Inverse Relations. 5 are inverses because their input and output are switched. For instance: f x x. x 5. f 4

Size: px
Start display at page:

Download "Inverse Relations. 5 are inverses because their input and output are switched. For instance: f x x. x 5. f 4"

Transcription

1 Inverse Functions

2 Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the original is the range of the inverse; and vice versa) 1 E: f 5 and f 5 are inverses because their input and output are switched. For instance: f 4 f f f ' 5 4

3 Tables and Graphs of Inverses Orginal Switch and y Inverse (0,5) (,16) (6,4) (10,0) (0,5) (18,16) (14,4) X Y X Y (4,14) (0,10) (4,6) Line of Symmetry: y = Switch and y (16,18) (16,) Although transformed, the graphs are identical

4 Inverse and Compositions In order for two functions to be inverses: f g AND g f

5 One-to-One Functions A function f() is one-to-one on a domain D if, for every value c, the equation f() = c has at most one solution for every in D. Or, for every a and b in D: f a f b a b unless Theorems: 1. A function has an inverse function if and only if it is one-to-one.. If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function.

6 The Horizontal Line Test If a horizontal line intersects a curve more than once, it s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.

7 The Horizontal Line Test If a horizontal line intersects a curve more than once, it s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.

8 Eample Without graphing, decide if the function below has an inverse function. f 6 3 If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function. See if the derivative is always one sign: f ' 6 Since the derivative is always negative, the inverse of f is a function.

9 Find the Inverse of a Function 1. Switch the and y of the function whose inverse you desire.. Solve for y to get the Inverse function 3. Make sure that the domains and ranges of your inverse and original function match up.

10 Eample Find the inverse of the following: d 4 3 Switch and y Only Half Parabola Solve for y 34 4 y 3 y Really y = Restrict the Domain! 3 Full Parabola y (too much) 4 d 3 y when 3 =3 Make sure to check with a table and graph on the calculator.

11 Logarithms v Eponentials

12 Definition of Logarithm The logarithm base a of b is the eponent you put on a to get b: lo g a b a > 0 if and only a i.e. Logs give you eponents! b if and b > 0 The logarithm to the base e, denoted ln, is called the natural logarithm.

13 Logarithm and Eponential Forms Logarithm Form 5 = log (3) Logs Give you Eponents Input Becomes Output 5 = 3 Base Stays the Base Eponential Form

14 Eamples Write each equation in eponential form 1.log 15 (5) = /3.Log 8 () = 1/3 15 /3 = 5 8 1/3 = Write each equation in logarithmic form 1.If 64 = 4 3.If 1/7 = 3 log 4 (64) = 3 Log 3 (1/7) =

15 Eample Complete the table if a is a positive real number and: f a Domain Range Continuous? One-to-One? Concavity Left End Behavior Right End Behavior 1 f a f All Reals All Positive Reals Yes Yes Always Up lim a 0 lim a log a All Positive Reals All Reals Yes Yes Always Down 0 lim log a lim log a

16 The Change of Base Formula The following formula allows you to evaluate any valid logarithm statement: log b a log log c c a b For a and b greater than 0 AND b 1. Eample: Evaluate log1.04 ln ln

17 Solve: Solving Equations with the Change of Base Formula Isolate the base and power Change the eponential equation to an logarithm equation Use the Change of Base Formula log log 454 log

18 Properties of Logarithms For a>0, b>0, m>0, m 1, and any real number n. Logarithm of 1: log 1 0 m Logarithm of the base: log 1 m m Power Property: log m a n nlog a m Product Property: ab a b log log log m m m Quotient Property: a log log a log b m b m m

19 Eample 1 Condense the epression: 1 3 log log 10 log log log 10 log log 10 log log

20 Eample Epand the epression: ln 3y ln 3 ln ln y ln 3 ln ln y

21 Eample 3 Solve the equation: 4 4 log log log 3 3

22 AP Reminders Do not forget the following relationships: ln e e ln e a e b e a b e e a b e ab

23 Inverse Trigonometry

24 Tangent Cosine Sine Trigonometric Functions Each one of these trigonometric functions fail the horizontal line test, so they are not one-to-one. Therefore, there inverses are not functions. Cosecant Secant Cotangent

25 Tangent Cosine Sine Trigonometric Functions with Restricted Domains D :, D : 0, D :, In order for their inverses to be functions, the domains of the trigonometric functions are restricted so that they become oneto-one. D :,0 0, D : 0,, D : 0, Cosecant Secant Cotangent

26 Trigonometric Functions with Restricted Domains Function Domain Range f () = sin f () = cos f () = tan f () = csc f () = sec f () = cot, 0,,,0 0, 0,, 0, 1,1 1,1,,, 1 1,, 1 1,

27 Tan -1 Cos -1 Sin -1 Inverse Trigonometric Functions Csc -1 Sec -1 Cot -1

28 Inverse Trigonometric Functions Function Domain Range f () = sin -1 f () = cos -1 f () = tan -1 f () = csc -1 f () = sec -1 f () = cot -1 1,1 1,1,,, 1 1,, 1 1,, 0,,,0 0, 0,, 0,

29 Alternate Names/Defintions for Inverse Trigonometric Functions Familiar Alternate Calculator f () = sin -1 f () = arcsin f () = sin -1 f () = cos -1 f () = arccos f () = tan -1 f () = arctan f () = cos -1 f () = tan -1 f () = csc -1 f () = arccsc f () = sin -1 1/ f () = sec -1 f () = arcsec f () = cos -1 1/ f () = cot -1 f () = arccot f () = -tan -1 + Arccot is different because it is always positive but tan can be negative.

30 Evaluate: Eample 1 sin 1 1 This epression asks us to find the angle whose sine is ½. Remember the range of the inverse of sine is,. 1 Since sin and 6, 6 sin 6 1 1

31 Eample Evaluate: csc 1 1 This epression asks us to find the angle whose cosecant is -1 (or sine is -1). Remember the range of the inverse of cosecant is,0 0,. Since csc 1 and 0, 1 csc 1

32 Eample 3 Evaluate: tan arcsin 1 3 The embedded epression asks us to find the angle whose sine is 1/3. Draw a picture (There are infinite varieties): 3 a Find the missing side length(s) a a 8 It does not even matter what the angle is, we only need to find: tan opp adj 1 4 Is the result positive or negative? Since arcsin 0, 1 3 tan 0

33 Eample 4 Evaluate: tan cos 1 ( 1 ) 6 The embedded epression asks us to find the angle whose cosine is -1/6. Draw a picture (There are infinite varieties): Ignore the negative for now. 6 1 Find the missing side length(s) o 1 6 o 35 o tan It does not even matter what the angle is, we only need to find: opp adj Is the result positive or negative? 1 1 Since cos, tan 0 6

34 Eample 3 Evaluate: costan 1 The embedded epression asks us to find the angle whose tangent is. Draw a possible picture (There are infinite varieties): h 1 Find the missing side length(s) 1 h h 1 It does not even matter what the angle is, we only need to find: cos adj hyp 1 1 Is the result positive or negative? Since - tan, 1 cos 0

35 White Board Challenge Evaluate without a calculator: sec 1 4

36 White Board Challenge Evaluate without a calculator: cot

37 White Board Challenge Evaluate without a calculator: arccos 3

38 White Board Challenge Evaluate without a calculator: cot csc 1 5 6

39 White Board Challenge Evaluate without a calculator: tan sin 1 1

THEOREM: THE CONSTANT RULE

THEOREM: THE CONSTANT RULE MATH /MYERS/ALL FORMULAS ON THIS REVIEW MUST BE MEMORIZED! DERIVATIVE REVIEW THEOREM: THE CONSTANT RULE The erivative of a constant function is zero. That is, if c is a real number, then c 0 Eample 1:

More information

TOTAL 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

TOTAL 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 information

Name: Math Analysis Chapter 3 Notes: Exponential and Logarithmic Functions

Name: Math Analysis Chapter 3 Notes: Exponential and Logarithmic Functions Name: Math Analysis Chapter 3 Notes: Eponential and Logarithmic Functions Day : Section 3-1 Eponential Functions 3-1: Eponential Functions After completing section 3-1 you should be able to do the following:

More information

Section Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.

Section 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 information

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Summer 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 information

A List of Definitions and Theorems

A 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 information

(ii) y = ln 1 ] t 3 t x x2 9

(ii) y = ln 1 ] t 3 t x x2 9 Study Guide for Eam 1 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its epression to be well-defined. Some eamples of the conditions are: What is inside

More information

CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.3: Derivative of a function pg.

CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.3: Derivative of a function pg. CALCULUS: Graphical,Numerical,Algebraic b Finne,Demana,Watts and Kenned Chapter : Derivatives.: Derivative of a function pg. 116-16 What ou'll Learn About How to find the derivative of: Functions with

More information

Trigonometric Functions. Copyright Cengage Learning. All rights reserved.

Trigonometric Functions. Copyright Cengage Learning. All rights reserved. 4 Trigonometric Functions Copyright Cengage Learning. All rights reserved. 4.3 Right Triangle Trigonometry Copyright Cengage Learning. All rights reserved. What You Should Learn Evaluate trigonometric

More information

CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS

CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS Course Number 5121 Department Mathematics Qualification Guidelines Successful completion of both semesters of Algebra

More information

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y. 90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y

More information

Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics. Unit C3. C3.1 Unit description

Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics. Unit C3. C3.1 Unit description Unit C3 Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics C3. Unit description Algebra and functions; trigonometry; eponentials and logarithms; differentiation;

More information

Name: Top Ten Things You ve Learned #10: Graphing Lines, Parabolas, and other Functions I. No Calculator: Sketch a graph of each equation

Name: Top Ten Things You ve Learned #10: Graphing Lines, Parabolas, and other Functions I. No Calculator: Sketch a graph of each equation Name: #: Graphing Lines, Parabolas, and other Functions I. No Calculator: Sketch a graph of each equation... - - - - - -.. 6. - - - - - - 7. 8. 9. - - - - - - ... a f - - - - - - II. Use a graphing calculator

More information

Math Section 4.3 Unit Circle Trigonometry

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

More information

Trigonometry Outline

Trigonometry Outline Trigonometr Outline Introduction Knowledge of the content of this outline is essential to perform well in calculus. The reader is urged to stud each of the three parts of the outline. Part I contains the

More information

Derivatives of Inverse Functions

Derivatives of Inverse Functions Derivatives of Inverse Functions Implicit differentiation enables us to determine the derivatives of inverse functions. determine the derivatives of arcsin, arccos, arctan, and ln. In this lecture, we

More information

6.5 Trigonometric Equations

6.5 Trigonometric Equations 6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or

More information

Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.

Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. Section 6.3 - Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve:

More information

AP Calculus AB SUMMER ASSIGNMENT. Dear future Calculus AB student

AP Calculus AB SUMMER ASSIGNMENT. Dear future Calculus AB student AP Calculus AB SUMMER ASSIGNMENT Dear future Calculus AB student We are ecited to work with you net year in Calculus AB. In order to help you be prepared for this class, please complete the summer assignment.

More information

Math Section 4.3 Unit Circle Trigonometry

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

More information

TRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)

TRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal) TRIG REVIEW NOTES Convert from radians to degrees: multiply by 0 180 Convert from degrees to radians: multiply by 0. 180 Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents

More information

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts

Math 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 information

5.3 Properties of Trigonometric Functions Objectives

5.3 Properties of Trigonometric Functions Objectives Objectives. Determine the Domain and Range of the Trigonometric Functions. 2. Determine the Period of the Trigonometric Functions. 3. Determine the Signs of the Trigonometric Functions in a Given Quadrant.

More information

Math 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts

Math 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 information

West Essex Regional School District. AP Calculus AB. Summer Packet

West Essex Regional School District. AP Calculus AB. Summer Packet West Esse Regional School District AP Calculus AB Summer Packet 05-06 Calculus AB Calculus AB covers the equivalent of a one semester college calculus course. Our focus will be on differential and integral

More information

Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts

Math 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 information

Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear.

Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear. Precalculus Review Functions to KNOW! 1. Polynomial Functions Types: General form Generic Graph and unique properties Constants Linear Quadratic Cubic Generalizations for Polynomial Functions - The domain

More information

Calculus with business applications, Lehigh U, Lecture 05 notes Summer

Calculus 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 information

AP Calculus Summer Packet

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

More information

The Chain Rule. This is a generalization of the (general) power rule which we have already met in the form: then f (x) = r [g(x)] r 1 g (x).

The Chain Rule. This is a generalization of the (general) power rule which we have already met in the form: then f (x) = r [g(x)] r 1 g (x). The Chain Rule This is a generalization of the general) power rule which we have already met in the form: If f) = g)] r then f ) = r g)] r g ). Here, g) is any differentiable function and r is any real

More information

Review of elements of Calculus (functions in one variable)

Review of elements of Calculus (functions in one variable) Review of elements of Calculus (functions in one variable) Mainly adapted from the lectures of prof Greg Kelly Hanford High School, Richland Washington http://online.math.uh.edu/houstonact/ https://sites.google.com/site/gkellymath/home/calculuspowerpoints

More information

4-3 Trigonometric Functions on the Unit Circle

4-3 Trigonometric Functions on the Unit Circle Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on

More information

Algebra/Pre-calc Review

Algebra/Pre-calc Review Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge

More information

7.3 Inverse Trigonometric Functions

7.3 Inverse Trigonometric Functions 58 transcendental functions 73 Inverse Trigonometric Functions We now turn our attention to the inverse trigonometric functions, their properties and their graphs, focusing on properties and techniques

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment Name: AP Calculus AB Summer Assignment Due Date: The beginning of class on the last class day of the first week of school. The purpose of this assignment is to have you practice the mathematical skills

More information

McKinney High School AP Calculus Summer Packet

McKinney High School AP Calculus Summer Packet McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work

More information

C. Finding roots of trinomials: 1st Example: x 2 5x = 14 x 2 5x 14 = 0 (x 7)(x + 2) = 0 Answer: x = 7 or x = -2

C. Finding roots of trinomials: 1st Example: x 2 5x = 14 x 2 5x 14 = 0 (x 7)(x + 2) = 0 Answer: x = 7 or x = -2 AP Calculus Students: Welcome to AP Calculus. Class begins in approimately - months. In this packet, you will find numerous topics that were covered in your Algebra and Pre-Calculus courses. These are

More information

Using the Definitions of the Trigonometric Functions

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

More information

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.

Using 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 information

Lesson 11 Inverse Trig Functions

Lesson 11 Inverse Trig Functions Unit : Trig Equations & Graphs Student ID #: Lesson 11 Inverse Trig Functions Goal: IX. use inverse trig to calculate an angle measure given a (special) ratio of sides Opener: Determine which values of

More information

Unit 10 Prerequisites for Next Year (Calculus)

Unit 10 Prerequisites for Next Year (Calculus) Unit 0 Prerequisites for Net Year (Calculus) The following Study Guide is the required INDEPENDENT review for you to work through for your final unit. You WILL have a test that covers this material after

More information

Unit 2 - The Trigonometric Functions - Classwork

Unit 2 - The Trigonometric Functions - Classwork Unit 2 - The Trigonometric Functions - Classwork Given a right triangle with one of the angles named ", and the sides of the triangle relative to " named opposite, adjacent, and hypotenuse (picture on

More information

1985 AP Calculus AB: Section I

1985 AP Calculus AB: Section I 985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of

More information

NOTES ON INVERSE TRIGONOMETRIC FUNCTIONS

NOTES ON INVERSE TRIGONOMETRIC FUNCTIONS NOTES ON INVERSE TRIGONOMETRIC FUNCTIONS MATH 5 (S). Definitions of Inverse Trigonometric Functions () y = sin or y = arcsin is the inverse function of y = sin on [, ]. The omain of y = sin = arcsin is

More information

Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions Chapter 5 Logarithmic, Exponential, an Other Transcenental Functions 5.1 The Natural Logarithmic Function: Differentiation 5.2 The Natural Logarithmic Function: Integration 5.3 Inverse Functions 5.4 Exponential

More information

FUNCTIONS AND MODELS

FUNCTIONS 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 information

TRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved.

TRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved. 12 TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. 12.2 The Trigonometric Functions Copyright Cengage Learning. All rights reserved. The Trigonometric Functions and Their Graphs

More information

Algebra/Trigonometry Review Notes

Algebra/Trigonometry Review Notes Algebra/Trigonometry Review Notes MAC 41 Calculus for Life Sciences Instructor: Brooke Quinlan Hillsborough Community College ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 1: POLYNOMIAL BASICS, POLYNOMIAL END BEHAVIOR,

More information

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (

More information

Analytic Trigonometry

Analytic Trigonometry 0 Analytic Trigonometry In this chapter, you will study analytic trigonometry. Analytic trigonometry is used to simplify trigonometric epressions and solve trigonometric equations. In this chapter, you

More information

2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2

2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2 29 April PreCalculus Final Review 1. Find the slope and y-intercept (if possible) of the equation of the line. Sketch the line: y = 3x + 13 2. Determine the domain of the function. Verify your result with

More information

Trigonometric Functions. Section 1.6

Trigonometric 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 information

1 Functions and Inverses

1 Functions and Inverses October, 08 MAT86 Week Justin Ko Functions and Inverses Definition. A function f : D R is a rule that assigns each element in a set D to eactly one element f() in R. The set D is called the domain of f.

More information

AP Calculus BC Summer Review

AP Calculus BC Summer Review AP Calculus BC 07-08 Summer Review Due September, 07 Name: All students entering AP Calculus BC are epected to be proficient in Pre-Calculus skills. To enhance your chances for success in this class, it

More information

Methods of Integration

Methods of Integration Methods of Integration Essential Formulas k d = k +C sind = cos +C n d = n+ n + +C cosd = sin +C e d = e +C tand = ln sec +C d = ln +C cotd = ln sin +C + d = tan +C lnd = ln +C secd = ln sec + tan +C cscd

More information

AP CALCULUS AB - SUMMER ASSIGNMENT 2018

AP CALCULUS AB - SUMMER ASSIGNMENT 2018 Name AP CALCULUS AB - SUMMER ASSIGNMENT 08 This packet is designed to help you review and build upon some of the important mathematical concepts and skills that you have learned in your previous mathematics

More information

Pre-Calculus (#9400)

Pre-Calculus (#9400) AASD MATHEMATICS CURRICULUM Pre-Calculus (#9400) Description This course is a foundation course for college-level mathematics classes. The topics covered include functions and their graphs; the circular

More information

Sect 7.4 Trigonometric Functions of Any Angles

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

More information

Calculus Summer TUTORIAL

Calculus Summer TUTORIAL Calculus Summer TUTORIAL The purpose of this tutorial is to have you practice the mathematical skills necessary to be successful in Calculus. All of the skills covered in this tutorial are from Pre-Calculus,

More information

June 9 Math 1113 sec 002 Summer 2014

June 9 Math 1113 sec 002 Summer 2014 June 9 Math 1113 sec 002 Summer 2014 Section 6.5: Inverse Trigonometric Functions Definition: (Inverse Sine) For x in the interval [ 1, 1] the inverse sine of x is denoted by either and is defined by the

More information

October 15 MATH 1113 sec. 51 Fall 2018

October 15 MATH 1113 sec. 51 Fall 2018 October 15 MATH 1113 sec. 51 Fall 2018 Section 5.5: Solving Exponential and Logarithmic Equations Base-Exponent Equality For any a > 0 with a 1, and for any real numbers x and y a x = a y if and only if

More information

Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach

Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach Section Notes Page Trigonometric Functions; Unit Circle Approach A unit circle is a circle centered at the origin with a radius of Its equation is x y = as shown in the drawing below Here the letter t

More information

Chapter 4 Trigonometric Functions

Chapter 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 information

As we know, the three basic trigonometric functions are as follows: Figure 1

As we know, the three basic trigonometric functions are as follows: Figure 1 Trigonometry Basic Functions As we know, the three basic trigonometric functions are as follows: sin θ = cos θ = opposite hypotenuse adjacent hypotenuse tan θ = opposite adjacent Where θ represents an

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment AP Calculus AB 07-08 Summer Assignment Welcome to AP Calculus AB! You are epected to complete the attached homework assignment during the summer. This is because of class time constraints and the amount

More information

Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters

Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters α( alpha), β ( beta), θ ( theta) as well as upper case letters A,B,

More information

Fox Lane High School Department of Mathematics

Fox Lane High School Department of Mathematics Fo Lane High School Department of Mathematics June 08 Hello Future AP Calculus AB Student! This is the summer assignment for all students taking AP Calculus AB net school year. It contains a set of problems

More information

1 Exponential Functions Limit Derivative Integral... 5

1 Exponential Functions Limit Derivative Integral... 5 Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................

More information

FUNDAMENTAL TRIGONOMETRIC INDENTITIES 1 = cos. sec θ 1 = sec. = cosθ. Odd Functions sin( t) = sint. csc( t) = csct tan( t) = tant

FUNDAMENTAL TRIGONOMETRIC INDENTITIES 1 = cos. sec θ 1 = sec. = cosθ. Odd Functions sin( t) = sint. csc( t) = csct tan( t) = tant NOTES 8: ANALYTIC TRIGONOMETRY Name: Date: Period: Mrs. Nguyen s Initial: LESSON 8.1 TRIGONOMETRIC IDENTITIES FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities sinθ 1 cscθ cosθ 1 secθ tanθ 1

More information

CK- 12 Algebra II with Trigonometry Concepts 1

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.

More information

AP Calculus AB Summer Assignment School Year

AP Calculus AB Summer Assignment School Year AP Calculus AB Summer Assignment School Year 018-019 Objective of the summer assignment: The AP Calculus summer assignment is designed to serve as a review for many of the prerequisite math skills required

More information

Summer Mathematics Prep

Summer Mathematics Prep Summer Mathematics Prep Entering Calculus Chesterfield County Public Schools Department of Mathematics SOLUTIONS Domain and Range Domain: All Real Numbers Range: {y: y } Domain: { : } Range:{ y : y 0}

More information

6.1 Antiderivatives and Slope Fields Calculus

6.1 Antiderivatives and Slope Fields Calculus 6. Antiderivatives and Slope Fields Calculus 6. ANTIDERIVATIVES AND SLOPE FIELDS Indefinite Integrals In the previous chapter we dealt with definite integrals. Definite integrals had limits of integration.

More information

Prentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Trigonometry (Grades 9-12)

Prentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Trigonometry (Grades 9-12) California Mathematics Content Standards for Trigonometry (Grades 9-12) Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric

More information

2. Pythagorean Theorem:

2. Pythagorean Theorem: Chapter 4 Applications of Trigonometric Functions 4.1 Right triangle trigonometry; Applications 1. A triangle in which one angle is a right angle (90 0 ) is called a. The side opposite the right angle

More information

Chapter 5 The Next Wave: MORE MODELING AND TRIGONOMETRY

Chapter 5 The Next Wave: MORE MODELING AND TRIGONOMETRY ANSWERS Mathematics (Mathematical Analysis) page 1 Chapter The Next Wave: MORE MODELING AND TRIGONOMETRY NW-1. TI-8, points; Casio, points a) An infinite number of them. b) 17p, - 7p c) Add p n to p, p

More information

A. Incorrect! For a point to lie on the unit circle, the sum of the squares of its coordinates must be equal to 1.

A. 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 information

Precalculus Midterm Review

Precalculus Midterm Review Precalculus Midterm Review Date: Time: Length of exam: 2 hours Type of questions: Multiple choice (4 choices) Number of questions: 50 Format of exam: 30 questions no calculator allowed, then 20 questions

More information

Inverse Trigonometric Functions. September 5, 2018

Inverse Trigonometric Functions. September 5, 2018 Inverse Trigonometric Functions September 5, 08 / 7 Restricted Sine Function. The trigonometric function sin x is not a one-to-one functions..0 0.5 Π 6, 5Π 6, Π Π Π Π 0.5 We still want an inverse, so what

More information

Trigonometric substitutions (8.3).

Trigonometric substitutions (8.3). Review for Eam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Eam covers: 7.4, 7.6, 7.7, 8-IT, 8., 8.2. Solving differential equations

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=

More information

Exercise Set 4.3: Unit Circle Trigonometry

Exercise Set 4.3: Unit Circle Trigonometry Eercise Set.: Unit Circle Trigonometr Sketch each of the following angles in standard position. (Do not use a protractor; just draw a quick sketch of each angle. Sketch each of the following angles in

More information

Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios

Exercise 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 information

Differential Calculus

Differential Calculus Differential Calculus. Compute the derivatives of the following functions a() = 4 3 7 + 4 + 5 b() = 3 + + c() = 3 + d() = sin cos e() = sin f() = log g() = tan h() = 3 6e 5 4 i() = + tan 3 j() = e k()

More information

Chapter 8: Trig Equations and Inverse Trig Functions

Chapter 8: Trig Equations and Inverse Trig Functions Haberman MTH Section I: The Trigonometric Functions Chapter 8: Trig Equations and Inverse Trig Functions EXAMPLE : Solve the equations below: a sin( t) b sin( t) 0 sin a Based on our experience with the

More information

CHAPTER 5: Analytic Trigonometry

CHAPTER 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 information

Calculus 1 (AP, Honors, Academic) Summer Assignment 2018

Calculus 1 (AP, Honors, Academic) Summer Assignment 2018 Calculus (AP, Honors, Academic) Summer Assignment 08 The summer assignments for Calculus will reinforce some necessary Algebra and Precalculus skills. In order to be successful in Calculus, you must have

More information

CHAPTER 3 DERIVATIVES (continued)

CHAPTER 3 DERIVATIVES (continued) CHAPTER 3 DERIVATIVES (continue) 3.3. RULES FOR DIFFERENTIATION A. The erivative of a constant is zero: [c] = 0 B. The Power Rule: [n ] = n (n-1) C. The Constant Multiple Rule: [c *f()] = c * f () D. The

More information

AP Calculus AB Summer Assignment

AP Calculus AB Summer Assignment AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted

More information

Trigonometry Exam 2 Review: Chapters 4, 5, 6

Trigonometry Exam 2 Review: Chapters 4, 5, 6 Trig Exam Review F07 O Brien Trigonometry Exam Review: Chapters,, 0% of the questions on Exam will come from Chapters through. The other 70 7% of the exam will come from Chapters through. There may be

More information

Basic Math Formulas. Unit circle. and. Arithmetic operations (ab means a b) Powers and roots. a(b + c)= ab + ac

Basic Math Formulas. Unit circle. and. Arithmetic operations (ab means a b) Powers and roots. a(b + c)= ab + ac Basic Math Formulas Arithmetic operations (ab means ab) Powers and roots a(b + c)= ab + ac a+b c = a b c + c a b + c d = ad+bc bd a b = a c d b d c a c = ac b d bd a b = a+b ( a ) b = ab (y) a = a y a

More information

Summer AP Assignment Coversheet Falls Church High School

Summer AP Assignment Coversheet Falls Church High School Summer AP Assignment Coversheet Falls Church High School Course: AP Calculus AB Teacher Name/s: Veronica Moldoveanu, Ethan Batterman Assignment Title: AP Calculus AB Summer Packet Assignment Summary/Purpose:

More information

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.

More information

Troy High School AP Calculus Summer Packet

Troy High School AP Calculus Summer Packet Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by

More information

Chapter 7, Continued

Chapter 7, Continued Math 150, Fall 008, c Benjamin Aurispa Chapter 7, Continued 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas Double-Angle Formulas Formula for Sine: Formulas for Cosine: Formula for Tangent: sin

More information

Pre Calc. Trigonometry.

Pre Calc. Trigonometry. 1 Pre Calc Trigonometry 2015 03 24 www.njctl.org 2 Table of Contents Unit Circle Graphing Law of Sines Law of Cosines Pythagorean Identities Angle Sum/Difference Double Angle Half Angle Power Reducing

More information

Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.2 Solving Quadratic Equations

Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.2 Solving Quadratic Equations Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.1.1 Solve Simple Equations Involving Absolute Value 0.2 Solving Quadratic Equations 0.2.1 Use the

More information

Transition to College Math

Transition to College Math Transition to College Math Date: Unit 3: Trigonometr Lesson 2: Angles of Rotation Name Period Essential Question: What is the reference angle for an angle of 15? Standard: F-TF.2 Learning Target: Eplain

More information

MTH 112: Elementary Functions

MTH 112: Elementary Functions 1/19 MTH 11: Elementary Functions Section 6.6 6.6:Inverse Trigonometric functions /19 Inverse Trig functions 1 1 functions satisfy the horizontal line test: Any horizontal line crosses the graph of a 1

More information

Core Mathematics 2 Unit C2 AS

Core Mathematics 2 Unit C2 AS Core Mathematics 2 Unit C2 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics C2.1 Unit description Algebra and functions; coordinate geometry in the (, y) plane; sequences

More information