Calculus with Analytic Geometry I Exam 10, Take Home Friday, November 8, 2013 Solutions.
|
|
- Norman Holland
- 6 years ago
- Views:
Transcription
1 All exercises are from Section 4.7 of the textbook. 1. Calculus with Analytic Geometry I Exam 10, Take Home Friday, November 8, 2013 Solutions. 2. Solution. The picture suggests using the angle θ as variable; θ = 0 means she rows all the way, θ = π/2, she walks all the way. There are several ways of getting all the distances; here is one. Let O be the center of the lake and consider the isosceles triangle AOB. The base angles must be equal, thus the angle AOB (angle at O) is π 2θ. The length AB is then given by AB 2 = cos(π 2θ) = 8 + 8cos(2θ) = 8(1 + cos 2 θ sin 2 θ) = 16 cos 2 θ, so that AB = 4 cos θ. Concerning the length of the arc from B to C, its length equals the radius times the subtended angle COB. This angle is again easily seen to be 2θ; thus the length of the arc is 4θ. We now see that the time to minimize is T (θ) = 4 cos θ 2 + 4θ 4 = 2 cos θ + θ, 0 θ π 2. We find critical points, if any. We have T (θ) = 2 sin θ + 1; setting it to 0 we get sin θ = 1/2. In the interval [0, π/2], sin θ = 1/2 only for θ = π/6. The easiest way of checking is to just evaluate T at the critical point and at the endpoints of the interval. We get T (0) = 2, T (π/6) = 2 cos(π/6) + π/6 = 3 + π , T (π/2) = π To minimize time, she should walk all the way. Here is a picture to go with the problem.
2 A convenient variable is x, where (x, 4 x 2 ) is the point of tangency. But we have to be careful in how we write the equation of the line, so for just a moment it may be better to denote the point of tangency by (x 0, 4 x 2 0). The value of the derivative at this point is d(4 x2 ) dx = 2x 0. The equation of the tangent x0 line is then y (4 x 2 0) = 2x 0 (x x 0 ), or y = 2x 0 x + x From this we get at once the values of the x and y intercepts of the tangent line; they are, respectively, x x 0 and x These are also the lengths of the legs of the triangle so that, writing again x for x 0, the area to minimize is A(x) = 1 ( x ) (4 + x 2 ) = 1 x 4 x3 + 2x + 4 x. The domain in which we have to consider this function is (0, ). We see that A (x) = 3 4 x ; setting to x2 0 and multiplying by 4x 2 (which changes nothing since x > 0) we get 3x 4 + 8x 2 16 = 0. This is a quadratic equation in x 2 with roots 4 and 4/3. Discarding the impossible root 4, we are left with x 2 = 4/3, thus x = ±2/ 3. The only critical point in the domain of relevance is x = 2/ 3. Now A (x) = 3 2 x + 8 x > 0 when 3 x > 0, so that we have a local minimum, hence an absolute minimum since it is the only critical point. The minimum area is thus ( ) 2 A 3 =
3 3. Solution. I ll denote by AB the length of the segment AB. The length of the rope is L = P Q csc θ 1 + ST csc θ 2. We have to relate θ 2 with θ 1. We can use that tan θ 2 = ST RT RT 2 + ST so that a standard trigonometry exercise shows that csc θ 2 = 2. Now ST RT = QT QR = QT P Q cot θ 1, so writing now θ for θ 1 we get for the length of the rope L = P Q sec θ + ( QT P Q cot θ) 2 + ST 2 But this gives a really nasty derivative to minimize, so let us try to find instead the point R that minimizes the length of the rope. I ll introduce for this purpose the variable x = QR and now we can express the length L of the rope in the somewhat simpler form The domain of the variable x is 0 x QT. Then L(x) = P Q 2 + x 2 + ST 2 + ( QT x) 2. L (x) = Setting to 0, and solving for x on gets x = x P Q 2 + x QT x 2 ST 2 + ( QT x) 2 QT P Q. Is this the x that minimizes L?. We notice that P Q + ST L QT (0) = ST 2 + QT < 0, QT 2 L ( QT ) = ST 2 + QT > 0; 2 since L is only zero at x, that means L is negative to the left of the critical point, positive to the right; the critical point is a local minimum and, being the only critical point in the interval, the point at which the absolute minimum occurs. With this value of x = QR we get tan θ 1 = P Q x = P Q + ST, QT
4 tan θ 2 = ST QT x P Q + ST =. QT The angles are equal. NOTE: There is a clever way of solving this problem without appealing to calculus. Can you find it? 4. Solution. The amount of water will be maximum when the cross-sectional area is maximum. The cross section is a trapezoid of bases of lengths (in cm) 10 and cos θ, height 10 sin θ. The area is thus the domain of A is 0 θ π/2. Now A(θ) = 100(1 + cos θ) sin θ. A (θ) = 100( sin 2 θ + (1 + cos θ) cos θ) = 100(cos θ + cos 2 θ sin 2 θ). To be able to solve for θ after setting A (θ)to0, it is a good idea to replace sin 2 θ by 1 cos 2 θ. With this substitution, we get A (θ) = 100(2 cos 2 θ + cos θ 1). Setting A (θ) to 0 yields a quadratic equation in cos θ that can be solved by the quadratic formula: 2 cos 2 θ + cos θ 1 = 0, so cos θ = 1 ± 3, 4 so θ = 1/2 or θ = 1. The second solution can t occur in the given domain, so cos θ = 1/2, hence θ = π/3 is the only critical point in the domain. Now The maximum value occurs when θ = π 3. A(0) = 0, A(π/3) = , A(π/2) = One can stop the domain at π/2 since it is obvious that a larger angle will give a worse cross-sectional area. But one can also consider a somewhat larger domain. However, one cannot take θ all the way to π because once θ = 5π/6 (150 ) the two sides of the gutter touch, the cross section is an equilateral triangle, and it doesn t make sense to keep bending the sides.
5 Solution. This exercise is way easier than it seems. It is a bit a lesson in anatomy. (a) A bit of trigonometry shows that the distance from B to the end of the horizontal blood vessel is b cot θ. The distance from A to B is thus a b cot θ. Trigonometry also shows that the distance from B to C is b csc θ. So assuming (as one has to, to get the required answer) that resistance is additive, the resistance from A to B is in fact ( a b cot θ R = C r b csc θ r 4 2 (b) I assume that the full question is: Given a major blood vessel (an artery, perhaps), a sub-vessel has to branch off to get blood to a point C at distance b from the major vessel, and it should branch off at a point at a distance less than a from the point nearest to C. So we do the usual. First of all, let us determine the domain of R as a function of θ. It should be clear that θ = 0 cannot work. In fact, the smallest value of θ has to have a tangent of b/a and the largest is π/2. The domain of R is arctan(b/a) θ π/2. Differentiating, ( +b csc R 2 θ (θ) = C 4 ). ) ( b csc θ cot θ csc θ 4 = Cb csc θ 4 cot θ ) 4 ; setting to 0, since csc θ is never 0, we get csc θ 4 cot θ 4 = 0, thus cot θ csc θ = r4 2 4 and, since cot θ/ csc θ = cos θ, we see that R (θ) = 0 if and only if cos θ = /r Is this in the domain? Since we can assume that a is fairly large compared to b, so arctan b/a is close to 0, the answer is probably yes. Justifying that this is a minimum (and not a maximum) is perhaps harder than usual, but not impossible. Here is one way. While the domain of R might not go all the way to θ = 0, the function R(θ) is define in (0, π ). If we replace cot, csc by their expressions in terms of sin, cos, the derivative becomes ( R 1 1 (θ) = Cb sin 2 θ 4 cos θ ) 4,
6 showing that the sign of R 1 (θ) is the same as the sign of 4 cos θ 4. Now 1 4 cos θ 4 = 1 θ= < 0 because r 2 < r 1, while 1 r 4 1 cos θ r 4 2 θ= π 2 = 1 r 4 1 > 0. The derivative goes from negative to positive, thus we have a minimum. (c) We have to find θ such that Thus θ = arccos cos θ = ( 2 3 r 1) 4 r 4 1 =
Chapter 06: Analytic Trigonometry
Chapter 06: Analytic Trigonometry 6.1: Inverse Trigonometric Functions The Problem As you recall from our earlier work, a function can only have an inverse function if it is oneto-one. Are any of our trigonometric
More information4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS
4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS MR. FORTIER 1. Trig Functions of Any Angle We now extend the definitions of the six basic trig functions beyond triangles so that we do not have to restrict
More informationUnit Circle. Return to. Contents
Unit Circle Return to Table of Contents 32 The Unit Circle The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. Quadrant II: x is negative and y is positive (0,1) 1 Quadrant
More information= 2 x. So, when 0 < x < 2 it is increasing but it is decreasing
Math 1A UCB, Spring 2010 A. Ogus Solutions 1 for Problem Set 10.5 # 5. y = x + x 1. Domain : R 2. x-intercepts : x + x = 0, i.e. x = 0, x =. y-intercept : y = 0.. symmetry : no. asymptote : no 5. I/D intervals
More informationMATH 127 SAMPLE FINAL EXAM I II III TOTAL
MATH 17 SAMPLE FINAL EXAM Name: Section: Do not write on this page below this line Part I II III TOTAL Score Part I. Multiple choice answer exercises with exactly one correct answer. Each correct answer
More informationChapter 5 Analytic Trigonometry
Chapter 5 Analytic Trigonometry Section 1 Section 2 Section 3 Section 4 Section 5 Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas
More informationMath 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts
Introduction Math 121: Calculus 1 - Fall 201/2014 Review of Precalculus Concepts Welcome to Math 121 - Calculus 1, Fall 201/2014! This problems in this packet are designed to help you review the topics
More informationGiven an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
More informationMTH 122: Section 204. Plane Trigonometry. Test 1
MTH 122: Section 204. Plane Trigonometry. Test 1 Section A: No use of calculator is allowed. Show your work and clearly identify your answer. 1. a). Complete the following table. α 0 π/6 π/4 π/3 π/2 π
More informationMath 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts
Introduction Math 11: Calculus 1 - Fall 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Fall 01/01! This problems in this packet are designed to help you review the topics from Algebra
More informationGiven an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
More informationMath 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts
Introduction Math 11: Calculus 1 - Winter 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Winter 01/01! This problems in this packet are designed to help you review the topics from
More informationSince x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C)
SAT II - Math Level 2 Test #01 Solution 1. x + = 2, then x² + = Since x + = 2, by squaring both side of the equation, (A) - (B) 0 (C) 2 (D) 4 (E) -2 we get x² + 2x 1 + 1 = 4, or simplifying it, x² + 2
More informationCrash Course in Trigonometry
Crash Course in Trigonometry Dr. Don Spickler September 5, 003 Contents 1 Trigonometric Functions 1 1.1 Introduction.................................... 1 1. Right Triangle Trigonometry...........................
More informationAnalytic Trigonometry
Chapter 5 Analytic Trigonometry Course Number Section 5.1 Using Fundamental Identities Objective: In this lesson you learned how to use fundamental trigonometric identities to evaluate trigonometric functions
More informationMAC Calculus II Spring Homework #6 Some Solutions.
MAC 2312-15931-Calculus II Spring 23 Homework #6 Some Solutions. 1. Find the centroid of the region bounded by the curves y = 2x 2 and y = 1 2x 2. Solution. It is obvious, by inspection, that the centroid
More informationTriangles and Vectors
Chapter 3 Triangles and Vectors As was stated at the start of Chapter 1, trigonometry had its origins in the study of triangles. In fact, the word trigonometry comes from the Greek words for triangle measurement.
More informationNotes on Radian Measure
MAT 170 Pre-Calculus Notes on Radian Measure Radian Angles Terri L. Miller Spring 009 revised April 17, 009 1. Radian Measure Recall that a unit circle is the circle centered at the origin with a radius
More informationMATH 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
More informationThe answers below are not comprehensive and are meant to indicate the correct way to solve the problem. sin
Math : Practice Final Answer Key Name: The answers below are not comprehensive and are meant to indicate the correct way to solve the problem. Problem : Consider the definite integral I = 5 sin ( ) d.
More informationA List of Definitions and Theorems
Metropolitan Community College Definition 1. Two angles are called complements if the sum of their measures is 90. Two angles are called supplements if the sum of their measures is 180. Definition 2. One
More informationCHAPTER 4. APPLICATIONS AND REVIEW IN TRIGONOMETRY
CHAPTER 4. APPLICATIONS AND REVIEW IN TRIGONOMETRY In the present chapter we apply the vector algebra and the basic properties of the dot product described in the last chapter to planar geometry and trigonometry.
More informationFunctions. Remark 1.2 The objective of our course Calculus is to study functions.
Functions 1.1 Functions and their Graphs Definition 1.1 A function f is a rule assigning a number to each of the numbers. The number assigned to the number x via the rule f is usually denoted by f(x).
More informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
More informationTrigonometric ratios:
0 Trigonometric ratios: The six trigonometric ratios of A are: Sine Cosine Tangent sin A = opposite leg hypotenuse adjacent leg cos A = hypotenuse tan A = opposite adjacent leg leg and their inverses:
More informationMAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS
MAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT100 is a fast-paced and thorough tour of precalculus mathematics, where the choice of topics is primarily motivated by the conceptual and technical knowledge
More informationCalculus I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationCalculus with Analytic Geometry I Exam 8 Take Home Part.
Calculus with Analytic Geometry I Exam 8 Take Home Part. INSTRUCTIONS: SHOW ALL WORK. Write clearly, using full sentences. Use equal signs appropriately; don t use them between quantities that are not
More informationMATH 103 Pre-Calculus Mathematics Dr. McCloskey Fall 2008 Final Exam Sample Solutions
MATH 103 Pre-Calculus Mathematics Dr. McCloskey Fall 008 Final Exam Sample Solutions In these solutions, FD refers to the course textbook (PreCalculus (4th edition), by Faires and DeFranza, published by
More informationChapter 3. Radian Measure and Circular Functions. Copyright 2005 Pearson Education, Inc.
Chapter 3 Radian Measure and Circular Functions Copyright 2005 Pearson Education, Inc. 3.1 Radian Measure Copyright 2005 Pearson Education, Inc. Measuring Angles Thus far we have measured angles in degrees
More informationSection Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.
Section 5.4 - Inverse Trigonometry In this section, we will define inverse since, cosine and tangent functions. RECALL Facts about inverse functions: A function f ) is one-to-one if no two different inputs
More informationsin cos 1 1 tan sec 1 cot csc 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
More informationMATH 2412 Sections Fundamental Identities. Reciprocal. Quotient. Pythagorean
MATH 41 Sections 5.1-5.4 Fundamental Identities Reciprocal Quotient Pythagorean 5 Example: If tanθ = and θ is in quadrant II, find the exact values of the other 1 trigonometric functions using only fundamental
More informationLesson 7.3 Exercises, pages
Lesson 7. Exercises, pages 8 A. Write each expression in terms of a single trigonometric function. cos u a) b) sin u cos u cot U tan U P DO NOT COPY. 7. Reciprocal and Quotient Identities Solutions 7 c)
More informationREQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS
REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS The Department of Applied Mathematics administers a Math Placement test to assess fundamental skills in mathematics that are necessary to begin the study
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More informationRadian measure and trigonometric functions on the reals
Trigonometry Radian measure and trigonometric functions on the reals The units for measuring angles tend to change depending on the context - in a geometry course, you re more likely to be measuring your
More informationChapter 4 Trigonometric Functions
SECTION 4.1 Special Right Triangles and Trigonometric Ratios Chapter 4 Trigonometric Functions Section 4.1: Special Right Triangles and Trigonometric Ratios Special Right Triangles Trigonometric Ratios
More informationFundamentals of Mathematics (MATH 1510)
Fundamentals of Mathematics () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University March 14-18, 2016 Outline 1 2 s An angle AOB consists of two rays R 1 and R
More informationSec 4 Maths. SET A PAPER 2 Question
S4 Maths Set A Paper Question Sec 4 Maths Exam papers with worked solutions SET A PAPER Question Compiled by THE MATHS CAFE 1 P a g e Answer all the questions S4 Maths Set A Paper Question Write in dark
More informationPractice Test - Chapter 4
Find the value of x. Round to the nearest tenth, if necessary. 1. An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the length of the side opposite.
More informationA. Incorrect! For a point to lie on the unit circle, the sum of the squares of its coordinates must be equal to 1.
Algebra - Problem Drill 19: Basic Trigonometry - Right Triangle No. 1 of 10 1. Which of the following points lies on the unit circle? (A) 1, 1 (B) 1, (C) (D) (E), 3, 3, For a point to lie on the unit circle,
More informationTopic 3 Part 1 [449 marks]
Topic 3 Part [449 marks] a. Find all values of x for 0. x such that sin( x ) = 0. b. Find n n+ x sin( x )dx, showing that it takes different integer values when n is even and when n is odd. c. Evaluate
More informationCalculus with business applications, Lehigh U, Lecture 05 notes Summer
Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 Trigonometric functions. Trigonometric functions often arise in physical applications with periodic motion. They do not arise often
More informationTrigonometry: Graphs of trig functions (Grade 10) *
OpenStax-CNX module: m39414 1 Trigonometry: Graphs of trig functions (Grade 10) * Free High School Science Texts Project This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution
More informationWYSE MATH STATE 2012 SOLUTIONS. 1. Ans E: Trapezoids need only have one pair of parallel sides. Parallelograms are, by definition, forced to have two.
WYSE MATH STATE 01 SOLUTIONS 1. Ans E: Trapezoids need only have one pair of parallel sides. Parallelograms are, by definition, forced to have two.. Ans A: All the cans can be arranged in 10 P 10 = 10!
More informationPrecalculus Summer Assignment 2015
Precalculus Summer Assignment 2015 The following packet contains topics and definitions that you will be required to know in order to succeed in CP Pre-calculus this year. You are advised to be familiar
More informationCoach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers
Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}
More informationName: Date: Period: Calculus Honors: 4-2 The Product Rule
Name: Date: Period: Calculus Honors: 4- The Product Rule Warm Up: 1. Factor and simplify. 9 10 0 5 5 10 5 5. Find ' f if f How did you go about finding the derivative? Let s Eplore how to differentiate
More informationLesson 28 Working with Special Triangles
Lesson 28 Working with Special Triangles Pre-Calculus 3/3/14 Pre-Calculus 1 Review Where We ve Been We have a new understanding of angles as we have now placed angles in a circle on a coordinate plane
More informationTOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12
NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus
More informationPartial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.
Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,
More informationAP CALCULUS Summer Assignment 2014
Name AP CALCULUS Summer Assignment 014 Welcome to AP Calculus. In order to complete the curriculum before the AP Exam in May, it is necessary to do some preparatory work this summer. The following assignment
More informationA2T Trig Packet Unit 1
A2T Trig Packet Unit 1 Name: Teacher: Pd: Table of Contents Day 1: Right Triangle Trigonometry SWBAT: Solve for missing sides and angles of right triangles Pages 1-7 HW: Pages 8 and 9 in Packet Day 2:
More informationUsing this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.
Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive
More informationExercise Set 4.1: Special Right Triangles and Trigonometric Ratios
Eercise Set.1: Special Right Triangles and Trigonometric Ratios Answer the following. 9. 1. If two sides of a triangle are congruent, then the opposite those sides are also congruent. 2. If two angles
More informationCHAPTER 5: Analytic Trigonometry
) (Answers for Chapter 5: Analytic Trigonometry) A.5. CHAPTER 5: Analytic Trigonometry SECTION 5.: FUNDAMENTAL TRIGONOMETRIC IDENTITIES Left Side Right Side Type of Identity (ID) csc( x) sin x Reciprocal
More informationPre-Calculus 40 Final Outline/Review:
2016-2017 Pre-Calculus 40 Final Outline/Review: Non-Calculator Section: 16 multiple choice (32 pts) and 6 open ended (24 pts). Calculator Section: 8 multiple choice (16 pts) and 11 open ended (36 pts).
More informationMATH 109 TOPIC 3 RIGHT TRIANGLE TRIGONOMETRY. 3a. Right Triangle Definitions of the Trigonometric Functions
Math 09 Ta-Right Triangle Trigonometry Review Page MTH 09 TOPIC RIGHT TRINGLE TRIGONOMETRY a. Right Triangle Definitions of the Trigonometric Functions a. Practice Problems b. 5 5 90 and 0 60 90 Triangles
More informationChapter 1: Packing your Suitcase
Chapter : Packing your Suitcase Lesson.. -. a. Independent variable = distance from end of tube to the wall. Dependent variable = width of field of view. e. The equation depends on the length and diameter
More information1. (10pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given that tan θ = 1. Draw a picture.
Trigonometry Exam 1 MAT 145, Spring 017 D. Ivanšić Name: Show all your work! 1. (10pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given that tan θ = 1. Draw a picture.
More informationExercise Set 6.2: Double-Angle and Half-Angle Formulas
Exercise Set : Double-Angle and Half-Angle Formulas Answer the following π 1 (a Evaluate sin π (b Evaluate π π (c Is sin = (d Graph f ( x = sin ( x and g ( x = sin ( x on the same set of axes (e Is sin
More informationEssex County College Division of Mathematics MTH-122 Assessments. Honor Code
Essex County College Division of Mathematics MTH-22 Assessments Last Name: First Name: Phone or email: Honor Code The Honor Code is a statement on academic integrity, it articulates reasonable expectations
More informationIntegrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61
Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up
More informationSummer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
More informationBook 4. June 2013 June 2014 June Name :
Book 4 June 2013 June 2014 June 2015 Name : June 2013 1. Given that 4 3 2 2 ax bx c 2 2 3x 2x 5x 4 dxe x 4 x 4, x 2 find the values of the constants a, b, c, d and e. 2. Given that f(x) = ln x, x > 0 sketch
More informationSection 5.4 The Other Trigonometric Functions
Section 5.4 The Other Trigonometric Functions Section 5.4 The Other Trigonometric Functions In the previous section, we defined the e and coe functions as ratios of the sides of a right triangle in a circle.
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.6 Inverse Functions and Logarithms In this section, we will learn about: Inverse functions and logarithms. INVERSE FUNCTIONS The table gives data from an experiment
More informationExam 1 Review SOLUTIONS
1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make
More informationA different parametric curve ( t, t 2 ) traces the same curve, but this time the par-
Parametric Curves: Suppose a particle is moving around in a circle or any curve that fails the vertical line test, then we cannot describe the path of this particle using an equation of the form y fx)
More informationQuick Review Sheet for A.P. Calculus Exam
Quick Review Sheet for A.P. Calculus Exam Name AP Calculus AB/BC Limits Date Period 1. Definition: 2. Steps in Evaluating Limits: - Substitute, Factor, and Simplify 3. Limits as x approaches infinity If
More informationChapter 1. Functions 1.3. Trigonometric Functions
1.3 Trigonometric Functions 1 Chapter 1. Functions 1.3. Trigonometric Functions Definition. The number of radians in the central angle A CB within a circle of radius r is defined as the number of radius
More informationMATH 255 Applied Honors Calculus III Winter Midterm 1 Review Solutions
MATH 55 Applied Honors Calculus III Winter 11 Midterm 1 Review Solutions 11.1: #19 Particle starts at point ( 1,, traces out a semicircle in the counterclockwise direction, ending at the point (1,. 11.1:
More informationTrig Identities, Solving Trig Equations Answer Section
Trig Identities, Solving Trig Equations Answer Section MULTIPLE CHOICE. ANS: B PTS: REF: Knowledge and Understanding OBJ: 7. - Compound Angle Formulas. ANS: A PTS: REF: Knowledge and Understanding OBJ:
More informationChapter 7 Trigonometric Identities and Equations 7-1 Basic Trigonometric Identities Pages
Trigonometric Identities and Equations 7- Basic Trigonometric Identities Pages 47 430. Sample answer: 45 3. tan, cot, cot tan cos cot, cot csc 5. Rosalinda is correct; there may be other values for which
More informationMTH 121 Fall 2007 Essex County College Division of Mathematics and Physics Worksheet #1 1
MTH Fall 007 Essex County College Division of Mathematics and Physics Worksheet # Preamble It is extremely important that you complete the following two items as soon as possible. Please send an email
More informationCALCULUS ASSESSMENT REVIEW
CALCULUS ASSESSMENT REVIEW DEPARTMENT OF MATHEMATICS CHRISTOPHER NEWPORT UNIVERSITY 1. Introduction and Topics The purpose of these notes is to give an idea of what to expect on the Calculus Readiness
More informationMATH section 3.1 Maximum and Minimum Values Page 1 of 7
MATH section. Maimum and Minimum Values Page of 7 Definition : Let c be a number in the domain D of a function f. Then c ) is the Absolute maimum value of f on D if ) c f() for all in D. Absolute minimum
More informationChapter 8B - Trigonometric Functions (the first part)
Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of
More informationThings You Should Know Coming Into Calc I
Things You Should Know Coming Into Calc I Algebraic Rules, Properties, Formulas, Ideas and Processes: 1) Rules and Properties of Exponents. Let x and y be positive real numbers, let a and b represent real
More informationCALCULUS I. Review. Paul Dawkins
CALCULUS I Review Paul Dawkins Table of Contents Preface... ii Review... 1 Introduction... 1 Review : Functions... Review : Inverse Functions...1 Review : Trig Functions...0 Review : Solving Trig Equations...7
More informationTrigonometric Functions. Section 1.6
Trigonometric Functions Section 1.6 Quick Review Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Radian
More informationMAT1035 Analytic Geometry
MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................
More informationAnswers for Ch. 6 Review: Applications of the Integral
Answers for Ch. 6 Review: Applications of the Integral. The formula for the average value of a function, which you must have stored in your magical mathematical brain, is b b a f d. a d / / 8 6 6 ( 8 )
More informationSection 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
More informationSolutionbank C2 Edexcel Modular Mathematics for AS and A-Level
Heinemann Solutionbank: Core Maths C file://c:\users\buba\kaz\ouba\c_mex.html Page of /0/0 Solutionbank C Exercise, Question The sector AOB is removed from a circle of radius 5 cm. The AOB is. radians
More informationMath 1501 Calc I Fall 2013 Lesson 9 - Lesson 20
Math 1501 Calc I Fall 2013 Lesson 9 - Lesson 20 Instructor: Sal Barone School of Mathematics Georgia Tech August 19 - August 6, 2013 (updated October 4, 2013) L9: DIFFERENTIATION RULES Covered sections:
More informationOld Math 120 Exams. David M. McClendon. Department of Mathematics Ferris State University
Old Math 10 Exams David M. McClendon Department of Mathematics Ferris State University 1 Contents Contents Contents 1 General comments on these exams 3 Exams from Fall 016 4.1 Fall 016 Exam 1...............................
More informationV. Graph Sketching and Max-Min Problems
V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.
More informationImplicit Differentiation and Inverse Trigonometric Functions
Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5
More informationCollege Prep Math Final Exam Review Packet
College Prep Math Final Exam Review Packet Name: Date of Exam: In Class 1 Directions: Complete each assignment using the due dates given by the calendar below. If you are absent from school, you are still
More informationSection 8.2 Vector Angles
Section 8.2 Vector Angles INTRODUCTION Recall that a vector has these two properties: 1. It has a certain length, called magnitude 2. It has a direction, indicated by an arrow at one end. In this section
More informationCalculus III: Practice Final
Calculus III: Practice Final Name: Circle one: Section 6 Section 7. Read the problems carefully. Show your work unless asked otherwise. Partial credit will be given for incomplete work. The exam contains
More informationMAC 1114: Trigonometry Notes
MAC 1114: Trigonometry Notes Instructor: Brooke Quinlan Hillsborough Community College Section 7.1 Angles and Their Measure Greek Letters Commonly Used in Trigonometry Quadrant II Quadrant III Quadrant
More information1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
More informationTrigonometry Exam II Review Problem Selected Answers and Solutions
Trigonometry Exam II Review Problem Selected Answers and Solutions 1. Solve the following trigonometric equations: (a) sin(t) = 0.2: Answer: Write y = sin(t) = 0.2. Then, use the picture to get an idea
More informationAlgebra II Standard Term 4 Review packet Test will be 60 Minutes 50 Questions
Algebra II Standard Term Review packet 2017 NAME Test will be 0 Minutes 0 Questions DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document.
More informationQuick Overview: Complex Numbers
Quick Overview: Complex Numbers February 23, 2012 1 Initial Definitions Definition 1 The complex number z is defined as: z = a + bi (1) where a, b are real numbers and i = 1. Remarks about the definition:
More informationMath 2 Trigonometry. People often use the acronym SOHCAHTOA to help remember which is which. In the triangle below: = 15
Math 2 Trigonometry 1 RATIOS OF SIDES OF A RIGHT TRIANGLE Trigonometry is all about the relationships of sides of right triangles. In order to organize these relationships, each side is named in relation
More informationMAC 2311 Calculus I Spring 2004
MAC 2 Calculus I Spring 2004 Homework # Some Solutions.#. Since f (x) = d dx (ln x) =, the linearization at a = is x L(x) = f() + f ()(x ) = ln + (x ) = x. The answer is L(x) = x..#4. Since e 0 =, and
More information