Math156 Review for Exam 4
|
|
|
- Donna Roberts
- 6 years ago
- Views:
Transcription
1 Math56 Review for Eam 4. What will be covered in this eam: Representing functions as power series, Taylor and Maclaurin series, calculus with parametric curves, calculus with polar coordinates.. Eam Rules: This is a closed note and close-tet eam. Formula sheet is not allowed. Calculators of any kind will not be allowed. Any electronic devices, including a cell phone, will not be allowed.. Epectations In this eam, students are epected to do each of the following: (A (representing functions as power series to know how to use both algebraic methods and calculus methods (term by term differentiation and integration within the interval of convergence to present a function as a power series. (B (Taylor and Maclaurin series know how to find Taylor and Maclaurin series of a function near a given point, and to use both algebraic methods and calculus methods in computing Taylor and Maclaurin series of a function near a given point. (C (calculus with parametric curves to know the skills of converting an (, y-coordinated function into parametric functions and vise versa; and to compute the differentiation and integration (including area and arc length computations of parametric functions. (D (calculus with polar coordinates to know the skills of converting an (, y-coordinated function into polar coordinated functions and vise versa; and to compute the differentiation and integration (including area and arc length computations of polar coordinated functions. 4. Summary Sheet (help yourself to make up your own summary materials Maclairin Series ( = n, r =. ( e = ( sin( = (4 cos( = n, r =. ( n n+ (n +!, r =. (5 tan ( = (6 ( + k = ( n n (n!, r =. ( n n+ ( k n (7 ln( = n= n +, r =. n, r =. n n, r =. Taylor Series The Taylor series of f( at a is Area in polar coordinates A = β Arc length in polar coordinates L = α r dθ. β α r + f (n (a ( a n. ( dr dθ. dθ
2 5. Warming Up Eercises By working on these and similar eercises, you will become more familiar with the related concepts and skills. It will help to warm you up and prepare well for the coming eam. Please note that there is no implication of any kind that any of these problem will be in the eam. Representing functions as power series: Find the power series representation for the functions and determine the interval of convergence. ( f( = + + = ( n. I = (, ( f( = 4 4 = 4n. I = (, ( f( = = n. I = (, (4 f( = = ( n. I = ( /, / (5 f( = 5 5 = 5 /5 = n. I = ( 5, 5 5n+ (6 f( = = ( 4 = ( n (4 n = ( /4, /4 (7 f( = = ( n n. I = (, 9n (8 f( = a a = n+. I = ( a, a an+ n= ( 4 n n+. I = Find the power series representation for the functions and determine the radius of convergence. d ( n ( f( = ln(5 ln 5 = 5 = n d = C 5 5 n5 n = n= n ln(5 n5 n. R = 5 ( f( = (. (Answer: ( = ( = n (n + n. R = / ( f( = n= n n+ n+, R = d = ( n d = (. (Answer: = n n+. f( = ( = d d (4 f( = tan (/. (Answer: f ( = (5 f( = 5 5 = 5 (6 f( = = ( /4, /4 n+ (n + n. f( = ( + (/ = ( n n+ n+ (n +. R = /5 = n. I = ( 5, 5 5n+ ( 4 = ( n (4 n = = n= n n n+ = ( 4 n n+. I =
3 (7 f( = = (8 f( = a a = ( n n ( n+. I = (, 9n. I = ( a, a an+ Find the Maclaurin series for each function f given below, and determine its radius of convergence. ( f( = tan (. (Ans: f( = tan = ( n 4n+ n +, r =. ( f( =. (Ans: f( = + + = ( n n+, r =. n+ ( f( = ln(. (Ans: f( = n +, r =. (4 f( = e n n+. (Ans: f( =, r =. (5 f( = sin( 4 ( n 8n+4. (Ans: f( =, r =. (n +! (6 f( = 4. (Ans: f( = (4n 6n+ n, r = 6. n= (7 f( = e / ( n n. (Ans: f( = n, r =. ( n π n n (8 f( = cos(π. (Ans: f( =, r =. (n! (9 f( = tan. (Ans: f( = ( n n+ n +, r =. ( f( = e ( n n+. (Ans: f( =, r =. ( n n n+ ( f( = cos(. (Ans: f( =, r =. (n! ( f( = +. (Ans: f( = + n 5 (n ( n, r =. n+/ n= Evaluate the integrals as an infinite series, and determine the radius of convergence. ( cos( d. (Ans: cos( d = ( n 6n+ (n! d = C + ( n 6n+ (6n + (n!, r =. sin( sin( ( d. (Ans: d = ( n n (n +! d = C + ( n n+ (n + (n +!, r =. n= e e n n ( d. (Ans: d = d = C + n, r =. Use Series to compute the limits. tan ( tan ( ( lim. (Ans: lim = lim cos( ( lim. (Ans: lim e cos( e = lim sin( 6 sin( 6 ( lim. (Ans: lim 5 tan( (4 lim 5 n= n= ( n n+ n + ( n+ n+ n+ n= n = lim n= =. =. ( n n+ 5 = (n +! 5! =. tan( +. (Ans: lim = lim =.
4 Find the Taylor series for each function f below centered at the given value a, and determine its radius of convergence. ( f( = + +, a =. (Ans: 4 + 5( + ( + (, r =. ( f( =, a =. (Ans: + ( + ( + + ( +, r =. ( f( = e, a =. (Ans: f( = e e e ( n =, r =. (4 f( = ln(, a =. (Ans: f( = ln(+( = ln +ln( = ln + Parametric Curves and Calculus with Parametric Curves n= ( n ( n n n, r =. Eliminate the parameter to find a Cartesian equation. ( = + t, y = t. (Ans. y = 9 ( +. ( = t, y = t. (Ans. = y /. Differentiation and Integration with Parametric Curves. y d ( Find d and d y d ( Find d and d if = t t, y = t. (Ans: if = t + ln(t, y = t ln(t. (Ans: d = t t, d y d = d(y d = (t + 4 9(t 4. d = t t +, d y d = d(y d = t (t +. ( Find an equation of the line tangent to the curve = t +, y = t / t at t =. (Ans: y 6 = ( 9. (4 Find an equation of the line tangent to the curve = cos(θ + sin(θ, y = sin(θ + cos(θ at θ =. (Ans: y = (. (5 Find the points on the curve = t + t t, y = t + t + where the tangent line is horizontal or vertical. (Ans: Horizontal at (, y = (,, vertical at ( 7, 6. (6 Find the points on the curve = cos(θ, y = sin(θ where the tangent line is horizontal or vertical. (Ans: Horizontal at (, y = (, ±, vertical at (±, and (±, ±. (7 Find the points on the curve = t, y = t t where the tangent line is horizontal or vertical. (Ans: Horizontal at (, y = (,, vertical at ( 7, 6. (8 Find the points on the curve = cos(θ, y = sin(θ where the tangent line is horizontal or vertical. (Ans: Horizontal at (, y = (6, ±6, vertical at (,. (9 Find equations of the tangents to the curve = t + and y = t + that pass through the point (4,. (Ans: y = 4, and y ( 5 = (. ( At what points on the curve = t + 4t and y = 6t is the tangent line parallel to the line = 7t and y = t 5? (Ans: (, y = ( 5, 6 or ( 8 7, ( Find the area bounded by the curve = t /t and y = t + /t and the line y =.5. (Ans: Set ( 5 t + /t = y =.5 to get t =.5 and t =. A = (.5 yd(t = t ( + t t dt = 5 4 ln. 4.5 ( Find the area bounded by the curve = cos(t and y = e t, t π and the lines y = and =. (Ans: A = (y d = (e t ( sin(tdt = eπ/. π/ ( Find the length of the curve = t +t and y = ln( + t, t. (Ans: L = t + t + ( + t dt = 4
5 u + u du with u = t +, Thus L = + ln( + + ln( +. (4 Find the length of the curve = e t + e t and y = 5 t, t. (Ans: L = (e t + e t dt = e e. Polar Coordinates and Calculus with Polar Coordinates Switching between polar coordinates and Cartesian coordinates. ( Find Cartesian equation for r = sin(θ. (Ans: + y =. ( Find Cartesian equation for r = sin(θ + cos(θ. (Ans: + y = + y. ( Find Cartesian equation for r = csc(θ. (Ans: y =. (4 Find Cartesian equation for r = tan(θ sec(θ. (Ans: = y. (5 Find polar equation for = y. (Ans: r = cot(θ csc(θ. (6 Find Cartesian equation for + y = 9. (Ans: r = 9 cos(θ+sin(θ. (7 Find polar equation for + y = c. (Ans: r = c cos(θ. (8 Find Cartesian equation for y =. (Ans: r = sec(θ. Note that cos(θ = cos (θ sin (θ. Differentiation and Integration with Polar Coordinates. ( Find the slope of tangent line to r = sin(θ at θ = π 6. (Ans: d = tan(θ, slope = tan( π =. ( Find the slope of tangent line to r = sin(θ at θ = π. (Ans: d = =. ( Find the slope of tangent line to r = /θ at θ = π. (Ans: (4 Find the slope of tangent line to r = sin(θ at θ = π/6. (Ans: slope =. cos(θ sin(θ sin(theta cos(θ, slope d = sin(θ + θ cos(θ, slope = π. cos(theta θ sin(θ d = cos(θ sin(θ + sin(θ cos(θ cos(θ cos(θ sin(θ sin(θ, (5 Find the points on r = cos(θ where the tangent line is horizontal or vertical. (Ans: Horizontal at (, π, (, π, vertical at (,, (, π/. 4 4 (6 Find the points on r = e θ where the tangent line is horizontal or vertical. (Ans: Horizontal at (e π(n /4, π(n 4, vertical at (eπ(n+/4, π(n + 4. (7 Find the points on r = + cos(θ where the tangent line is horizontal or vertical. (Ans: Horizontal at (, π, (, π, (, 5π, vertical at (,, (, π, (, 4π. (8 Find the points on r = e sin(θ where the tangent line is horizontal or vertical. (Ans: Horizontal at ( 4 4, π, ( 4 4, 4π (,, vertical at ( 4 4, π 6, ( 4 4, 7π 6 (,. (9 Find the area of the region bounded by r = π/4 θ, θ π 4. (Ans: A = θ ( Find the area of the region bounded by r = e θ/, π θ π. (Ans: A = π e θ π dθ = π 64. ( Find the area of the region bounded by r = sin(θ, π/ θ π/. (Ans: A = dθ = eπ e π. π/ π/ sin (θ dθ = 5
6 π + 8. ( Find the area of the region bounded by r = sin(θ, θ π. (Ans: A = π sin(θ dθ =. ( Find the area of the region inside r = 4 sin(θ and outside r =. (Ans: 4 sin(θ = to get bounds. π/ (4 sin(θ A = dθ = 4 π +. π/6 (4 Find the area of the region inside r = sin(θ and outside r =. (Ans: sin(θ = to get bounds. π ( sin(θ A = dθ = π 4 +. π (5 Find the area of the region inside r = cos(θ and outside r = + cos(θ. (Ans: cos(θ = + cos(θ to π/ ( cos(θ ( cos(θ get bounds. A = dθ = 9π. (4 Find the area of the region inside r = + sin(θ and outside r = sin(θ. (Ans: + sin(θ = sin(θ to π ( + sin(θ ( sin(θ get bounds. A = dθ = π. 8 (5 Find the area of the region enclosed by one loop of r = sin(θ. (Ans: A = (6 Find the area of the region enclosed by one loop of r = 4 sin(θ. (Ans: A = π/ π/ (7 Find the area of the region enclosed by one loop of r = + sin(θ (inner loop. (Ans: A = π. sin (θ dθ = π 8. 4 sin (θ dθ = 4π. π/ 7π/6 ( + sin(θ dθ = 6
Math 113 Final Exam Practice
Math Final Exam Practice The Final Exam is comprehensive. You should refer to prior reviews when studying material in chapters 6, 7, 8, and.-9. This review will cover.0- and chapter 0. This sheet has three
Power Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.
.8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x
MATH 1080 Test 2 -Version A-SOLUTIONS Fall a. (8 pts) Find the exact length of the curve on the given interval.
MATH 8 Test -Version A-SOLUTIONS Fall 4. Consider the curve defined by y = ln( sec x), x. a. (8 pts) Find the exact length of the curve on the given interval. sec x tan x = = tan x sec x L = + tan x =
APPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
There are some trigonometric identities given on the last page.
MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during
MATH QUIZ 3 1/2. sin 1 xdx. π/2. cos 2 (x)dx. x 3 4x 10 x 2 x 6 dx.
NAME: I.D.: MATH 56 - QUIZ 3 Instruction: Each problem is worth of point in this take home project. Circle your answers and show all your work CLEARLY. Use additional paper if needed. Solutions with answer
Math Calculus II Homework # Due Date Solutions
Math 35 - Calculus II Homework # - 007.08.3 Due Date - 007.09.07 Solutions Part : Problems from sections 7.3 and 7.4. Section 7.3: 9. + d We will use the substitution cot(θ, d csc (θ. This gives + + cot
You can learn more about the services offered by the teaching center by visiting
MAC 232 Exam 3 Review Spring 209 This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources
Part I: Multiple Choice Mark the correct answer on the bubble sheet provided. n=1. a) None b) 1 c) 2 d) 3 e) 1, 2 f) 1, 3 g) 2, 3 h) 1, 2, 3
Math (Calculus II) Final Eam Form A Fall 22 RED KEY Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.. Which of the following series converge absolutely? ) ( ) n 2) n 2 n (
Odd Answers: Chapter Eight Contemporary Calculus 1 { ( 3+2 } = lim { 1. { 2. arctan(a) 2. arctan(3) } = 2( π 2 ) 2. arctan(3)
Odd Answers: Chapter Eight Contemporary Calculus PROBLEM ANSWERS Chapter Eight Section 8.. lim { A 0 } lim { ( A ) ( 00 ) } lim { 00 A } 00.. lim {. arctan() A } lim {. arctan(a). arctan() } ( π ). arctan()
1. (4 % each, total 20 %) Answer each of the following. (No need to show your work for this problem). 3 n. n!? n=1
NAME: EXAM 4 - Math 56 SOlutions Instruction: Circle your answers and show all your work CLEARLY Partial credit will be given only when you present what belongs to part of a correct solution (4 % each,
a k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
Monroe Township High School Mathematics Department
To: AP Calculus AB Re: Summer Project 017 Date: June 017 Monroe Township High School Mathematics Department To help begin your study of Calculus, you will be required to complete a review project this
Chapter 9 Overview: Parametric and Polar Coordinates
Chapter 9 Overview: Parametric and Polar Coordinates As we saw briefly last year, there are axis systems other than the Cartesian System for graphing (vector coordinates, polar coordinates, rectangular
MATH 100 REVIEW PACKAGE
SCHOOL OF UNIVERSITY ARTS AND SCIENCES MATH 00 REVIEW PACKAGE Gearing up for calculus and preparing for the Assessment Test that everybody writes on at. You are strongly encouraged not to use a calculator
EXAM. Practice for Second Exam. Math , Fall Nov 4, 2003 ANSWERS
EXAM Practice for Second Eam Math 135-006, Fall 003 Nov 4, 003 ANSWERS i Problem 1. In each part, find the integral. A. d (4 ) 3/ Make the substitution sin(θ). d cos(θ) dθ. We also have Then, we have d/dθ
y = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx
Millersville University Name Answer Key Mathematics Department MATH 2, Calculus II, Final Examination May 4, 2, 8:AM-:AM Please answer the following questions. Your answers will be evaluated on their correctness,
Math Worksheet 1 SHOW ALL OF YOUR WORK! f(x) = (x a) 2 + b. = x 2 + 6x + ( 6 2 )2 ( 6 2 )2 + 7 = (x 2 + 6x + 9) = (x + 3) 2 2
Names Date. Consider the function Math 0550 Worksheet SHOW ALL OF YOUR WORK! f() = + 6 + 7 (a) Complete the square and write the function in the form f() = ( a) + b. f() = + 6 + 7 = + 6 + ( 6 ) ( 6 ) +
Trigonometric Ratios. θ + k 360
Trigonometric Ratios These notes are intended as a summary of section 6.1 (p. 466 474) in your workbook. You should also read the section for more complete explanations and additional examples. Coterminal
AP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:
AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period: Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented
Differential Equaitons Equations
Welcome to Multivariable Calculus / Dierential Equaitons Equations The Attached Packet is or all students who are planning to take Multibariable Multivariable Calculus/ Dierential Equations in the all.
1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
Practice Problems: Exam 2 MATH 230, Spring 2011 Instructor: Dr. Zachary Kilpatrick Show all your work. Simplify as much as possible.
Practice Problems: Exam MATH, Spring Instructor: Dr. Zachary Kilpatrick Show all your work. Simplify as much as possible.. Write down a table of x and y values associated with a few t values. Then, graph
Fall 2016, MA 252, Calculus II, Final Exam Preview Solutions
Fall 6, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card, and
Parametric Equations and Polar Coordinates
Parametric Equations and Polar Coordinates Parametrizations of Plane Curves In previous chapters, we have studied curves as the graphs of functions or equations involving the two variables x and y. Another
Fall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes
Math 8 Fall Hour Exam /8/ Time Limit: 5 Minutes Name (Print): This exam contains 9 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information
9.1. Click here for answers. Click here for solutions. PARAMETRIC CURVES
SECTION 9. PARAMETRIC CURVES 9. PARAMETRIC CURVES A Click here for answers. S Click here for solutions. 5 (a) Sketch the curve b using the parametric equations to plot points. Indicate with an arrow the
ACS MATHEMATICS GRADE 10 WARM UP EXERCISES FOR IB HIGHER LEVEL MATHEMATICS
ACS MATHEMATICS GRADE 0 WARM UP EXERCISES FOR IB HIGHER LEVEL MATHEMATICS DO AS MANY OF THESE AS POSSIBLE BEFORE THE START OF YOUR FIRST YEAR IB HIGHER LEVEL MATH CLASS NEXT SEPTEMBER Write as a single
AP Calculus Summer Packet
AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept
MATH 162. FINAL EXAM ANSWERS December 17, 2006
MATH 6 FINAL EXAM ANSWERS December 7, 6 Part A. ( points) Find the volume of the solid obtained by rotating about the y-axis the region under the curve y x, for / x. Using the shell method, the radius
BC Exam 1 - Part I 28 questions No Calculator Allowed - Solutions C = 2. Which of the following must be true?
BC Exam 1 - Part I 8 questions No Calculator Allowed - Solutions 6x 5 8x 3 1. Find lim x 0 9x 3 6x 5 A. 3 B. 8 9 C. 4 3 D. 8 3 E. nonexistent ( ) f ( 4) f x. Let f be a function such that lim x 4 x 4 I.
Curriculum Map for Mathematics HL (DP1)
Curriculum Map for Mathematics HL (DP1) Unit Title (Time frame) Sequences and Series (8 teaching hours or 2 weeks) Permutations & Combinations (4 teaching hours or 1 week) Standards IB Objectives Knowledge/Content
18.01 Final Answers. 1. (1a) By the product rule, (x 3 e x ) = 3x 2 e x + x 3 e x = e x (3x 2 + x 3 ). (1b) If f(x) = sin(2x), then
8. Final Answers. (a) By the product rule, ( e ) = e + e = e ( + ). (b) If f() = sin(), then f (7) () = 8 cos() since: f () () = cos() f () () = 4 sin() f () () = 8 cos() f (4) () = 6 sin() f (5) () =
MATH 152, Fall 2017 COMMON EXAM II - VERSION A
MATH 15, Fall 17 COMMON EXAM II - VERSION A LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited.. TURN OFF cell phones
Example 2.1. Draw the points with polar coordinates: (i) (3, π) (ii) (2, π/4) (iii) (6, 2π/4) We illustrate all on the following graph:
Section 10.3: Polar Coordinates The polar coordinate system is another way to coordinatize the Cartesian plane. It is particularly useful when examining regions which are circular. 1. Cartesian Coordinates
Math Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
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
Math 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2
Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
Exam 4 SCORE. MA 114 Exam 4 Spring Section and/or TA:
Exam 4 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may
Friday 09/15/2017 Midterm I 50 minutes
Fa 17: MATH 2924 040 Differential and Integral Calculus II Noel Brady Friday 09/15/2017 Midterm I 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets.
Examiner: D. Burbulla. Aids permitted: Formula Sheet, and Casio FX-991 or Sharp EL-520 calculator.
University of Toronto Faculty of Applied Science and Engineering Solutions to Final Examination, June 216 Duration: 2 and 1/2 hrs First Year - CHE, CIV, CPE, ELE, ENG, IND, LME, MEC, MMS MAT187H1F - Calculus
Sect 7.4 Trigonometric Functions of Any Angles
Sect 7.4 Trigonometric Functions of Any Angles Objective #: Extending the definition to find the trigonometric function of any angle. Before we can extend the definition our trigonometric functions, we
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3 2, 5 2 C) - 5 2
Test Review (chap 0) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) Find the point on the curve x = sin t, y = cos t, -
Pre- Calculus Mathematics Trigonometric Identities and Equations
Pre- Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
Limits. Final Exam Study Guide. Calculus I. 1. Basic Limits I: Evaluate each limit exactly. (a) lim. (c) lim. 2t 15 3 (g) lim. (e) lim. (f) lim.
Limits 1. Basic Limits I: Evaluate each limit eactly. 3 ( +5 8) (c) lim(sin(α) 5cos(α)) α π 6 (e) lim t t 15 3 (g) lim t 0 t (4t 3 8t +1) t 1 (tan(θ) cot(θ)+1) θ π 4 (f) lim 16 ( 5 (h) lim t 0 3 t ). Basic
SECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Summer 208 Remarks The Fundamental Theorem of Calculus (FTC) will make the evaluation of definite integrals
Mathematics 132 Calculus for Physical and Life Sciences 2 Exam 3 Review Sheet April 15, 2008
Mathematics 32 Calculus for Physical and Life Sciences 2 Eam 3 Review Sheet April 5, 2008 Sample Eam Questions - Solutions This list is much longer than the actual eam will be (to give you some idea of
Go over the illustrated examples in each section.
Math 242 Fall 2009 Please before you start the practice problems, go over the illustrated examples in each section first. Cover up the solutions and try to work out the answers on your own. Practice Problems
Math 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
Math Worksheet 1. f(x) = (x a) 2 + b. = x 2 6x = (x 2 6x + 9) = (x 3) 2 1
Names Date Math 00 Worksheet. Consider the function f(x) = x 6x + 8 (a) Complete the square and write the function in the form f(x) = (x a) + b. f(x) = x 6x + 8 ( ) ( ) 6 6 = x 6x + + 8 = (x 6x + 9) 9
Math 1310 Final Exam
Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space
Math 2300 Calculus II University of Colorado Final exam review problems
Math 300 Calculus II University of Colorado Final exam review problems. A slope field for the differential equation y = y e x is shown. Sketch the graphs of the solutions that satisfy the given initial
10.1 Curves Defined by Parametric Equation
10.1 Curves Defined by Parametric Equation 1. Imagine that a particle moves along the curve C shown below. It is impossible to describe C by an equation of the form y = f (x) because C fails the Vertical
Midterm Exam #1. (y 2, y) (y + 2, y) (1, 1)
Math 5B Integral Calculus March 7, 7 Midterm Eam # Name: Answer Key David Arnold Instructions. points) This eam is open notes, open book. This includes any supplementary tets or online documents. You are
Math Test #3 Info and Review Exercises
Math 181 - Test #3 Info and Review Exercises Fall 2018, Prof. Beydler Test Info Date: Wednesday, November 28, 2018 Will cover sections 10.1-10.4, 11.1-11.7. You ll have the entire class to finish the test.
Math 113/113H Winter 2006 Departmental Final Exam
Name KEY Instructor Section No. Student Number Math 3/3H Winter 26 Departmental Final Exam Instructions: The time limit is 3 hours. Problems -6 short-answer questions, each worth 2 points. Problems 7 through
Math 121: Exam 2 Review Sheet
Eam Information Math 11: Eam Review Sheet Eam will be given on Friday, March 1 from 8-8:50 am. The eam will cover chapters.3-.6 and 3.1-3.5. The eam is closed book, closed notes, and without calculator.
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
Mathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx.
Mathematics 14 Fall Term 26 Solutions to Final Exam 1. Evaluate sin(ln t) dt. Solution. We first make the substitution t = e x, for which dt = e x. This gives sin(ln t) dt = e x sin(x). To evaluate the
Math 2153, Exam III, Apr. 17, 2008
Math 53, Exam III, Apr. 7, 8 Name: Score: Each problem is worth 5 points. The total is 5 points. For series convergence or ivergence, please write own the name of the test you are using an etails of using
University of Toronto FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, JUNE, 2012 First Year - CHE, CIV, IND, LME, MEC, MSE
University of Toronto FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, JUNE, 212 First Year - CHE, CIV, IND, LME, MEC, MSE MAT187H1F - CALCULUS II Exam Type: A Examiner: D. Burbulla INSTRUCTIONS:
RED. Math 113 (Calculus II) Final Exam Form A Fall Name: Student ID: Section: Instructor: Instructions:
Name: Student ID: Section: Instructor: Math 3 (Calculus II) Final Exam Form A Fall 22 RED Instructions: For questions which require a written answer, show all your work. Full credit will be given only
Review (2) Calculus II (201-nyb-05/05,06) Winter 2019
Review () Calculus II (-nyb-5/5,6) Winter 9 Note. You should also review the integrals eercises on the first review sheet. Eercise. Evaluate each of the following integrals. a. sin 3 ( )cos ( csc 3 (log)cot
Math 1501 Calc I Summer 2015 QUP SOUP w/ GTcourses
Math 1501 Calc I Summer 2015 QUP SOUP w/ GTcourses Instructor: Sal Barone School of Mathematics Georgia Tech May 22, 2015 (updated May 22, 2015) Covered sections: 3.3 & 3.5 Exam 1 (Ch.1 - Ch.3) Thursday,
July 21 Math 2254 sec 001 Summer 2015
July 21 Math 2254 sec 001 Summer 2015 Section 8.8: Power Series Theorem: Let a n (x c) n have positive radius of convergence R, and let the function f be defined by this power series f (x) = a n (x c)
PreCalculus First Semester Exam Review
PreCalculus First Semester Eam Review Name You may turn in this eam review for % bonus on your eam if all work is shown (correctly) and answers are correct. Please show work NEATLY and bo in or circle
Using the Definitions of the Trigonometric Functions
1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities Signs and Ranges of Function Values Pythagorean Identities Quotient Identities February 1, 2013 Mrs. Poland Objectives Objective
Amherst College, DEPARTMENT OF MATHEMATICS Math 11, Final Examination, May 14, Answer Key. x 1 x 1 = 8. x 7 = lim. 5(x + 4) x x(x + 4) = lim
Amherst College, DEPARTMENT OF MATHEMATICS Math, Final Eamination, May 4, Answer Key. [ Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value,
Math 113 Final Exam Practice Problem Solutions. f(x) = ln x x. lim. lim. x x = lim. = lim 2
Math 3 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
Spring 2015, MA 252, Calculus II, Final Exam Preview Solutions
Spring 5, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card,
Practice problems from old exams for math 132 William H. Meeks III
Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are
Section 6.2 Trigonometric Functions: Unit Circle Approach
Section. Trigonometric Functions: Unit Circle Approach The unit circle is a circle of radius centered at the origin. If we have an angle in standard position superimposed on the unit circle, the terminal
Math 1272 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Exam ) This fraction appears in Problem 5 of the undated-? exam; a solution can be found in that solution set. (E) ) This integral appears in Problem 6 of the Fall 4 exam;
HKUST. MATH1014 Calculus II. Directions:
HKUST MATH114 Calculus II Midterm Eamination (Sample Version) Name: Student ID: Lecture Section: Directions: This is a closed book eamination. No Calculator is allowed in this eamination. DO NOT open the
Mathematics 111 (Calculus II) Laboratory Manual
Mathematics (Calculus II) Laboratory Manual Department of Mathematics & Statistics University of Regina nd edition prepared by Patrick Maidorn, Fotini Labropulu, and Robert Petry University of Regina Department
Lesson 33 - Trigonometric Identities. Pre-Calculus
Lesson 33 - Trigonometric Identities Pre-Calculus 1 (A) Review of Equations An equation is an algebraic statement that is true for only several values of the variable The linear equation 5 = 2x 3 is only
Math 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
Review Problems for the Final
Review Problems for the Final Math -3 5 7 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the
Math 120: Precalculus Autumn 2017 A List of Topics for the Final
Math 120: Precalculus Autumn 2017 A List of Topics for the Final Here s a fairly comprehensive list of things you should be comfortable doing for the final. Really Old Stuff 1. Unit conversion and rates
Calculus 152 Take Home Test 2 (50 points)
Calculus 5 Take Home Test (5 points) Due Tuesday th November. The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics
Parametric Curves. Calculus 2 Lia Vas
Calculus Lia Vas Parametric Curves In the past, we mostly worked with curves in the form y = f(x). However, this format does not encompass all the curves one encounters in applications. For example, consider
IYGB Mathematical Methods 1
IYGB Mathematical Methods Practice Paper B Time: 3 hours Candidates may use any non programmable, non graphical calculator which does not have the capability of storing data or manipulating algebraic expressions
Examiner: D. Burbulla. Aids permitted: Formula Sheet, and Casio FX-991 or Sharp EL-520 calculator.
University of Toronto Faculty of Applied Science and Engineering Solutions to Final Examination, June 017 Duration: and 1/ hrs First Year - CHE, CIV, CPE, ELE, ENG, IND, LME, MEC, MMS MAT187H1F - Calculus
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.
Math 323 Exam 1 Practice Problem Solutions
Math Exam Practice Problem Solutions. For each of the following curves, first find an equation in x and y whose graph contains the points on the curve. Then sketch the graph of C, indicating its orientation.
(x 3)(x + 5) = (x 3)(x 1) = x + 5
RMT 3 Calculus Test olutions February, 3. Answer: olution: Note that + 5 + 3. Answer: 3 3) + 5) = 3) ) = + 5. + 5 3 = 3 + 5 3 =. olution: We have that f) = b and f ) = ) + b = b + 8. etting these equal
MAT 500: AP Calculus AB
017 Summer Assignment MAT 500: Summer Assignment MAT 500: Dear students, Welcome to! I can t wait to start the year with all of you, and have my eye on the prize; Tuesday May 15, 018 at 8 a.m.! It seems
1. Evaluate the integrals. a. (9 pts) x e x/2 dx. Solution: Using integration by parts, let u = x du = dx and dv = e x/2 dx v = 2e x/2.
MATH 8 Test -SOLUTIONS Spring 4. Evaluate the integrals. a. (9 pts) e / Solution: Using integration y parts, let u = du = and dv = e / v = e /. Then e / = e / e / e / = e / + e / = e / 4e / + c MATH 8
Differential Equations: Homework 8
Differential Equations: Homework 8 Alvin Lin January 08 - May 08 Section.6 Exercise Find a general solution to the differential equation using the method of variation of parameters. y + y = tan(t) r +
Without fully opening the exam, check that you have pages 1 through 11.
MTH 33 Solutions to Final Exam May, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show
Math 226 Calculus Spring 2016 Exam 2V1
Math 6 Calculus Spring 6 Exam V () (35 Points) Evaluate the following integrals. (a) (7 Points) tan 5 (x) sec 3 (x) dx (b) (8 Points) cos 4 (x) dx Math 6 Calculus Spring 6 Exam V () (Continued) Evaluate
These items need to be included in the notebook. Follow the order listed.
* Use the provided sheets. * This notebook should be your best written work. Quality counts in this project. Proper notation and terminology is important. We will follow the order used in class. Anyone
Multiple Choice Answers. MA 114 Calculus II Spring 2013 Final Exam 1 May Question
MA 114 Calculus II Spring 2013 Final Exam 1 May 2013 Name: Section: Last 4 digits of student ID #: This exam has six multiple choice questions (six points each) and five free response questions with points
Advanced Higher Grade
Prelim Eamination / 5 (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours Read Carefully. Full credit will be given only where the solution contains appropriate woring.. Calculators
Chapter 10 Conics, Parametric Equations, and Polar Coordinates Conics and Calculus
Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 1. Parabola A parabola is the set of all points x, y ( ) that are equidistant from a fixed line and a fixed point
Math 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find:
Math B Final Eam, Solution Prof. Mina Aganagic Lecture 2, Spring 20 The eam is closed book, apart from a sheet of notes 8. Calculators are not allowed. It is your responsibility to write your answers clearly..
Find the rectangular coordinates for each of the following polar coordinates:
WORKSHEET 13.1 1. Plot the following: 7 3 A. 6, B. 3, 6 4 5 8 D. 6, 3 C., 11 2 E. 5, F. 4, 6 3 Find the rectangular coordinates for each of the following polar coordinates: 5 2 2. 4, 3. 8, 6 3 Given the
Warmup for AP Calculus BC
Nichols School Mathematics Department Summer Work Packet Warmup for AP Calculus BC Who should complete this packet? Students who have completed Advanced Functions or and will be taking AP Calculus BC in
Chapter 29 BC Calculus Practice Test
Chapter 9 BC Calculus Practice Test The Eam AP Calculus BC Eam SECTION I: Multiple-Choice Questions DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time hour and 5 minutes Number