...MTH 1002: Calculus 2 Spring 2014 Instructor: Dr. C. Knoll Office: Bldg 406 (Academic Quad) Email: cknoll@fit.edu Dr. A. Gibbins Bldg 406 (Academic Quad) agibbins@fit.edu Dr. D. Zaffran Bldg 405 (Academic Quad) dzaffran@fit.edu Grading Policy: Online Homework ( 50 Practice Tests ( 50 Quizzes ( 200 Tests ( 300 Final Exam ( 200 TOTAL ( 800 Grading Scale: A: 90 – 100; B: 80 – 89; C: 70 – 79; D: 60 – 69; F: below 60 Late work will not be accepted without an excused absence. Only students with excused absences will be allowed to take make-up exams, quizzes, labs, etc. There will be absolutely no exceptions (consult your student handbook). An excused absence requires official documentation, e.g. a doctor’s note (in the case of illness). ATTENDANCE IS REQUIRED and will be taken at all lectures and labs. Required Text: Single Variable Calculus: Early Transcendentals, 7th ed., by Stewart. Online Homework URL: Available through the Angel link on the FloridaTech homepage or at www.webassign.net using the Course ID: fit 9672 0423 The LectureTopics will correspond to the following sections from the textbook: 5.1: Area Between Two Curves 5.2: Volumes by Slicing & Disks and Washers 5.3: Volumes by Cylindrical Shells 7.1: Integration by Parts 7.2: Trigonometric Integrals 7.3: Trigonometric Substitution 7.4: Partial Fraction Decomposition 7.5: Strategy for Integration 7.6: Integration...
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...TUTORIAL 3 SEQUENCES AND SERIES 3.1 Sequences and Series 1. Find the first four terms and 100th term of the sequence. (a) [pic] (b) [pic] (c) [pic] 2. Find the nth term of a sequence whose first several terms are given. (a) [pic]…… (b) 0, 2, 0, 2, 0, 2 …… 3. Find the sum. (a) [pic] (b) [pic] 4. Write the sum using sigma notation. (a) [pic] (b)[pic] 5. Find the nth term for each of the following sequences. Hence, determine whether the respective sequence is divergent or convergent. For a convergent sequence, state its limits. [pic] [pic] 6. For the sequence [pic]find the nth term and show that the above sequence is convergent and determine its limits. 7. The tenth term of an arithmetic sequence is [pic], and the second term is [pic]. Find the first term. 8. The first term of an arithmetic sequence is 1, and the common difference is 4. Is 11937 a term of this sequence? If so, which term is it? 9. The common ratio in a geometric sequence is [pic], and the fourth term is [pic]. Find the third term. 10. Which term of the geometric sequence 2, 6, 18, … is 118098? 11. Express the repeating decimal as fraction. (a) 0.777… (b) [pic] (c) [pic] 12. Find the sum of the first ten terms of the sequence. [pic] 13. The sum of the first three terms of a geometric series is 52, and the common ratio is r = 3. Find the first term. 14. A person has two parents, four grandparents, eight great-grandparents,...
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...Arithmetic and geometric progressions mcTY-apgp-2009-1 This unit introduces sequences and series, and gives some simple examples of each. It also explores particular types of sequence known as arithmetic progressions (APs) and geometric progressions (GPs), and the corresponding series. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • recognise the difference between a sequence and a series; • recognise an arithmetic progression; • find the n-th term of an arithmetic progression; • find the sum of an arithmetic series; • recognise a geometric progression; • find the n-th term of a geometric progression; • find the sum of a geometric series; • find the sum to infinity of a geometric series with common ratio |r| < 1. Contents 1. Sequences 2. Series 3. Arithmetic progressions 4. The sum of an arithmetic series 5. Geometric progressions 6. The sum of a geometric series 7. Convergence of geometric series www.mathcentre.ac.uk 1 c mathcentre 2009 2 3 4 5 8 9 12 1. Sequences What is a sequence? It is a set of numbers which are written in some particular order. For example, take the numbers 1, 3, 5, 7, 9, . . . . Here, we seem to have a rule. We have a sequence of odd numbers. To put this another way, we start with the number 1, which is an odd number, and then each successive number is obtained...
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...Related rates Rules and identities:Power rule, Product rule, Quotient rule, Chain rule | [show]Integral calculus | IntegralLists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order | [show]Vector calculus | Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem | [show]Multivariable calculus | Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian | | Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more...
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...April 8, 2016 1 Contents 1 Algebra 1.1 Rules of Basic Operations . . . . . . . . . . . . . . . . . . . . . 1.2 Rules of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Allowed and Disallowed Calculator Functions During the Exam 1.5 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Arithmetic Sequences and Series . . . . . . . . . . . . . . . . . 1.7 Sum of Finite Arithmetic Series (u1 + · · · + un ) . . . . . . . . . 1.8 Partial Sum of Finite Arithmetic Series (uj + · · · + un ) . . . . . 1.9 Geometric Sequences and Series . . . . . . . . . . . . . . . . . . 1.10 Sum of Finite Geometric Series . . . . . . . . . . . . . . . . . . 1.11 Sum of Infinite Geometric Series . . . . . . . . . . . . . . . . . 1.11.1 Example Involving Sum of Infinite Geometric Series . . 1.12 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.1 Sigma Notation for Arithmetic Series . . . . . . . . . . . 1.12.2 Sigma Notation for Geometric Series . . . . . . . . . . . 1.12.3 Sigma Notation for Infinite Geometric Series . . ....
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...MCR 3U Exam Review Unit 1 1. Evaluate each of the following. a) b) c) 2. Simplify. Express each answer with positive exponents. a) b) c) 3. Simplify and state restrictions a) b) c) d) 4. Is Justify your response. 5. Is Justify your response. Unit 2 1. Simplify each of the following. a) b) c) 2. Solve. a) b) c) 3. Solve. Express solutions in simplest radical form. a) b) 4. Find the maximum or minimum value of the function and the value of x when it occurs. a) b) 5. Write a quadratic equation, in standard form, with the roots a) and and that passes through the point (3, 1). b) and and that passes through the point (-1, 4). 6. The sum of two numbers is 20. What is the least possible sum of their squares? 7. Two numbers have a sum of 22 and their product is 103. What are the numbers ,in simplest radical form. Unit 3 1. Determine which of the following equations represent functions. Explain. Include a graph. a) b) c) d) 2. State the domain and range for each relation in question 1. 3. If and , determine the following: a) b) 4. Let . Determine the values of x for which a) b) Recall the base graphs. 5. Graph . State the domain and range. Describe...
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...CMSC 131 Summer 2005 Quiz 2 Worksheet The second Quiz of the course will be on Friday, Jun 17. The following list provides more information about the quiz: • You will have 25 minutes to complete the quiz. • It will be a written quiz (not using any computer). • It will be closed-book, closed-notes, and no calculator is allowed. • Answers must be neat and legible. We recommend that you use pencil and eraser. • The quiz will be based on the exercises you will find below. The quiz will ask you to write pseudocode for a particular problem. • We have provided previous semesters’ quizzes at the end. Take a look at them so you get an idea of the pseudocode we expect. The following exercises cover the material to be covered in Quiz #2. Solutions to these exercises will not be provided, but you are welcome to discuss your solutions with the TA and the instructor during office hours. Keep in mind that in the following exercises you are being asked to provide only pseudocode. 1. Write pseudocode for a program that computes the average of a set of values after the highest and lowest scores have been removed. 2. Write pseudocode for a program that reads a sequence of integer values and determines whether it is a decreasing sequence. A decreasing sequence is one where each value is greater than or equal to the next element in the sequence. The program will first read the number of values to process followed by the values...
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...Matlab Assignment 7 Make the Matalb assignment discussed in the last class (least square regression estimates). Make sure that you program a function with proper comments and at least one test for sound input. Test your function with some input vectors, for example: y=[1,2,4,23,4,6,3,2] and x=[5,4,3,2,6,5,4,3] You can take any other input vectors. m.file command: function [alpha_estimate, beta_estimate] = my_regression(y,x) n = length(x) a = sum(x); b = sum(y); c = sum(x)/n; d = sum(y)/n; e = sum(x.*y); f = sum(x.*x); alpha_estimate = d-(e-n*c*d)/(f-n*c^2)*c; beta_estimate = (e-n*c*d)/(f-n*c^2); disp('alpha =') disp(alpha_estimate) disp('beta =') disp(beta_estimate) % Purpose of the function: This function is used to calculate the % coefficients of the regression formula. % Input: value of y and x % Output: alpha_estimate and beta_estimate % How to run the function: % I use n to represent the length of vector x and y % a to represent the sum of vector x % b to represent the sum of vector y % c to represent the avergae of vector x % d to represent the average of vector y % e to represent the sum of vector x and y % f to represent the sum of square of vector x %then calculating: alpha_estimate = d-(e-n*c*d)/(f-n*c^2)*c; % beta_estimate = (e-n*c*d)/(f-n*c^2); % Author: Hengya Jin % Date of last change: 11/27/2013 end Check: >> y=[1,2,4,23,4...
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...A Taylor series for the function arctan The integral If we invert y = arctan(x) to obtain x = tan y, then, by differentiating with respect to y, we find dx/dy = sec2 y = 1 + tan2 y = 1 + x2 . Thus we have (ignoring the constant of integration) y = arctan(x) = dx . 1 + x2 (1) If we now differentiate y = arctan(x/a) with respect to x, where a is a constant, we have, by the chain rule, y = 1 a 1 a = 2 . 1 + (x/a)2 x + a2 (2) Thus we obtain the indefinite integral 1 dx = arctan(x/a). x 2 + a2 a (3) The Taylor Series By expanding the integrand in (3) as a geometric series 1/(1 − r) = 1 + r + r 2 + . . ., |r| < 1, and then integrating, we can obtain a series to represent the function arctan(x/a). We use the dummy variable t for the integration on [0, x] and we first write x x arctan(x/a) = a 0 dt 1 = t2 + a 2 a 0 dt 1 + (t/a)2 (4) Substituting the geometric series with r = −(t/a)2 , we find 1 arctan(x/a) = a x ∞ (−t2 /a2 )n dt = 0 n=0 (−1)n x 2n + 1 a n=0 ∞ 2n+1 . (5) The radius of convergence of this series is the same as that of the original geometric series, namely R = 1, or, in terms of x, |x/a| < 1. The series is a convergent alternating series at the right-hand end point x = a; and it can be shown that sum equals the value of arctan(1) = π/4 (as we might hope). Thus we have the nice (but slowly converging) series for π given by π 1 1 1 = 1 − + − + .... (6) 4 3 5 7 By choosing partial sums of (5) we obtain a sequence of Taylor polynomial...
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...replace=TRUE) > table(rolls2) rolls2 1 2 3 4 5 6 8166 8027 8068 7868 7912 7959 (b) Next we form this into a 2-column matrix (thus with 24,000 rows): > two.rolls=matrix(rolls2,nrow=24000,ncol=2) (c) Here we compute the sum of each (2-roll) row: > sum.rolls=apply(two.rolls,1,sum) > table(sum.rolls) sum.rolls 2 3 4 5 6 7 8 9 10 11 742 1339 2006 2570 3409 4013 3423 2651 1913 1291 1 12 643 Note table() gives us the frequency table for the 24,000 row sums. (d) Next we form the vector of sums into a 24-row matrix (thus with 1,000 columns): > twodozen=matrix(sum.rolls,nrow=24,ncol=1000,byrow=TRUE) (e) To find the 1,000 column minima use > min.pair=apply(twodozen,2,min) (f) Finally compute the number of columns whose minimum is 2, that is the number of series of 24 rolls of two dice with at least one sum of 2: > sum(min.pair==2) [1] 518 6. > p1.est=sum(min.roll==1)/4000 > p2.est=sum(min.pair==2)/48000 7. Here we repeat the above procedure 25 times, keeping track of the sums in both cases each time: > > > + + + + + + + + + + + + + + results1=0 results2=0 for (i in 25){...
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...#include <stdio.h> //Declare Prototype float getavg(int ageSum, int n); int main() { //Define variable family as number of loops and i as counter int family, i; printf("Enter the number of family members being submitted:"); scanf("%d", &family); //Declare rest of variables to hold the family names,states, ages, the sum of their ages, and the average of their ages char familynames[family][30], familystate[family][30]; int familymemberages[family], sum = 0; float averageAge; //Start loop and counter for (i=0; i < family; i++) { printf("\nPlease enter the following details for family member: \n" ); //Ask user to input first name printf("First name: "); scanf("%s", familynames[i]); //Store name in familynames //Ask user to input age printf("Age (in years): "); scanf("%d", &familymemberages[i]); //Store age in familymemberages //Ask user to input state they reside in printf("State of residence (by Abbreviation; Pa,Ny, Tx, etc...): "); scanf("%s", familystate[i]); //Store state in familystate //Sum of ages adding of all familymemberages sum += familymemberages[i]; } //Define the average age averageAge = getavg(sum, family); //Display the average age printf("\n\nThe average age among your family members is %.2f years\n", averageAge); //Display the list of members in Tx printf("List of members who live in Texas are: \n"); //Compare each charactors in string for Tx ...
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...1a) Assume that the total number of instructions = 100 Based on the information given in the question, there will be 3.6 instructions of store(1.0) and load(2.6) for every 100 instructions. As per the code sequence for calculating the effective address, there will be one ADD instruction with each and every LW instruction. In the new method of calculating effective address, we are modifying the given code sequence form in such a way that there will be only one instruction instead of two instructions(ADD and LW). This results in decrease in 3.6 in the instruction count. So, the instruction count new will be 100-3.6 = 96.4 So, the ratio of instruction count (enhanced MIPS) to the instruction count (original MIPS) is 96.4/100 = 0.964 1b) speedup = Told/Tnew Assume that the old clockcycle time is t According to the question the new clockcycle time is 1.05t Told = CPI x IC(original) x Clockcycle time Told = CPI x 100 x 1t Tnew = CPI x IC(enhanced) x Clockcycle time Tnew = CPI x 96.4 x 1.05t Speedup = Told/Tnew = 0.987 From the above speedup value we can say that the original system is 1.01 times faster than the enhanced system. 2a) The depth of the pipeline depends on the maximum CPI of executing a instruction. Here, the CPI of load/store is 6. So, the pipeline will have 6 stages. 2b) Average Instruction time = CPI x IC(original) x clockcycle time In the pipelined processor, the average CPI of a pipeline will be 1 Average execution time for 100 instructions = The first five cycles...
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...Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8] If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The number added to each term is constant (always the same). The fixed amount is called the common difference, d,referring to the fact that the difference between two successive terms yields the constant value that was added. To find the common difference, subtract the first term from the second term. 1. Find the common difference for this arithmetic sequence 5, 9, 13, 17 ... | 1. The common difference, d, can be found by subtracting the first term from the second term, which in this problem yields 4. Checking shows that 4 is the difference between all of the entries. | 2. Find the common difference for the arithmetic sequence whose formula is an = 6n + 3 | 2. The formula indicates that 6 is the value being added (with increasing multiples) as the terms increase. A listing of the terms will also show what is happening in the sequence (start with n = 1). 9, 15, 21, 27, 33, ... The list shows the common difference to be 6. | 3. Find the 10th term of the sequence ...
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...I would explain to someone that learning is much more unique and different than what we had expected it to be. I would remind them that learning is a way for us to have a better understanding of our society while it expands. It is the things that motivate us give, us purpose and meaning to move forward to our goals and achieve them. This gives a chance to push ourselves over limits and to thrive to reach goals. My views of learning have changed because of this class only because I had no idea there was such a thing as learning patterns that help us succeed in our abilities. This class has helped remind me that not only does my motivation and purpose help keep me thriving to reach my dreams, but also the discipline I must pursue in order to stay focused. Over this course I have learned I am a dynamic learner. My sequence and precision (28 and 25) are my highest learning pattern because I use them daily and helping me organize and keep order plus control and my daily activities. What technical reasoning (18) means I need a little work on it, but I do not avoid it. Don't get me wrong I do have a passion to figure out how things work when I need to fix them, but on the other hand I get to bust rated and have no patience for when things do not go my way. My confluence (22) is where I like to think out of the box and come up with any ideas that could be resourceful for future situations. Most people think that influence is a risk taker and which we are to a certain point but I do not...
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...Course 1 1. Series 1.1 Numerical series Definitions ➢ Let (an)n(1 a sequence of real numbers. The infinite sum (a1+a2+…+an+…) is called (numerical) series of general term an and is denoted by [pic] (or [pic]). ➢ The finite sum Sn= a1+a2+…+an=[pic], n(1 is named partial sum of order n, associated to the series[pic]. ➢ The series [pic] is convergent if and only if (iff) the sequence (Sn)n(1 is convergent (( S=[pic](R). In this case the limit S is named the sum of series[pic]. ➢ The series [pic] is divergent iff the sequence (Sn)n(1 is divergent (the limit [pic] is +(, -( or doesn’t exists). Exercises Study the nature (convergent or divergent) and if the case, compute the sum of the following series: a)[pic]; b) [pic](geometric series); c) [pic]. Properties 1) If in a series the order of a finite number of terms is changed then we get a series having the same nature as the first one. 2) If in a series a finite number of terms is added or subtracted the result is a series with the same nature as the first one. 3) The remainders of a convergent series, Rn= S-Sn form a sequence convergent to 0. 4) If the series [pic]is convergent then the sequence of partial sums (Sn)n(1 is bounded. 5) If the series [pic] and [pic]are convergent and have the sums A and, respectively B then the series [pic] is convergent and has the sum ((A+(B),( (, ((R. 6) If the series [pic]is convergent then[pic]=0. Remarks 1) (Divergence criterion) If [pic]is a series such that...
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