MATH1013 Calculus I. Edmund Y. M. Chiang. Department of Mathematics Hong Kong University of Science & Technology.
|
|
- Hilary Francis
- 5 years ago
- Views:
Transcription
1 1 Based on Stewart, James, Single Variable Calculus, Early Transcendentals, 7th edition, Brooks/Coles, 2012 Briggs, Cochran and Gillett: Calculus for Scientists and Engineers: Early Transcendentals, Pearson 2013 MATH1013 Calculus I Integration I (Chap. 5 ( 5.1, 5.2)) 1 Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology November 24, 2014
2 Definite integrals Riemann Sums Integrable functions
3 Area under curve We shall consider continuous functions defined on a closed interval only. The aim is to develop a theory that can be used to find area of the region under the given function.
4 Example We consider the problem of finding the area under the straight line f (x) = x for the interval 0 x 1. We divide the interval [0, 1] into five subintervals of equal width. By a partition of [0, 1] with five points: {x 0, x 1, x 2, x 3, x 4, x 5 } of [0, 1]. So we have the subintervals: [x 0, x 1 ] = [0, 1/5] [x 1, x 2 ] = [1/5, 2/5] [x 2, x 3 ] = [2/5, 3/5] [x 3, x 4 ] = [3/5, 4/5] [x 4, x 5 ] = [4/5, 5/5]
5 Right Riemann sum The f attains maximum values in each of the above intervals: f (x 1 ) = f (1/5) = 1/5, f (x 2 ) = f (2/5) = 2/5, f (x 3 ) = f (3/5) = 3/5, f (x 4 ) = f (4/5) = 4/5, f (x 5 ) = f (5/5) = 5/5 = 1. Let us sum the areas of the five rectangles with height at the right-end point of [x i 1, x i ] and with base 1/n. Thus we have ( ) ( ) ( ) ( ) ( ) S 5 = f f f f f 5 5 = 1 ( ) = = 3 25, is called an right Riemann sum of f over [0, 1] with respect to the above partition. The approximation 3/25 is larger than the actual area.
6 Left Riemann sum The f attains minimum values in each of the above intervals: f (x 0 ) = f (0/5) = 0/5, f (x 1 ) = f (1/5) = 1/5, f (x 2 ) = f (2/5) = 2/5, f (x 3 ) = f (3/5) = 3/5, f (x 4 ) = f (4/5) = 4/5. Let us sum the areas of the five rectangles height at the left-end point in [x i 1, x i ] and with base 1/n. Thus we have ( ) ( ) ( ) ( ) ( ) S 5 = f f f f f 5 5 = 1 ( ) = = 2 5, is called an left Riemann sum of f over [0, 1] with respect to the above partition. The approximation 2/25 is smaller than the actual area.
7 Left and Right Riemann sums figures Suppose the value of the area that we are to find is A. Then we clearly see from the figure below that the following inequalities must hold: 2 5 = S 5 A S 5 = 3 5.
8 Left and Right Riemann sums: general case It is also clear that the above argument that we divide [0, 1] into five equal intervals is nothing special. Hence the same argument applies for any number of intervals. Hence we must have S n A S n, (2) where n is any positive integer. We partition [0, 1] into n equal subintervals: {x 0, x 1, x 2,..., x n 1, x n } = {0, 1 n, 2 n,..., n 1 n, n n }. and the width of each of the subintervals of 1/n.
9 Hence the upper sum is S n = f ( 1 n Right Riemann sum: general case ) 1 n + f ( 2 n ) 1 n + + f ( n 1 n = 1 1 n n n n + + n 1 1 n n + n 1 n n = 1 ( ) (n 1) + n n 2 = 1 n(n + 1) n 2 2 = 1 ( 1 ) n ) 1 n + f ( n ) 1 n n Note that we have used the fundamental formula that (n 1) + n = n(n + 1) 2
10 Hence the lower sum is S n = f ( 0 ) 1 n n + f ( 1 n = 1 n Left Riemann sum: general case ) 1 n + + f ( n 2 n 0 n n n + + n 2 1 n n + n 1 1 n n = 1 ( ) (n 2) + n 1 n 2 = 1 n 2 (n 1)n 2 We deduce that 1 2 ( 1 1 n = 1 2 ( 1 ) 1. n ) A 1 2 ) 1 n + f ( n 1 n ( 1 ) 1 +, for all n. n ) 1 n Letting n + gives that 1 2 A 1 2. That is A = 1/2 and lim S n = 1/2 = lim S n + n + n.
11 Riemann sums We consider a closed interval [a, b] and let P = {x 0, x 1, x 2,..., x n 1, x n }, x i = x i x i 1 = b a n, i = 1,, n to be any points lying inside [a, b]. Let f (x) be a continuous function defined on [a, b]. Then we define a Riemann sum of f (x) over [a, b] with respect to partition P to be the sum f (x 1 ) x 1 + f (x 2 ) x f (x n) x n where xi is an arbitrary point lying in [x i 1, x i ], i = 1, 2,, n. Depending on the choices of xi, we have 1. Left Riemann sum if xi = x i 1 ; 2. Right Riemann sum if x i = x i ; 3. Mid-point Riemann sum if x i = (x i 1 + x i )/2.
12 Riemann sum figure Figure: (Briggs, et al, Figure 5.8)
13 Left Riemann sum figure Figure: (Briggs, et al, Figure 5.9)
14 Right Riemann sum figure Figure: (Briggs, et al, Figure 5.10)
15 Mid-point Riemann sum figure Figure: (Briggs, et al, Figure 5.11)
16 So we have Riemann sum of Sine We compute various Riemann sums under the sine curve from x = 0 to x = π/2 with six intervals. So we have x = b a n Left Riemann sum gives = π/2 0 6 = π 12. f (x 1 ) x 1 + f (x 2 ) x f (x n) x n Right Riemann sum gives f (x 1 ) x 1 + f (x 2 ) x f (x n) x n Mid-point Riemann sum gives f (x 1 ) x 1 + f (x 2 ) x f (x n) x n 1.003
17 Partition of Riemann sum of Sine Figure: (Briggs, et al, Figure 5.2)
18 Partition of Riemann sum of Sine Figure: (Briggs, et al, Theorem 5.1)
19 Partition 50 Figure: (Briggs, et al, Figure 5.15)
20 Example (Briggs, et al, p. 372) After choosing a partition that divides [0, 2] into 50 subintervals: x k = x k x k 1 = = 1 25 = The Right Riemann Sum is given by 50 k=1 50 = f (xk ) x 50 k = f (x k ) (x k x k 1 ) k=1 k=1 f ( k ) (0.04) = k=1 [( k ) 3 ] + 1 (0.04) 25 = [ 1 (50 51) 2 ] + 50 (0.04) =
21 Example (Briggs, et al, p. 372) II The Left Riemann Sum is given by 49 k=0 49 = f (xk ) x 49 k+1 = f (x k ) (x k+1 x k ) k=0 k=0 f ( k ) (0.04) = k=0 [( k ) 3 ] + 1 (0.04) 25 = [ 1 (49 50) 2 ] + 50 (0.04) =
22 Example (Briggs, et al, p. 388) III After choosing a partition that divides [0, 2] into n subintervals: x k = x k x k 1 = 2 0 n The Right Riemann Sum is given by = 2 n. n f (xk ) x k = k=1 as n. n f (x k ) 2 n k=1 = 2 n [( 2k ) 3 ] + 1 = 2 ( 2 3 n n n n 3 k=1 = 2 ( 2 3 n n 3 n2 (n + 1) 2 ) + n 4 [ ( = ) 2 ] + 1 6, n n k 3 + k=1 n k=1 ) 1
23 Example (Briggs, et al, p. 389) III The Left Riemann Sum is given by n 1 f (xk ) x n 1 k+1 = f (x k ) 2 n k=0 k=0 = 2 n 1 [( 2k ) 3 ] + 1 = 2 ( 2 3 n 1 n 1 ) n n n n 3 k k=0 k=0 k=0 = 2 ( 2 3 n n 3 n2 (n 1) 2 ) [ ( + n = ) 2 ] + 1 6, 4 n In fact, we have ( ) 2 ( + 2 A ) n n to hold for every integer n. So A = 6.
24 Example (Stewart, p. 362) Let f (x) = x 2 and find the area A under f over the interval [0, 1]. Choose the partition P = {0, 1 n, 2 n,, n 1 n, n n = 1}. Show that the following inequalities 1 3 ( 1 )( n 2n ) A 1 3 ( 1 )( 1 ) n 2n by computing the left Riemann sum and right Riemann sum. Then show that the area is A = 1 3. We omit the details.
25 Some left/right Riemann sums Figure: (Stewart, Figures )
26 Exercises (Briggs, et al: p. 376) Write the following sum in summation notation: (Stewart: p. 369, Q. 4) Estimate the area under f (x) = x over [0, 4] using four approximating rectangles. 4 (Briggs, et al, p. 377) Given that f (1 + k) 1 is a k=1 Riemann sum of a certain function f over an interval [a, b] with a partition of n subdivisions. Identify the f, [a, b] and n.
27 Upper/Lower Riemann sums Riemann originally considered for a partition P n = {x 0,, x n } Area = lim n R n = [f (x 1 ) x + f (x 2 ) x + f (x n ) x] Area = lim n L n = [f (x 0 ) x + f (x 1 ) x + f (x n 1 ) x] and more generally, the upper and lower sums defined respectively by Area = lim n U n = lim n [f (x 1 ) x + f (x 2 ) x + f (x n) x] where f (xk ) attains the largest value in the k th interval [x k 1, x k ]; Area = lim n L n = lim n [f (x 1 ) x + f (x 2 ) x + f (x n ) x] where f (x k ) attains the smallest value in the k th interval [x k 1, x k ]
28 Upper Riemann sum figure Figure: (Stewart, Figures )
29 Upper/lower Riemann figures Figure: (Stewart, Figures )
30 Definite Integrals definition Figure: (Stewart Figure p. 372)
31 Definite Integral notation Figure: (Publisher Figure 5.21)
32 Integrable functions Theorem (Compare Stewart p. 373) Let f be a continuous function except on a finite number of discontinuities over the interval [a, b]. Then f is integrable on [a, b]. That is, lim δx 0 k=1 n f (xk ) x k = b a f (x) dx, exists irrespective to the x k and the partition [x k 1, x k ] chosen. So Since f (x) = x 2 is continuous over [0, 1] so it is integrable and 1 0 x 2 dx = 1 2 according to a previous calculation. Since f (x) = x 3 is continuous over [0, 1] so it is integrable and 1 0 x 3 dx = 1 according to a previous calculation. 3
33 Special partition and sample points Figure: (Stewart p. 374)
34 Piecewise continuous functions The following function has a finite number of discontinuities and so is integrable. However, we note that part 2 of the area is negative: Figure: (Briggs, et al, Figure 5.23)
35 Negative area Figure: (Briggs, et al, Figure 5.18)
36 Negative area Figure: (Briggs, et al, Figure 5.17)
37 Figure: (Briggs, et al, Figure 5.20) Net area
38 Figure: (Briggs, et al, Figure 5.24) Recognizing integral
39 Computing net area Figure: (Briggs, et al, Figure 5.31)
40 Exercises 1. Write down the right Riemann sum for 2. Interpret the sum lim integral. 3. Let n k=1 f (x) = x 2 dx; n 3 n(1 + 3k as a certain Riemann n ) { 2x 2, if x 2; x + 4, if x > 2. Compute both the net area and actual area of 5 0 f (t) dt.
41 One can analyze the the sum as n k=1 3 n(1 + 3k n ) = = Hints to Exercises n k=1 n k=1 1 (1 + 3 k n ) 3 n 1 (1 + k 3 n ) 3 n = n f (k 3 n ) 3 n k=1 which represents a right Riemann sum for f (x) = 1, and for 1 + x the integration range from a = 0 and b = 3. But then lim n 0 k=1 3 3 n(1 + 3k n ) = x dx = ln 1 + x = (ln 4 ln 1) = ln
42 Hints to Exercises The net area of the last example is given by 2 0 (2x 2) dx ( x + 4) dx = (x 2 2x) ( x 2 /2 + 4x) 5 2 = ( ) ( ) + (20 8) = = 3 2. The actual area is given by (2x 2) dx + (2x 2) dx + 5 ( x + 4) dx + ( x + 4) dx = (x 2 2x) (x 2 2x) ( x 2 /2 + 4x) 4 ( x /2 + 4x) 5 4 = = 9 2.
43 Properties of Definite Integral Figure: (Briggs, et al, Table 5.4)
44 Sum of integrals Figure: (Briggs, et al, Figure 5.29)
45 Comparison properties of Definite Integral Figure: (Stewart, p. 381)
MATH1013 Calculus I. Introduction to Functions 1
MATH1013 Calculus I Introduction to Functions 1 Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology May 9, 2013 Integration I (Chapter 4) 2013 1 Based on Briggs,
More informationMATH1013 Calculus I. Functions I 1
1 Steward, James, Single Variable Calculus, Early Transcendentals, 7th edition, Brooks/Coles, 2012 Based on Briggs, Cochran and Gillett: Calculus for Scientists and Engineers: Early Transcendentals, Pearson
More informationMATH1013 Calculus I. Derivatives V ( 4.7, 4.9) 1
1 Based on Stewart, James, Single Variable Calculus, Early Transcendentals, 7th edition, Brooks/Coles, 2012 Briggs, Cochran and Gillett: Calculus for Scientists and Engineers: Early Transcendentals, Pearson
More informationMATH1013 Calculus I. Revision 1
MATH1013 Calculus I Revision 1 Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology November 27, 2014 2013 1 Based on Briggs, Cochran and Gillett: Calculus for Scientists
More information1 Approximating area under curves and Riemann sums
Professor Jennifer Balakrishnan, jbala@bu.edu What is on today 1 Approximating area under curves and Riemann sums 1 1.1 Riemann sums................................... 1 1.2 Area under the velocity curve..........................
More informationMATH1013 Calculus I. Derivatives II (Chap. 3) 1
MATH1013 Calculus I Derivatives II (Chap. 3) 1 Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology October 16, 2013 2013 1 Based on Briggs, Cochran and Gillett: Calculus
More informationToday s Agenda. Upcoming Homework Section 5.1: Areas and Distances Section 5.2: The Definite Integral
Today s Agenda Upcoming Homework Section 5.1: Areas and Distances Section 5.2: The Definite Integral Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 18 November 2015 1 / 13 Upcoming
More informationMA 137 Calculus 1 with Life Science Applications. (Section 6.1)
MA 137 Calculus 1 with Life Science Applications (Section 6.1) Alberto Corso alberto.corso@uky.edu Department of Mathematics University of Kentucky December 2, 2015 1/17 Sigma (Σ) Notation In approximating
More informationThe University of Sydney Math1003 Integral Calculus and Modelling. Semester 2 Exercises and Solutions for Week
The University of Sydney Math3 Integral Calculus and Modelling Semester 2 Exercises and Solutions for Week 2 2 Assumed Knowledge Sigma notation for sums. The ideas of a sequence of numbers and of the limit
More informationMATH Calculus of One Variable, Part I Spring 2019 Textbook: Calculus. Early Transcendentals. by Briggs, Cochran, Gillett, Schulz.
MATH 1060 - Calculus of One Variable, Part I Spring 2019 Textbook: Calculus. Early Transcendentals. by Briggs, Cochran, Gillett, Schulz. 3 rd Edition Testable Skills Unit 3 Important Students should expect
More informationChapter 5 - Integration
Chapter 5 - Integration 5.1 Approximating the Area under a Curve 5.2 Definite Integrals 5.3 Fundamental Theorem of Calculus 5.4 Working with Integrals 5.5 Substitution Rule for Integrals 1 Q. Is the area
More informationFIRST 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() Chapter 8 November 9, / 1
Example 1: An easy area problem Find the area of the region in the xy-plane bounded above by the graph of f(x) = 2, below by the x-axis, on the left by the line x = 1 and on the right by the line x = 5.
More informationMATH 2413 TEST ON CHAPTER 4 ANSWER ALL QUESTIONS. TIME 1.5 HRS.
MATH 1 TEST ON CHAPTER ANSWER ALL QUESTIONS. TIME 1. HRS. M1c Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Use the summation formulas to rewrite the
More informationScience One Integral Calculus. January 8, 2018
Science One Integral Calculus January 8, 2018 Last time a definition of area Key ideas Divide region into n vertical strips Approximate each strip by a rectangle Sum area of rectangles Take limit for n
More informationAPPLICATIONS OF INTEGRATION
6 APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6.5 Average Value of a Function In this section, we will learn about: Applying integration to find out the average value of a function. AVERAGE
More informationF (x) is an antiderivative of f(x) if F (x) = f(x). Lets find an antiderivative of f(x) = x. We know that d. Any ideas?
Math 24 - Calculus for Management and Social Science Antiderivatives and the Indefinite Integral: Notes So far we have studied the slope of a curve at a point and its applications. This is one of the fundamental
More informationWe saw in Section 5.1 that a limit of the form. arises when we compute an area.
INTEGRALS 5 INTEGRALS Equation 1 We saw in Section 5.1 that a limit of the form n lim f ( x *) x n i 1 i lim[ f ( x *) x f ( x *) x... f ( x *) x] n 1 2 arises when we compute an area. n We also saw that
More informationRAMs.notebook December 04, 2017
RAMsnotebook December 04, 2017 Riemann Sums and Definite Integrals Estimate the shaded area Area between a curve and the x-axis How can you improve your estimate? Suppose f(x) 0 x [a, b], then the area
More informationMATH 1014 Tutorial Notes 8
MATH4 Calculus II (8 Spring) Topics covered in tutorial 8:. Numerical integration. Approximation integration What you need to know: Midpoint rule & its error Trapezoid rule & its error Simpson s rule &
More information1 Definition of the Riemann integral
MAT337H1, Introduction to Real Analysis: notes on Riemann integration 1 Definition of the Riemann integral Definition 1.1. Let [a, b] R be a closed interval. A partition P of [a, b] is a finite set of
More informationThe Integral of a Function. The Indefinite Integral
The Integral of a Function. The Indefinite Integral Undoing a derivative: Antiderivative=Indefinite Integral Definition: A function is called an antiderivative of a function on same interval,, if differentiation
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationMAT137 - Term 2, Week 2
MAT137 - Term 2, Week 2 This lecture will assume you have watched all of the videos on the definition of the integral (but will remind you about some things). Today we re talking about: More on the definition
More informationScience One Integral Calculus
Science One Integral Calculus January 018 Happy New Year! Differential Calculus central idea: The Derivative What is the derivative f (x) of a function f(x)? Differential Calculus central idea: The Derivative
More informationMath 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition
Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition 1 Appendix A : Numbers, Inequalities, and Absolute Values Sets A set is a collection of objects with an important
More informationconverges as well if x < 1. 1 x n x n 1 1 = 2 a nx n
Solve the following 6 problems. 1. Prove that if series n=1 a nx n converges for all x such that x < 1, then the series n=1 a n xn 1 x converges as well if x < 1. n For x < 1, x n 0 as n, so there exists
More informationdy = f( x) dx = F ( x)+c = f ( x) dy = f( x) dx
Antiderivatives and The Integral Antiderivatives Objective: Use indefinite integral notation for antiderivatives. Use basic integration rules to find antiderivatives. Another important question in calculus
More informationSample Final Questions: Solutions Math 21B, Winter y ( y 1)(1 + y)) = A y + B
Sample Final Questions: Solutions Math 2B, Winter 23. Evaluate the following integrals: tan a) y y dy; b) x dx; c) 3 x 2 + x dx. a) We use partial fractions: y y 3 = y y ) + y)) = A y + B y + C y +. Putting
More informationThe total differential
The total differential The total differential of the function of two variables The total differential gives the full information about rates of change of the function in the -direction and in the -direction.
More informationChapter 7: Applications of Integration
Chapter 7: Applications of Integration Fall 214 Department of Mathematics Hong Kong Baptist University 1 / 21 7.1 Volumes by Slicing Solids of Revolution In this section, we show how volumes of certain
More informationMATH1013 Calculus I. Limits (part I) 1
MATH1013 Calculus I Limits (part I) 1 Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology March 5, 2014 2013 1 Based on Briggs, Cochran and Gillett: Calculus for
More informationAP Calculus. Area Accumulation and Approximation
AP Calculus Area Accumulation and Approximation Student Handout 26 27 EDITION Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss
More informationDavid M. Bressoud Macalester College, St. Paul, Minnesota Given at Allegheny College, Oct. 23, 2003
David M. Bressoud Macalester College, St. Paul, Minnesota Given at Allegheny College, Oct. 23, 2003 The Fundamental Theorem of Calculus:. If F' ( x)= f ( x), then " f ( x) dx = F( b)! F( a). b a 2. d dx
More informationGiven a sequence a 1, a 2,...of numbers, the finite sum a 1 + a 2 + +a n,wheren is an nonnegative integer, can be written
A Summations When an algorithm contains an iterative control construct such as a while or for loop, its running time can be expressed as the sum of the times spent on each execution of the body of the
More informationQuestions 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 informationArc 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 informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) (a) (b) (c) (d) (e)...
Math 55, Exam III November 5, The Honor Code is in effect for this examination. All work is to be your own. No calculators. The exam lasts for hour and 5 min. Be sure that your name is on every page in
More informationChapter 4 Integration
Chapter 4 Integration SECTION 4.1 Antiderivatives and Indefinite Integration Calculus: Chapter 4 Section 4.1 Antiderivative A function F is an antiderivative of f on an interval I if F '( x) f ( x) for
More information16.4. Power Series. Introduction. Prerequisites. Learning Outcomes
Power Series 6.4 Introduction In this Section we consider power series. These are examples of infinite series where each term contains a variable, x, raised to a positive integer power. We use the ratio
More informationOBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph.
4.1 The Area under a Graph OBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph. 4.1 The Area Under a Graph Riemann Sums (continued): In the following
More informationArkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan. Solutions to Assignment 7.6. sin. sin
Arkansas Tech University MATH 94: Calculus II Dr. Marcel B. Finan Solutions to Assignment 7.6 Exercise We have [ 5x dx = 5 ] = 4.5 ft lb x Exercise We have ( π cos x dx = [ ( π ] sin π x = J. From x =
More informationMATH : 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 informationArea. A(2) = sin(0) π 2 + sin(π/2)π 2 = π For 3 subintervals we will find
Area In order to quantify the size of a -dimensional object, we use area. Since we measure area in square units, we can think of the area of an object as the number of such squares it fills up. Using this
More informationAbsolute 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 information1 5 π 2. 5 π 3. 5 π π x. 5 π 4. Figure 1: We need calculus to find the area of the shaded region.
. Area In order to quantify the size of a 2-dimensional object, we use area. Since we measure area in square units, we can think of the area of an object as the number of such squares it fills up. Using
More informationf (x) dx = F (b) F (a), where F is any function whose derivative is
Chapter 7 Riemann Integration 7.1 Introduction The notion of integral calculus is closely related to the notion of area. The earliest evidence of integral calculus can be found in the works of Greek geometers
More informationMath 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
More informationMath 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 informationSince the exponential function is a continuous function, it suffices to find the limit: lim sin x ln x. sin x ln x = ln x
Math 180 Written Homework Assignment #9 Due Tuesday, November 18th at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180 students,
More informationTHE UNIVERSITY OF BRITISH COLUMBIA Sample Questions for Midterm 1 - January 26, 2012 MATH 105 All Sections
THE UNIVERSITY OF BRITISH COLUMBIA Sample Questions for Midterm 1 - January 26, 2012 MATH 105 All Sections Closed book examination Time: 50 minutes Last Name First Signature Student Number Special Instructions:
More informationAbsolute 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 informationPRELIMINARIES FOR HYPERGEOMETRIC EQUATION. We will only consider differential equations with regular singularities in this lectures.
PRELIMINARIES FOR HYPERGEOMETRIC EQUATION EDMUND Y.-M. CHIANG Abstract. We give a brief introduction to some preliminaries for Gauss hypergeometric equations. We will only consider differential equations
More informationMath Calculus I
Math 165 - Calculus I Christian Roettger 382 Carver Hall Mathematics Department Iowa State University www.iastate.edu/~roettger November 13, 2011 4.1 Introduction to Area Sigma Notation 4.2 The Definite
More informationSolutions Final Exam May. 14, 2014
Solutions Final Exam May. 14, 2014 1. (a) (10 points) State the formal definition of a Cauchy sequence of real numbers. A sequence, {a n } n N, of real numbers, is Cauchy if and only if for every ɛ > 0,
More informationMATH5011 Real Analysis I. Exercise 1 Suggested Solution
MATH5011 Real Analysis I Exercise 1 Suggested Solution Notations in the notes are used. (1) Show that every open set in R can be written as a countable union of mutually disjoint open intervals. Hint:
More informationSolutions to Homework 11
Solutions to Homework 11 Read the statement of Proposition 5.4 of Chapter 3, Section 5. Write a summary of the proof. Comment on the following details: Does the proof work if g is piecewise C 1? Or did
More information7.1 Indefinite Integrals Calculus
7.1 Indefinite Integrals Calculus Learning Objectives A student will be able to: Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve differential
More informationThe Integral Test. P. Sam Johnson. September 29, P. Sam Johnson (NIT Karnataka) The Integral Test September 29, / 39
The Integral Test P. Sam Johnson September 29, 207 P. Sam Johnson (NIT Karnataka) The Integral Test September 29, 207 / 39 Overview Given a series a n, we have two questions:. Does the series converge?
More informationMath 229: Introduction to Analytic Number Theory The product formula for ξ(s) and ζ(s); vertical distribution of zeros
Math 9: Introduction to Analytic Number Theory The product formula for ξs) and ζs); vertical distribution of zeros Behavior on vertical lines. We next show that s s)ξs) is an entire function of order ;
More informationScience One Integral Calculus. January 9, 2019
Science One Integral Calculus January 9, 2019 Recap: What have we learned so far? The definite integral is defined as a limit of Riemann sums Riemann sums can be constructed using any point in a subinterval
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) (a) (b) (c) (d) (e)...
. Math 00, Exam November 0, 0. The Honor Code is in e ect 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 informationSteps for finding area using Summation
Steps for finding area using Summation 1) Identify a o and a 0 = starting point of the given interval [a, b] where n = # of rectangles 2) Find the c i 's Right: Left: 3) Plug each c i into given f(x) >
More informationChapter 8. Infinite Series
8.4 Series of Nonnegative Terms Chapter 8. Infinite Series 8.4 Series of Nonnegative Terms Note. Given a series we have two questions:. Does the series converge? 2. If it converges, what is its sum? Corollary
More informationJUST 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 informationFinal. due May 8, 2012
Final due May 8, 2012 Write your solutions clearly in complete sentences. All notation used must be properly introduced. Your arguments besides being correct should be also complete. Pay close attention
More informationDistance and Velocity
Distance and Velocity - Unit #8 : Goals: The Integral Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite integral and
More informationCalculus Honors Curriculum Guide Dunmore School District Dunmore, PA
Calculus Honors Dunmore School District Dunmore, PA Calculus Honors Prerequisite: Successful completion of Trigonometry/Pre-Calculus Honors Major topics include: limits, derivatives, integrals. Instruction
More informationc) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0
Some Math 35 review problems With answers 2/6/2005 The following problems are based heavily on problems written by Professor Stephen Greenfield for his Math 35 class in spring 2005. His willingness to
More informationGoal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) RECTANGULAR APPROXIMATION METHODS
AP Calculus 5. Areas and Distances Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) Exercise : Calculate the area between the x-axis and the graph of y = 3 2x.
More informationMath 180 Written Homework Assignment #10 Due Tuesday, December 2nd at the beginning of your discussion class.
Math 18 Written Homework Assignment #1 Due Tuesday, December 2nd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 18 students, but
More information18.01 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 informationMATH CALCULUS I 4.1: Area and Distance
MATH 12002 - CALCULUS I 4.1: Area and Distance Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 8 The Area and Distance Problems
More informationCALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) Fundamental Theorem of Calculus (Part I)
CALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) Fundamental Theorem of Calculus (Part I) CALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) APPLICATION (1, 4) 2
More informationWorksheet 7, Math 10560
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
More information5.3 Definite Integrals and Antiderivatives
5.3 Definite Integrals and Antiderivatives Objective SWBAT use properties of definite integrals, average value of a function, mean value theorem for definite integrals, and connect differential and integral
More informationSeparable Differential Equations
Separable Differential Equations MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Fall 207 Background We have previously solved differential equations of the forms: y (t) = k y(t) (exponential
More informationMAC 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 informationLectures. Section Theoretical (Definitions & Theorem) Examples Exercises HW
King Abdul-Aziz University Academic year 1437-1438 Department of Mathematics 2016-2017 Math 110 (S & E) Syllabus / Term (1) Book: Calculus Early Transcendentals by James Stewart 7 th edition Lectures Chapter
More informationRelationship Between Integration and Differentiation
Relationship Between Integration and Differentiation Fundamental Theorem of Calculus Philippe B. Laval KSU Today Philippe B. Laval (KSU) FTC Today 1 / 16 Introduction In the previous sections we defined
More informationDay 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value
AP Calculus Unit 6 Basic Integration & Applications Day 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value b (1) v( t) dt p( b) p( a), where v(t) represents the velocity and
More informationF (x) = P [X x[. DF1 F is nondecreasing. DF2 F is right-continuous
7: /4/ TOPIC Distribution functions their inverses This section develops properties of probability distribution functions their inverses Two main topics are the so-called probability integral transformation
More informationCalculus I Sample Exam #01
Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6
More informationINTEGRATION: AREAS AND RIEMANN SUMS MR. VELAZQUEZ AP CALCULUS
INTEGRATION: AREAS AND RIEMANN SUMS MR. VELAZQUEZ AP CALCULUS APPROXIMATING AREA For today s lesson, we will be using different approaches to the area problem. The area problem is to definite integrals
More informationJosh Engwer (TTU) Area Between Curves 22 January / 66
Area Between Curves Calculus II Josh Engwer TTU 22 January 2014 Josh Engwer (TTU) Area Between Curves 22 January 2014 1 / 66 Continuity & Differentiability of a Function (Notation) Definition Given function
More informationMULTIVARIABLE CALCULUS BRIGGS PDF
MULTIVARIABLE CALCULUS BRIGGS PDF ==> Download: MULTIVARIABLE CALCULUS BRIGGS PDF MULTIVARIABLE CALCULUS BRIGGS PDF - Are you searching for Multivariable Calculus Briggs Books? Now, you will be happy that
More informationJUST THE MATHS UNIT NUMBER NUMERICAL MATHEMATICS 6 (Numerical solution) of (ordinary differential equations (A)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 17.6 NUMERICAL MATHEMATICS 6 (Numerical solution) of (ordinary differential equations (A)) by A.J.Hobson 17.6.1 Euler s unmodified method 17.6.2 Euler s modified method 17.6.3
More informationMATH 140B - HW 5 SOLUTIONS
MATH 140B - HW 5 SOLUTIONS Problem 1 (WR Ch 7 #8). If I (x) = { 0 (x 0), 1 (x > 0), if {x n } is a sequence of distinct points of (a,b), and if c n converges, prove that the series f (x) = c n I (x x n
More informationCompletion Date: Monday February 11, 2008
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #4 Completion Date: Monday February, 8 Department of Mathematical and Statistical Sciences University of Alberta Question. [Sec..9,
More informationChapter 6: The Definite Integral
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 6: The Definite Integral v v Sections: v 6.1 Estimating with Finite Sums v 6.5 Trapezoidal Rule v 6.2 Definite Integrals 6.3 Definite Integrals and Antiderivatives
More informationElementary Analysis Math 140D Fall 2007
Elementary Analysis Math 140D Fall 2007 Bernard Russo Contents 1 Friday September 28, 2007 1 1.1 Course information............................ 1 1.2 Outline of the course........................... 1
More informationFUNCTIONS THAT PRESERVE THE UNIFORM DISTRIBUTION OF SEQUENCES
transactions of the american mathematical society Volume 307, Number 1, May 1988 FUNCTIONS THAT PRESERVE THE UNIFORM DISTRIBUTION OF SEQUENCES WILLIAM BOSCH ABSTRACT. In this paper, necessary and sufficient
More informationCalculus II - Fall 2013
Calculus II - Fall Midterm Exam II, November, In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit.. Find the area between
More informationIn Praise of y = x α sin 1x
In Praise of y = α sin H. Turgay Kaptanoğlu Countereamples have great educational value, because they serve to illustrate the limits of mathematical facts. Every mathematics course should include countereamples
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Background We have seen that some power series converge. When they do, we can think of them as
More information5.4 Continuity: Preliminary Notions
5.4. CONTINUITY: PRELIMINARY NOTIONS 181 5.4 Continuity: Preliminary Notions 5.4.1 Definitions The American Heritage Dictionary of the English Language defines continuity as an uninterrupted succession,
More informationCALCULUS II MATH Dr. Hyunju Ban
CALCULUS II MATH 2414 Dr. Hyunju Ban Introduction Syllabus Chapter 5.1 5.4 Chapters To Be Covered: Chap 5: Logarithmic, Exponential, and Other Transcendental Functions (2 week) Chap 7: Applications of
More informationHUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK
HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK COURSE / SUBJECT A P C a l c u l u s ( A B ) KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS Limits and Continuity Derivatives
More informationTAYLOR AND MACLAURIN SERIES
TAYLOR AND MACLAURIN SERIES. Introduction Last time, we were able to represent a certain restricted class of functions as power series. This leads us to the question: can we represent more general functions
More informationAbstract. 2. We construct several transcendental numbers.
Abstract. We prove Liouville s Theorem for the order of approximation by rationals of real algebraic numbers. 2. We construct several transcendental numbers. 3. We define Poissonian Behaviour, and study
More information