusb.edu Text: Calculus of a Single Variable, Larson/Hostetler/Edwards

More information

MATHEMATICS AP Calculus (BC) Standard: Number, Number Sense and Operations

MATHEMATICS AP Calculus (BC) Standard: Number, Number Sense and Operations Standard: Number, Number Sense and Operations Computation and A. Develop an understanding of limits and continuity. 1. Recognize the types of nonexistence of limits and why they Estimation are nonexistent.

More information

10.1 Sequences. A sequence is an ordered list of numbers: a 1, a 2, a 3,..., a n, a n+1,... Each of the numbers is called a term of the sequence.

10.1 Sequences. A sequence is an ordered list of numbers: a 1, a 2, a 3,..., a n, a n+1,... Each of the numbers is called a term of the sequence. 10.1 Sequences A sequence is an ordered list of numbers: a 1, a 2, a 3,..., a n, a n+1,... Each of the numbers is called a term of the sequence. Notation: A sequence {a 1, a 2, a 3,...} can be denoted

More information

CK- 12 Algebra II with Trigonometry Concepts 1

CK- 12 Algebra II with Trigonometry Concepts 1 14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, 3. 3. 5 9,,,,,,, 4 4 4 4 3 5 3, and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1 4.ans-1401-01 5.

More information

Math 115 HW #5 Solutions

Math 115 HW #5 Solutions Math 5 HW #5 Solutions From 29 4 Find the power series representation for the function and determine the interval of convergence Answer: Using the geometric series formula, f(x) = 3 x 4 3 x 4 = 3(x 4 )

More information

1. Taylor Polynomials of Degree 1: Linear Approximation. Reread Example 1.

1. Taylor Polynomials of Degree 1: Linear Approximation. Reread Example 1. Math 114, Taylor Polynomials (Section 10.1) Name: Section: Read Section 10.1, focusing on pages 58-59. Take notes in your notebook, making sure to include words and phrases in italics and formulas in blue

More information

A) 13 B) 9 C) 22 D) log 9

A) 13 B) 9 C) 22 D) log 9 Math 70 Exam 2 Review Name Be sure to complete these problems before the review session. Participation in our review session will count as a quiz grade. Please bring any questions you have ready to ask!

More information

Calculus II Practice Test Problems for Chapter 7 Page 1 of 6

Calculus II Practice Test Problems for Chapter 7 Page 1 of 6 Calculus II Practice Test Problems for Chapter 7 Page of 6 This is a set of practice test problems for Chapter 7. This is in no way an inclusive set of problems there can be other types of problems on

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Math 00 Calculus Lectures Chapter 8 Series Numeration of sections corresponds to the text James Stewart, Essential Calculus, Early Transcendentals, Second edition. Section 8. Sequences A sequence is a

More information

Absolute Convergence and the Ratio Test

Absolute Convergence and the Ratio Test Absolute Convergence and the Ratio Test MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Bacground Remar: All previously covered tests for convergence/divergence apply only

More information

Representation of Functions as Power Series.

Representation of Functions as Power Series. MATH 0 - A - Spring 009 Representation of Functions as Power Series. Our starting point in this section is the geometric series: x n = + x + x + x 3 + We know this series converges if and only if x

More information

Math Analysis Chapter 5 Notes: Analytic Trigonometric

Math Analysis Chapter 5 Notes: Analytic Trigonometric Math Analysis Chapter 5 Notes: Analytic Trigonometric Day 9: Section 5.1-Verifying Trigonometric Identities Fundamental Trig Identities Reciprocal Identities: 1 1 1 sin u = cos u = tan u = cscu secu cot

More information

Absolute Convergence and the Ratio & Root Tests

Absolute Convergence and the Ratio & Root Tests Absolute Convergence and the Ratio & Root Tests Math114 Department of Mathematics, University of Kentucky February 20, 2017 Math114 Lecture 15 1/ 12 ( 1) n 1 = 1 1 + 1 1 + 1 1 + Math114 Lecture 15 2/ 12

More information

Questions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.

Questions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt. Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,

More information

Calculus I Sample Final exam

Calculus I Sample Final exam Calculus I Sample Final exam Solutions [] Compute the following integrals: a) b) 4 x ln x) Substituting u = ln x, 4 x ln x) = ln 4 ln u du = u ln 4 ln = ln ln 4 Taking common denominator, using properties

More information

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 4 Solutions

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 4 Solutions Math 0: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 30 Homework 4 Solutions Please write neatly, and show all work. Caution: An answer with no work is wrong! Problem A. Use Weierstrass (ɛ,δ)-definition

More information

Section 5.8. Taylor Series

Section 5.8. Taylor Series Difference Equations to Differential Equations Section 5.8 Taylor Series In this section we will put together much of the work of Sections 5.-5.7 in the context of a discussion of Taylor series. We begin

More information

SUBJECT: ADDITIONAL MATHEMATICS CURRICULUM OUTLINE LEVEL: 3 TOPIC OBJECTIVES ASSIGNMENTS / ASSESSMENT WEB-BASED RESOURCES. Online worksheet.

SUBJECT: ADDITIONAL MATHEMATICS CURRICULUM OUTLINE LEVEL: 3 TOPIC OBJECTIVES ASSIGNMENTS / ASSESSMENT WEB-BASED RESOURCES. Online worksheet. TERM 1 Simultaneous Online worksheet. Week 1 Equations in two Solve two simultaneous equations where unknowns at least one is a linear equation, by http://www.tutorvista.com/mat substitution. Understand

More information

Ê 7, 45 Ê 7 Ë 7 Ë. Time: 100 minutes. Name: Class: Date:

Ê 7, 45 Ê 7 Ë 7 Ë. Time: 100 minutes. Name: Class: Date: Class: Date: Time: 100 minutes Test1 (100 Trigonometry) Instructor: Koshal Dahal SHOW ALL WORK, EVEN FOR MULTIPLE CHOICE QUESTIONS, TO RECEIVE FULL CREDIT. 1. Find the terminal point P (x, y) on the unit

More information

MATH 1231 MATHEMATICS 1B Calculus Section 4.4: Taylor & Power series.

MATH 1231 MATHEMATICS 1B Calculus Section 4.4: Taylor & Power series. MATH 1231 MATHEMATICS 1B 2010. For use in Dr Chris Tisdell s lectures. Calculus Section 4.4: Taylor & Power series. 1. What is a Taylor series? 2. Convergence of Taylor series 3. Common Maclaurin series

More information

5.9 Representations of Functions as a Power Series

5.9 Representations of Functions as a Power Series 5.9 Representations of Functions as a Power Series Example 5.58. The following geometric series x n + x + x 2 + x 3 + x 4 +... will converge when < x

More information

DIFFERENTIATION RULES

DIFFERENTIATION RULES 3 DIFFERENTIATION RULES DIFFERENTIATION RULES We have: Seen how to interpret derivatives as slopes and rates of change Seen how to estimate derivatives of functions given by tables of values Learned how

More information

FIRST YEAR CALCULUS W W L CHEN

FIRST YEAR CALCULUS W W L CHEN FIRST YER CLCULUS W W L CHEN c W W L Chen, 994, 28. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

More information

Math 12 Final Exam Review 1

Math 12 Final Exam Review 1 Math 12 Final Exam Review 1 Part One Calculators are NOT PERMITTED for this part of the exam. 1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 θ 2π? 2 b) Draw these angles

More information

Assignment. Disguises with Trig Identities. Review Product Rule. Integration by Parts. Manipulating the Product Rule. Integration by Parts 12/13/2010

Assignment. Disguises with Trig Identities. Review Product Rule. Integration by Parts. Manipulating the Product Rule. Integration by Parts 12/13/2010 Fitting Integrals to Basic Rules Basic Integration Rules Lesson 8.1 Consider these similar integrals Which one uses The log rule The arctangent rule The rewrite with long division principle Try It Out

More information

Fall 2014: the Area Problem. Example Calculate the area under the graph f (x) =x 2 over the interval 0 apple x apple 1.

Fall 2014: the Area Problem. Example Calculate the area under the graph f (x) =x 2 over the interval 0 apple x apple 1. Fall 204: the Area Problem Example Calculate the area under the graph f (x) =x 2 over the interval 0 apple x apple. Velocity Suppose that a particle travels a distance d(t) at time t. Then Average velocity

More information

MIDLAND ISD ADVANCED PLACEMENT CURRICULUM STANDARDS AP CALCULUS BC

MIDLAND ISD ADVANCED PLACEMENT CURRICULUM STANDARDS AP CALCULUS BC Curricular Requirement 1: The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineated in the Calculus BC

More information

p324 Section 5.2: The Natural Logarithmic Function: Integration

p324 Section 5.2: The Natural Logarithmic Function: Integration p324 Section 5.2: The Natural Logarithmic Function: Integration Theorem 5.5: Log Rule for Integration Let u be a differentiable function of x 1. 2. Example 1: Using the Log Rule for Integration ** Note:

More information

MATH 1A - FINAL EXAM DELUXE - SOLUTIONS. x x x x x 2. = lim = 1 =0. 2) Then ln(y) = x 2 ln(x) 3) ln(x)

MATH 1A - FINAL EXAM DELUXE - SOLUTIONS. x x x x x 2. = lim = 1 =0. 2) Then ln(y) = x 2 ln(x) 3) ln(x) MATH A - FINAL EXAM DELUXE - SOLUTIONS PEYAM RYAN TABRIZIAN. ( points, 5 points each) Find the following limits (a) lim x x2 + x ( ) x lim x2 + x x2 + x 2 + + x x x x2 + + x x 2 + x 2 x x2 + + x x x2 +

More information

Math Requirements for applicants by Innopolis University

Math Requirements for applicants by Innopolis University Math Requirements for applicants by Innopolis University Contents 1: Algebra... 2 1.1 Numbers, roots and exponents... 2 1.2 Basics of trigonometry... 2 1.3 Logarithms... 2 1.4 Transformations of expressions...

More information

Math 116 Second Midterm November 14, 2012

Math 116 Second Midterm November 14, 2012 Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that

More information

Solutions to Exam 2, Math 10560

Solutions to Exam 2, Math 10560 Solutions to Exam, Math 6. Which of the following expressions gives the partial fraction decomposition of the function x + x + f(x = (x (x (x +? Solution: Notice that (x is not an irreducile factor. If

More information

FINAL REVIEW FOR MATH The limit. a n. This definition is useful is when evaluating the limits; for instance, to show

FINAL REVIEW FOR MATH The limit. a n. This definition is useful is when evaluating the limits; for instance, to show FINAL REVIEW FOR MATH 500 SHUANGLIN SHAO. The it Define a n = A: For any ε > 0, there exists N N such that for any n N, a n A < ε. This definition is useful is when evaluating the its; for instance, to

More information

MATH115. Sequences and Infinite Series. Paolo Lorenzo Bautista. June 29, De La Salle University. PLBautista (DLSU) MATH115 June 29, / 16

MATH115. Sequences and Infinite Series. Paolo Lorenzo Bautista. June 29, De La Salle University. PLBautista (DLSU) MATH115 June 29, / 16 MATH115 Sequences and Infinite Series Paolo Lorenzo Bautista De La Salle University June 29, 2014 PLBautista (DLSU) MATH115 June 29, 2014 1 / 16 Definition A sequence function is a function whose domain

More information

B L U E V A L L E Y D I S T R I C T C U R R I C U L U M & I N S T R U C T I O N Mathematics AP Calculus BC

B L U E V A L L E Y D I S T R I C T C U R R I C U L U M & I N S T R U C T I O N Mathematics AP Calculus BC B L U E V A L L E Y D I S T R I C T C U R R I C U L U M & I N S T R U C T I O N Mathematics AP Calculus BC Weeks ORGANIZING THEME/TOPIC CONTENT CHAPTER REFERENCE FOCUS STANDARDS & SKILLS Analysis of graphs.

More information

Math WW08 Solutions November 19, 2008

Math WW08 Solutions November 19, 2008 Math 352- WW08 Solutions November 9, 2008 Assigned problems 8.3 ww ; 8.4 ww 2; 8.5 4, 6, 26, 44; 8.6 ww 7, ww 8, 34, ww 0, 50 Always read through the solution sets even if your answer was correct. Note

More information

Convergence of sequences and series

Convergence of sequences and series Convergence of sequences and series A sequence f is a map from N the positive integers to a set. We often write the map outputs as f n rather than f(n). Often we just list the outputs in order and leave

More information

Saxon Calculus Scope and Sequence

Saxon Calculus Scope and Sequence hmhco.com Saxon Calculus Scope and Sequence Foundations Real Numbers Identify the subsets of the real numbers Identify the order properties of the real numbers Identify the properties of the real number

More information

Arkansas Council of Teachers of Mathematics 2013 State Contest Calculus Exam

Arkansas Council of Teachers of Mathematics 2013 State Contest Calculus Exam 0 State Contest Calculus Eam In each of the following choose the BEST answer and shade the corresponding letter on the Scantron Sheet. Answer all multiple choice questions before attempting the tie-breaker

More information

Department of Mathematical Sciences. Math 226 Calculus Spring 2016 Exam 2V2 DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO

Department of Mathematical Sciences. Math 226 Calculus Spring 2016 Exam 2V2 DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO Department of Mathematical Sciences Math 6 Calculus Spring 6 Eam V DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO NAME (Printed): INSTRUCTOR: SECTION NO.: When instructed, turn over this cover page

More information

Assignment 16 Assigned Weds Oct 11

Assignment 16 Assigned Weds Oct 11 Assignment 6 Assigned Weds Oct Section 8, Problem 3 a, a 3, a 3 5, a 4 7 Section 8, Problem 4 a, a 3, a 3, a 4 3 Section 8, Problem 9 a, a, a 3, a 4 4, a 5 8, a 6 6, a 7 3, a 8 64, a 9 8, a 0 56 Section

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com PhysicsAndMathsTutor.com physicsandmathstutor.com June 2005 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (b) Solve, for 0 θ < 360, the equation 2 tan 2 θ + secθ = 1, giving your

More information

Announcements. Topics: Homework:

Announcements. Topics: Homework: Announcements Topics: - sections 7.1 (differential equations), 7.2 (antiderivatives), and 7.3 (the definite integral +area) * Read these sections and study solved examples in your textbook! Homework: -

More information

SOLVED PROBLEMS ON TAYLOR AND MACLAURIN SERIES

SOLVED PROBLEMS ON TAYLOR AND MACLAURIN SERIES SOLVED PROBLEMS ON TAYLOR AND MACLAURIN SERIES TAYLOR AND MACLAURIN SERIES Taylor Series of a function f at x = a is ( f k )( a) ( x a) k k! It is a Power Series centered at a. Maclaurin Series of a function

More information

MAC Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below.

MAC Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below. MAC 23. Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below. (x, y) y = 3 x + 4 a. x = 6 b. x = 4 c. x = 2 d. x = 5 e. x = 3 2. Consider the area of the

More information

Introduction to Series and Sequences Math 121 Calculus II Spring 2015

Introduction to Series and Sequences Math 121 Calculus II Spring 2015 Introduction to Series and Sequences Math Calculus II Spring 05 The goal. The main purpose of our study of series and sequences is to understand power series. A power series is like a polynomial of infinite

More information

cos t 2 sin 2t (vi) y = cosh t sinh t (vii) y sin x 2 = x sin y 2 (viii) xy = cot(xy) (ix) 1 + x = sin(xy 2 ) (v) g(t) =

cos t 2 sin 2t (vi) y = cosh t sinh t (vii) y sin x 2 = x sin y 2 (viii) xy = cot(xy) (ix) 1 + x = sin(xy 2 ) (v) g(t) = MATH1003 REVISION 1. Differentiate the following functions, simplifying your answers when appropriate: (i) f(x) = (x 3 2) tan x (ii) y = (3x 5 1) 6 (iii) y 2 = x 2 3 (iv) y = ln(ln(7 + x)) e 5x3 (v) g(t)

More information

Study # 1 11, 15, 19

Study # 1 11, 15, 19 Goals: 1. Recognize Taylor Series. 2. Recognize the Maclaurin Series. 3. Derive Taylor series and Maclaurin series representations for known functions. Study 11.10 # 1 11, 15, 19 f (n) (c)(x c) n f(c)+

More information

GRE Math Subject Test #5 Solutions.

GRE Math Subject Test #5 Solutions. GRE Math Subject Test #5 Solutions. 1. E (Calculus) Apply L Hôpital s Rule two times: cos(3x) 1 3 sin(3x) 9 cos(3x) lim x 0 = lim x 2 x 0 = lim 2x x 0 = 9. 2 2 2. C (Geometry) Note that a line segment

More information

Table of Contents. Module 1

Table of Contents. Module 1 Table of Contents Module Order of operations 6 Signed Numbers Factorization of Integers 7 Further Signed Numbers 3 Fractions 8 Power Laws 4 Fractions and Decimals 9 Introduction to Algebra 5 Percentages

More information

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e)...

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e)... Math, Exam III November 6, 7 The Honor Code is in effect for this examination. All work is to be your own. No calculators. The exam lasts for hour and min. Be sure that your name is on every page in case

More information

f (x) = k=0 f (0) = k=0 k=0 a k k(0) k 1 = a 1 a 1 = f (0). a k k(k 1)x k 2, k=2 a k k(k 1)(0) k 2 = 2a 2 a 2 = f (0) 2 a k k(k 1)(k 2)x k 3, k=3

f (x) = k=0 f (0) = k=0 k=0 a k k(0) k 1 = a 1 a 1 = f (0). a k k(k 1)x k 2, k=2 a k k(k 1)(0) k 2 = 2a 2 a 2 = f (0) 2 a k k(k 1)(k 2)x k 3, k=3 1 M 13-Lecture Contents: 1) Taylor Polynomials 2) Taylor Series Centered at x a 3) Applications of Taylor Polynomials Taylor Series The previous section served as motivation and gave some useful expansion.

More information

Math 180, Final Exam, Fall 2012 Problem 1 Solution

Math 180, Final Exam, Fall 2012 Problem 1 Solution Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.

More information

1 Super-Brief Calculus I Review.

1 Super-Brief Calculus I Review. CALCULUS II MATH 66 SPRING 206 (COHEN) LECTURE NOTES Remark 0.. Much of this set of lecture notes is adapted from a combination of Rogawski s Calculus Early Transcendentals and Briggs and Cochran s Calculus,

More information

Integration. Tuesday, December 3, 13

Integration. Tuesday, December 3, 13 4 Integration 4.3 Riemann Sums and Definite Integrals Objectives n Understand the definition of a Riemann sum. n Evaluate a definite integral using properties of definite integrals. 3 Riemann Sums 4 Riemann

More information

Lecture 5: Integrals and Applications

Lecture 5: Integrals and Applications Lecture 5: Integrals and Applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde 1 1 / 21 Outline The

More information

Graphing Radicals Business 7

Graphing Radicals Business 7 Graphing Radicals Business 7 Radical functions have the form: The most frequently used radical is the square root; since it is the most frequently used we assume the number 2 is used and the square root

More information

Math 308 Week 8 Solutions

Math 308 Week 8 Solutions Math 38 Week 8 Solutions There is a solution manual to Chapter 4 online: www.pearsoncustom.com/tamu math/. This online solutions manual contains solutions to some of the suggested problems. Here are solutions

More information

AP Calculus Testbank (Chapter 9) (Mr. Surowski)

AP Calculus Testbank (Chapter 9) (Mr. Surowski) AP Calculus Testbank (Chapter 9) (Mr. Surowski) Part I. Multiple-Choice Questions n 1 1. The series will converge, provided that n 1+p + n + 1 (A) p > 1 (B) p > 2 (C) p >.5 (D) p 0 2. The series

More information

Prentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Trigonometry (Grades 9-12)

Prentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Trigonometry (Grades 9-12) California Mathematics Content Standards for Trigonometry (Grades 9-12) Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric

More information

AS PURE MATHS REVISION NOTES

AS PURE MATHS REVISION NOTES AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are

More information

Lecture 4: Integrals and applications

Lecture 4: Integrals and applications Lecture 4: Integrals and applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: autumn 2013 Lejla Batina Version: autumn 2013 Calculus en Kansrekenen 1 / 18

More information

Taylor and Maclaurin Series. Copyright Cengage Learning. All rights reserved.

Taylor and Maclaurin Series. Copyright Cengage Learning. All rights reserved. 11.10 Taylor and Maclaurin Series Copyright Cengage Learning. All rights reserved. We start by supposing that f is any function that can be represented by a power series f(x)= c 0 +c 1 (x a)+c 2 (x a)

More information

Math 175: Chapter 6 Review: Trigonometric Functions

Math 175: Chapter 6 Review: Trigonometric Functions Math 175: Chapter 6 Review: Trigonometric Functions In order to prepare for a test on Chapter 6, you need to understand and be able to work problems involving the following topics. A. Can you sketch an

More information

Taylor and Maclaurin Series. Approximating functions using Polynomials.

Taylor and Maclaurin Series. Approximating functions using Polynomials. Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear

More information

6.2 Trigonometric Integrals and Substitutions

6.2 Trigonometric Integrals and Substitutions Arkansas Tech University MATH 9: Calculus II Dr. Marcel B. Finan 6. Trigonometric Integrals and Substitutions In this section, we discuss integrals with trigonometric integrands and integrals that can

More information

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

More information

ADDITONAL MATHEMATICS

ADDITONAL MATHEMATICS ADDITONAL MATHEMATICS 00 0 CLASSIFIED TRIGONOMETRY Compiled & Edited B Dr. Eltaeb Abdul Rhman www.drtaeb.tk First Edition 0 5 Show that cosθ + + cosθ = cosec θ. [3] 0606//M/J/ 5 (i) 6 5 4 3 0 3 4 45 90

More information

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1. Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

More information

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

More information

Los Angeles Unified School District Secondary Mathematics Branch

Los Angeles Unified School District Secondary Mathematics Branch Math Analysis AB or Trigonometry/Math Analysis AB (Grade 10, 11 or 12) Prerequisite: Algebra 2AB 310601 Math Analysis A 310602 Math Analysis B 310505 Trigonometry/Math Analysis A 310506 Trigonometry/Math

More information

10.7 Trigonometric Equations and Inequalities

10.7 Trigonometric Equations and Inequalities 0.7 Trigonometric Equations and Inequalities 857 0.7 Trigonometric Equations and Inequalities In Sections 0. 0. and most recently 0. we solved some basic equations involving the trigonometric functions.

More information

AP Calculus BC. Functions, Graphs, and Limits

AP Calculus BC. Functions, Graphs, and Limits AP Calculus BC The Calculus courses are the Advanced Placement topical outlines and prepare students for a successful performance on both the Advanced Placement Calculus exam and their college calculus

More information

y sin n x dx cos x sin n 1 x n 1 y sin n 2 x cos 2 x dx y sin n xdx cos x sin n 1 x n 1 y sin n 2 x dx n 1 y sin n x dx

y sin n x dx cos x sin n 1 x n 1 y sin n 2 x cos 2 x dx y sin n xdx cos x sin n 1 x n 1 y sin n 2 x dx n 1 y sin n x dx SECTION 7. INTEGRATION BY PARTS 57 EXAPLE 6 Prove the reduction formula N Equation 7 is called a reduction formula because the eponent n has been reduced to n and n. 7 sin n n cos sinn n n sin n where

More information

*n23494b0220* C3 past-paper questions on trigonometry. 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (2)

*n23494b0220* C3 past-paper questions on trigonometry. 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (2) C3 past-paper questions on trigonometry physicsandmathstutor.com June 005 1. (a) Given that sin θ + cos θ 1, show that 1 + tan θ sec θ. (b) Solve, for 0 θ < 360, the equation tan θ + secθ = 1, giving your

More information

Find all points where the function is discontinuous. 1) Find all vertical asymptotes of the given function. x(x - 1) 2) f(x) =

Find all points where the function is discontinuous. 1) Find all vertical asymptotes of the given function. x(x - 1) 2) f(x) = Math 90 Final Review Find all points where the function is discontinuous. ) Find all vertical asymptotes of the given function. x(x - ) 2) f(x) = x3 + 4x Provide an appropriate response. 3) If x 3 f(x)

More information

Taylor and Maclaurin Series. Approximating functions using Polynomials.

Taylor and Maclaurin Series. Approximating functions using Polynomials. Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear

More information

Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain.

Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain. Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain. For example f(x) = 1 1 x = 1 + x + x2 + x 3 + = ln(1 + x) = x x2 2

More information

Techniques of Integration

Techniques of Integration 0 Techniques of Integration ½¼º½ ÈÓÛ Ö Ó Ò Ò Ó Ò Functions consisting of products of the sine and cosine can be integrated by using substitution and trigonometric identities. These can sometimes be tedious,

More information

Convergence Tests. Academic Resource Center

Convergence Tests. Academic Resource Center Convergence Tests Academic Resource Center Series Given a sequence {a 0, a, a 2,, a n } The sum of the series, S n = A series is convergent if, as n gets larger and larger, S n goes to some finite number.

More information

18.01 Final Exam. 8. 3pm Hancock Total: /250

18.01 Final Exam. 8. 3pm Hancock Total: /250 18.01 Final Exam Name: Please circle the number of your recitation. 1. 10am Tyomkin 2. 10am Kilic 3. 12pm Coskun 4. 1pm Coskun 5. 2pm Hancock Problem 1: /25 Problem 6: /25 Problem 2: /25 Problem 7: /25

More information