PASS MOCK EXAM FOR PRACTICE ONLY
|
|
- Erik French
- 7 years ago
- Views:
Transcription
1 PASS MOCK EXAM FOR PRACTICE ONLY Course: MATH 4 ABCDEF Facilitator: Stephen Kimbell December 6 th 25 - (3:-6:) and (8:-2:) in ME 338 (Mackenzie Building 3 rd floor 3 rd block) It is most beneficial to you to write this mock final UNDER EXAM CONDITIONS. This means: Complete the mock final in 3 hour(s). Work on your own. Keep your notes and textbook closed. Attempt every question. After the time limit, go back over your work with a different colour or on a separate piece of paper and try to do the questions you are unsure of. Record your ideas in the margins to remind yourself of what you were thinking when you take it up at PASS. The purpose of this mock exam is to give you practice answering questions in a timed setting and to help you to gauge which aspects of the course content you know well and which are in need of further development and review. Use this mock exam as a learning tool in preparing for the actual exam. Please note: Come to the PASS session with your mock exam complete. There, you can work with other students to review your work. Often, there is not enough time to review the entire exam in the PASS session. Decide which questions you most want to review the Facilitator may ask students to vote on which questions they want to discuss. Facilitators do not bring copies of the mock exam to the session. Please print out and complete the exam before you attend. Facilitators do not produce or distribute an answer key for mock exams. Facilitators help students to work together to compare and assess the answers they have. If you are not able to attend the PASS session, you can work alone or with others in the class. *** I based the mock exam on my exam in 23 with a few minor changes. I think it will be a great representation for what you will see in your exam. I included a few extra longer answer problems if you are curious or interested in how calculus can be applied in engineering! Good Luck writing the Mock Exam!! DISCLAIMER: PASS handouts are designed as a study aid only for use in PASS workshops. Handouts may contain errors, intentional or otherwise. It is up to the student to verify the information contained within.
2 Multiple Choice Calculus Questions f(+h) f(3). Let f(x) = x + x 3. Calculate L = lim h + h a) b) c) - d) DNE 2. Let f(x) = cos (x) x 2 for x and f(x) = A, for x =. What value of A will make f continuous at a) b) /2 c) - d) 3. Evaluate L = lim x2 2x 3 x 3 x 2 9 a) b) 3/2 c) 2/3 d) DNE 4. Evaluate L = lim x sin (tx) sin (5x) a) /5 b) 5 c) t/5 d) DNE 5. Two functions f and g are defined by f(x) = 3x 2 and g(x) = cos(x). What is the value of f(g()) a) -3 b) 3 c) -3.2 d) 6. Find the derivative of the function y = 8x ln (5x+) a) 8(5x+) ln(5x+) 4x (5x+)(ln(5x+)) 2 b) ln (4x) c) 8 ln(5x+) 4x ln(5x+)(5x+) 2 d) 8 5ln(5x+) 7. Find the derivative of the function y = 5x 2 e 3x a) xe 3x (2x + 3) b) 5xe 3x (2x + 3) c) ex 3x (3x + 2) d) 5xe 3x (3x + 2) 8. Find the derivative of the function y = ln(e x2 + ) a) 2xex2 e x2 + b) 2x e x2 c) 2xe x2 ln(e x2 +) d) 2e x2 (e x2 +) 2 9. Find any local maximum or minimum points of the given function y = x 3 3x 2 + a) Minimum at (,), maximum at (2,-3) b) Minimum at (2,-3), maximum at (,) c) Minimum at (2,-3), maximum at (-2,-9) and (,) d) Minimum at (2,-3). Which of the following statements is true? a) f(x) = 2e x is concave down for all x and has no POI b) f(x) = x 5 + is concave up for all x and has no POI c) f(x) = x is concave up for x <,concave down for x > and has a POI at (,5) d) f(x) = (x 5) 3 is concave up for x > 5,concave down for x < 5 and has a POI at (5,)
3 . Evaluate sec2 (lnx) dx x a) tan(lnx) + C b) ln(secx) + C c) 2sec(lnx) + C d) ln(tanx) + C π 2. Evaluate the definite integral sin 2 x 2 cos2 x dx 2 a) π 6 b) π 2 c) π 4 d) π 8 3. Evaluate I = e 4x sin ( x 2 ) dx a) 2 e4x (3 sin ( x ) + 4 cos 2 (x)) + C b) 2 23 e4x (2 sin ( x ) + 3 cos 2 (x )) + C 2 c) 65 e4x (2 sin ( x ) + 6 cos 2 (x)) + C d) 2 5 e4x (sin ( x ) cos 2 (x )) + C 2 4. Evaluate the definite integral e x 3 (x 2 + 2x)dx 3 a) b) 27e-36 c) e- d) 2e+4 e 5. Evaluate the definite integral (xlnx) 2 dx a) e 4 b) e3 2 c) 5e d) e Evaluate I = 4 x 4 dx a) ln x ln x + 2 tan x + C b) ln x tan x + C c) ln x 4ln x + 2 tan x + C d) 2ln x + ln x + + tan x + C 7. Let f(x) = sin(sin3x). Evaluate f (π/2) a) b) c) 2 d) 3 arcsin (5x) 8. Evaluate the following limit: L = lim x x 2 a) 5 b) /5 c) d) DNE 9. Given that f is such that its inverse F exists, f ( 5) = 4, F(2) = 5, find the value of the derivative of F at x=2. a) 4 b) /4 c) 5 d) /5 2. Let y be given implicitly as a differentiable function of x by 2x = xy + y 2. Then the slope of the tangent line to the curve y = y(x) at the point (x, y) where x =, y = is equal to: a) 2 b) /2 c) 3 d) /3
4 f(5+h) f(5) 2. Let f(x) = 2 x 5. Calculate L = lim h h a) L= b) L=5 c) L=-5 d) DNE 22. Let f(x) = x Evaluate the second derivate of f at x= a) 4 b) c) /2 d) DNE 23. Find an expression for the volume V of the solid of revolution obtained by rotating the region bounded by the graph of y = x 3, y = x 2 +, x = and x = about the y axis. a) πx(x 2 x 3 )dx d) 2πx 2 ( x 2 x 3 )dx b) 2πx( + x 2 x 3 )dx c) 2πx( x 2 x 3 )dx 24. Evaluate the improper integral 3 2 x x e dx a) 2 b) 2 c) 6 d) 25. Find the area of the region enclosed by the curves y x, y 2x x 2 2 a) /3 b) 9 c) 2/3 d) 6 END OF MATH 4 CORE CONTENT
5 26. Number of Cycles to Failure due to Fatigue BONUS ENGINEERING QUESTIONS Parts in machines and structures have natural imperfections and may have small cracks created during production. These small cracks can propagate under cyclical stresses the structure or machine may be exposed. For example an engine turbine blade is under cyclical stresses during takeoff and landing when differing thrust levels are required. Due to changes in rotational speed, temperature, and pressure, resulting cyclical stresses are put on the blade. This is known as fatigue. The differential equation that describes the growth of the crack in terms of the number of cycles is: da dn = A(K)m Let s say we want to find how many cycles before the part will fail. How can we solve this? Hint: what are you solving for, also this involves INTEGRATION. dn da = A(K) m = A(Yσ πa) m Givens: Y =.45, parameters: A =2.*, m = 3., initial crack size: a o =.3* 3 m, stress: σ= MPa K = Yσ πa, stiffness of material = 55 MPa m, largest allowable crack size before fast fracture: a f =? N f =?
6 27. Differential Equation solution using Partial Fractions: With differential equations, sometimes the equations are too complicated to simply deal with it normally using conventional integration and differentiation in the time domain. Instead, you will learn in later years that you can transform the equation into a simpler algebraic form using the Laplace transform and convert the problem into the s-domain. This is what we will do here to solve the following differential equation. x + 2x + 5x = 3, x() =, x () = If we take the Laplace transform, we can convert the equation into the form: s 2 X(s) + 2sX(s) + 5X(s) = 3 s And then we can solve for the solution in the Laplace domain: X(s): X(s) = 3 s(s 2 +2s+5) However, at this point we want to convert back to the time domain using the inverse Laplace function; however because of the nature of the solution it is too difficult to determine the inverse directly. First, we have to use PARTIAL FRACTIONS to split up the fraction into an easier form to take the inverse. Your job is to find the partial fraction expansion of the solution X(s):
7 28. Secular Variation in Orbital Elements due to J2 First of all, Orbital mechanics is based upon inertial 2 body Keplerian conditions. Put more simply, basic orbital mechanics considers only the effect of one larger body (earth/sun/moon etc) and the smaller satellite body. Additionally, the kinematics of the motion of orbits is defined by the simple, perfect condition known as a Keplerian orbit. However, in real situations, there are perturbing forces that change the orbital elements. For example atmospheric drag, solar radiation, and most importantly J2 or the effect of the oblateness of the earth. Therefore, calculations must be made to accommodate for these perturbing accelerations. Secular Variation is the term that describes the change in an orbital element over one orbit. In other words it T can be described as a = da 2π da dt, it can also be written as a = dθ. dt θ dt From analysing orbital conditions & parameters of the perturbing acceleration, θ = μa a 2, da dt = 2a2 μa f θ Given: T= hrs convert to s f θ =.5sinθ a = km (semi major axis) μ = 3986km (gravitational parameter) Find the secular change in semi major axis a:
Techniques of Integration
CHPTER 7 Techniques of Integration 7.. Substitution Integration, unlike differentiation, is more of an art-form than a collection of algorithms. Many problems in applied mathematics involve the integration
More informationcorrect-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationLimits and Continuity
Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More information100. In general, we can define this as if b x = a then x = log b
Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,
More informationCalculus 1: Sample Questions, Final Exam, Solutions
Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.
More informationAP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:
AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More information14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style
Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of
More information2 Integrating Both Sides
2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation
More informationx 2 y 2 +3xy ] = d dx dx [10y] dy dx = 2xy2 +3y
MA7 - Calculus I for thelife Sciences Final Exam Solutions Spring -May-. Consider the function defined implicitly near (,) byx y +xy =y. (a) [7 points] Use implicit differentiation to find the derivative
More informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More informationPractice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
More informationLecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)
Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More information5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
More informationAverage rate of change of y = f(x) with respect to x as x changes from a to a + h:
L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,
More informationSection 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a
More informationMicroeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
More informationParticular Solutions. y = Ae 4x and y = 3 at x = 0 3 = Ae 4 0 3 = A y = 3e 4x
Particular Solutions If the differential equation is actually modeling something (like the cost of milk as a function of time) it is likely that you will know a specific value (like the fact that milk
More informationTo give it a definition, an implicit function of x and y is simply any relationship that takes the form:
2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to
More informationLinear and quadratic Taylor polynomials for functions of several variables.
ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationSection 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
More informationVisualizing Differential Equations Slope Fields. by Lin McMullin
Visualizing Differential Equations Slope Fields by Lin McMullin The topic of slope fields is new to the AP Calculus AB Course Description for the 2004 exam. Where do slope fields come from? How should
More informationAP Calculus BC 2006 Free-Response Questions
AP Calculus BC 2006 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More informationFind the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.
SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.
More informationAP Calculus BC Exam. The Calculus BC Exam. At a Glance. Section I. SECTION I: Multiple-Choice Questions. Instructions. About Guessing.
The Calculus BC Exam AP Calculus BC Exam SECTION I: Multiple-Choice Questions At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Grade 50% Writing Instrument Pencil required
More informationNonhomogeneous Linear Equations
Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where
More informationThe Derivative. Philippe B. Laval Kennesaw State University
The Derivative Philippe B. Laval Kennesaw State University Abstract This handout is a summary of the material students should know regarding the definition and computation of the derivative 1 Definition
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More informationNotes and questions to aid A-level Mathematics revision
Notes and questions to aid A-level Mathematics revision Robert Bowles University College London October 4, 5 Introduction Introduction There are some students who find the first year s study at UCL and
More informationSolutions to Homework 5
Solutions to Homework 5 1. Let z = f(x, y) be a twice continously differentiable function of x and y. Let x = r cos θ and y = r sin θ be the equations which transform polar coordinates into rectangular
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More informationMath 113 HW #7 Solutions
Math 3 HW #7 Solutions 35 0 Given find /dx by implicit differentiation y 5 + x 2 y 3 = + ye x2 Answer: Differentiating both sides with respect to x yields 5y 4 dx + 2xy3 + x 2 3y 2 ) dx = dx ex2 + y2x)e
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
More informationThe Mathematics Diagnostic Test
The Mathematics iagnostic Test Mock Test and Further Information 010 In welcome week, students will be asked to sit a short test in order to determine the appropriate lecture course, tutorial group, whether
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationCalculus AB 2014 Scoring Guidelines
P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official
More information4 More Applications of Definite Integrals: Volumes, arclength and other matters
4 More Applications of Definite Integrals: Volumes, arclength and other matters Volumes of surfaces of revolution 4. Find the volume of a cone whose height h is equal to its base radius r, by using the
More informationINTEGRATING FACTOR METHOD
Differential Equations INTEGRATING FACTOR METHOD Graham S McDonald A Tutorial Module for learning to solve 1st order linear differential equations Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationTo differentiate logarithmic functions with bases other than e, use
To ifferentiate logarithmic functions with bases other than e, use 1 1 To ifferentiate logarithmic functions with bases other than e, use log b m = ln m ln b 1 To ifferentiate logarithmic functions with
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationDon't Forget the Differential Equations: Finishing 2005 BC4
connect to college success Don't Forget the Differential Equations: Finishing 005 BC4 Steve Greenfield available on apcentral.collegeboard.com connect to college success www.collegeboard.com The College
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More informationWEIGHTLESS WONDER Reduced Gravity Flight
WEIGHTLESS WONDER Reduced Gravity Flight Instructional Objectives Students will use trigonometric ratios to find vertical and horizontal components of a velocity vector; derive a formula describing height
More informationInverse Functions and Logarithms
Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a one-to-one function if it never takes on the same value twice; that
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More informationCourse outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems)
Course outline, MA 113, Spring 2014 Part A, Functions and limits 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems) Functions, domain and range Domain and range of rational and algebraic
More informationSECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise
More information2. Orbits. FER-Zagreb, Satellite communication systems 2011/12
2. Orbits Topics Orbit types Kepler and Newton laws Coverage area Influence of Earth 1 Orbit types According to inclination angle Equatorial Polar Inclinational orbit According to shape Circular orbit
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationAP Calculus AB 2013 Free-Response Questions
AP Calculus AB 2013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded
More informationSecond-Order Linear Differential Equations
Second-Order Linear Differential Equations A second-order linear differential equation has the form 1 Px d 2 y dx 2 dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. We saw in Section 7.1
More informationMath 432 HW 2.5 Solutions
Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/
More information(b)using the left hand end points of the subintervals ( lower sums ) we get the aprroximation
(1) Consider the function y = f(x) =e x on the interval [, 1]. (a) Find the area under the graph of this function over this interval using the Fundamental Theorem of Calculus. (b) Subdivide the interval
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationChapter 11. Techniques of Integration
Chapter Techniques of Integration Chapter 6 introduced the integral. There it was defined numerically, as the limit of approximating Riemann sums. Evaluating integrals by applying this basic definition
More informationIntegration by substitution
Integration by substitution There are occasions when it is possible to perform an apparently difficult piece of integration by first making a substitution. This has the effect of changing the variable
More informationHomework #1 Solutions
MAT 303 Spring 203 Homework # Solutions Problems Section.:, 4, 6, 34, 40 Section.2:, 4, 8, 30, 42 Section.4:, 2, 3, 4, 8, 22, 24, 46... Verify that y = x 3 + 7 is a solution to y = 3x 2. Solution: From
More informationa b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12.
MA123 Elem. Calculus Fall 2015 Exam 2 2015-10-22 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACT-approved calculator during
More informationGRAPHING IN POLAR COORDINATES SYMMETRY
GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,
More informationSeparable First Order Differential Equations
Separable First Order Differential Equations Form of Separable Equations which take the form = gx hy or These are differential equations = gxĥy, where gx is a continuous function of x and hy is a continuously
More informationMicroeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS
DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding
More informationSolving DEs by Separation of Variables.
Solving DEs by Separation of Variables. Introduction and procedure Separation of variables allows us to solve differential equations of the form The steps to solving such DEs are as follows: dx = gx).
More informationDifferential Equations
40 CHAPTER 15 Differential Equations In many natural conditions the rate at which the amount of an object changes is directly proportional to the amount of the object itself. For example: 1) The marginal
More informationAP Calculus AB Syllabus
Course Overview and Philosophy AP Calculus AB Syllabus The biggest idea in AP Calculus is the connections among the representations of the major concepts graphically, numerically, analytically, and verbally.
More informationAP Calculus BC 2013 Free-Response Questions
AP Calculus BC 013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract -
More informationON THE EXPONENTIAL FUNCTION
ON THE EXPONENTIAL FUNCTION ROBERT GOVE AND JAN RYCHTÁŘ Abstract. The natural exponential function is one of the most important functions students should learn in calculus classes. The applications range
More informationAP Calculus AB 2012 Free-Response Questions
AP Calculus AB 1 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in
More informationis identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
More information2.2 Separable Equations
2.2 Separable Equations 73 2.2 Separable Equations An equation y = f(x, y) is called separable provided algebraic operations, usually multiplication, division and factorization, allow it to be written
More informationAP PHYSICS C Mechanics - SUMMER ASSIGNMENT FOR 2016-2017
AP PHYSICS C Mechanics - SUMMER ASSIGNMENT FOR 2016-2017 Dear Student: The AP physics course you have signed up for is designed to prepare you for a superior performance on the AP test. To complete material
More informationAP Calculus BC 2001 Free-Response Questions
AP Calculus BC 001 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More information2.2 Derivative as a Function
2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x
More informationCHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises
CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =
More informationAstronomy 110 Homework #04 Assigned: 02/06/2007 Due: 02/13/2007. Name:
Astronomy 110 Homework #04 Assigned: 02/06/2007 Due: 02/13/2007 Name: Directions: Listed below are twenty (20) multiple-choice questions based on the material covered by the lectures this past week. Choose
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More information12.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following:
Section 1.6 Logarithmic and Exponential Equations 811 1.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following: Solve Quadratic Equations (Section
More informationMath 115 HW #8 Solutions
Math 115 HW #8 Solutions 1 The function with the given graph is a solution of one of the following differential equations Decide which is the correct equation and justify your answer a) y = 1 + xy b) y
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More informationImplicit Differentiation
Implicit Differentiation Sometimes functions are given not in the form y = f(x) but in a more complicated form in which it is difficult or impossible to express y explicitly in terms of x. Such functions
More information