1 Fundamental Concepts From Algebra & Precalculus
|
|
|
- Peter Johnson
- 5 years ago
- Views:
Transcription
1 Fundamental Concepts From Algebra & Precalculus. Review Exercises.. Simplify eac expression.. 5 7) [ 5)) ]. ) 5) 7) [ 5) 8 6)] [ ] ) 8) [ )6)] 7) ) 6 8 ) ) 7) x 5x 5 6x 0. ) 5x 9 5x 5x ). 8x 6 y ) x 5 6 y ) 6. x y ) x 5 6 y ).. x y 7 y 5 8x y 6 ) ) x y y ) 5. xy y 7x ) 6. x y 5 )9x 6 y) / x y / 7. x y ) xy ) 8. 7x 6 y ) 8 x y ) x6 5 y 5 ) 5.. Factor te expression completely and simplify). 9. 6y + y 0. 9x y + 6xy. x. y. 8a b 6. x z 7 6. x + x x + 5x u u y 6y + y 0. x x + 6x. x x x 7x +. 6x 7xy 5y. 6x 9y 5. 6x 6x x 5x 7. x + x x 8. 5 x + 5x 0x 0 9. x + x 5x y y. 6x y 7 y x + 9. x x ) xx ) 5x + 5. x x + ))x + ) ) + x + ) x ). 6x 5) )x + )x) + x + ) )6x 5) 6)
2 5. 5x + ) x)x ) + x + ) 5 )x ) 6. x ) )9 x ) 5 + x ) 5)9 x ) 6x ) 7. x ) )x + ) / + x ) )x + ) / 8. x + ) / x x + ) / 9. x x ) + xx ) 6x x + y) 00x + y) 5. x 7x x 5x x +.. Complete te Square witout canging te value of te expression). 5. x + 6x 5. t 9t 55. y 8y x + x x x x + x Perform te Indicated Operations and completely simplify) x + 6. x x x x 0 8 y 0y 6 5y 6. x x 9 + xx ) 6. x x x 0 + x x x x + x + x x 8 x ) x x + x x + ) ) + ) x + x x + x 70. x + x + x x + 7. r s + s r r s s r 7. x + x + + x x + 7. x x x + 6 x x x 9 x x + 9x 6x + x x 5x + x 75. x 6 x + x 0 x + 0x + x + 8x x 5x x x x 6x x + x 7x x + ) x + ) x x) 78. x x + ) 79. x + ) )x 5)) + x 5) )x + ) ) 80. x + ) 6 )x 5) ) + 6x + ) 5 )x 5) 8. x + 9) )x + 6) + x + 9) x)x + 6) 8. x)x ) x )x ) x) [ x ) ] x x 5) x )x 5) x) [ x 5) ] 8. x x ) x ) x ) x) [ x ) ] 86. x + ) x )x + ) x) [ x + ) ] )x + ) )x + ) x + ) )x + ) ) [ x + ) ]
3 ..5 Rationalize te Numerator and completely simplify). 87. x y x y 88. x + ) + x +..6 Solve te Equation. 89. x x = x + 5x = 9. x = x ) 9. x + x 0) = 5x ) 9. x 6x + = 9. x + )x + ) = 95. x = 5x + 8x 96. x + 8x = 9x 97. x x = x + 8x + = x x + = x + ) 5 x ) = 5 0. x )x + ) = 0. x ) x + ) = 0. z z + ) z + ) = 0 0. x + x x x = x + = 6 + x x + 7 = x + x x = 5x + x S = x 5 x = x + 5 6t t + t = t + p q + p q) for q. f = p + q for q. R = + R + R + R for R. 0 8x = x. x x = 5. x = + 5x x + = x 7. x 7x + = 0 given tat is a root 8. x + 7x x 8 = 0 given tat is a root..7 Solve te Absolute Value Inequality - write te solution as an interval. 9. x + 5 < 7 0. < 6 x ). x > 9. < x + 7. x +. x Polynomial Division 5. x 0x x + ) x + ) 6. x ) x ) 7. x + 0x + 8x 7x + ) x) 8. x + 6x + 6x 0x ) x ) 9. x + 6x + 6x 0x x + x 0. x + x x 5 x + x + 5. x 5 8x + x + x x +. x + 6x + x x +
4 ..9 Function Notation - evaluate te function as indicated, and simplify.. fx) = 7x a) f ) b) f ) c) ft) f ) d) fx + ). fx) = x x 5 a) f0) b) f 5 ) c) f) f ) d) fx + ) 5. gx) = x + x + 6. fx) = a) g0) b) g ) c) g) + g 5) d) fx ) { x + 8, x < 0 0 x, x 0 a) f) b) f 0) c) f0) d) f6) f ), x < 7. fx) = 5x, x < 0x, x a) f ) b) f) c) f) + f) d) f 5) + f ) 8. fx) = x + 5 a) fx + ) f) b) fx + ) fx) c) fx ) f) x d) fx + ) fx) In exercises 9-, find and simplify te difference quotient fx + ) fx) for eac function. 9. fx) = x + x fx) = x x +. fx) = x. fx) = x..0 Function Operations In exercises -6, below find a) f g)x) b) g f)x) c) f g)). fx) = x +, gx) = x. fx) = x, gx) = x + 5. fx) = 6x, gx) = x fx) = 5x, gx) = x + x In exercises 7-50, A) Find an equation for f, te inverse function. B) Verify tat your equation is correct by sowing tat ff x)) = x and f fx)) = x. 7. fx) = x 8. fx) = 8x + 9. fx) = x fx) = x x + Solve eac inequality in exercises 5-5, and express eac solution set in interval notation. 5. x x 0 5. x + x 5. x x + x x 0 5. x x x x 0 < 0 algebra review.dvi
5 . Answers to Review Exercises.. Answers for: Simplify eac expression x x 0. 75x 7. x y. y x). x y. y 9 5. y x 6. x y 5 7. x 8y 8. 9 x y.. Answers for: Factor te expression completely. 9. y y + ) 0. xyx + y). x + )x ). + y) y). 9a + b )9a b ). x + )x + )x ) 5. z )z + 6z + 9) 6. x + ) 7. 55x + )x + ) 8. u + 7)u ) 9. y 6)y + ) 0. x )x + ). x 5)x 9). x )x ). x 5y)x + y). 6x + 7y)6x 7y) 5. 8x ) 6. 5xx + )x ) 7. x + )x + )x ) 8. 5 x + ) x ) 9. x 5)x + 5)x + ) 0. y y + )y ). x + )x )y ). x )6x 5)x + ). x + ) 6x 7x + 6). x + )6x 5) x 0x + 6) 5. x + ) x ) x + 0x + 8) 6. 8x ) 9 x ) x 5x ) 7. 7x ) x + 5) x + ) / 8. x + 9 x + ) / 9. x )x + )x ) 50. x + y) x + y 0)x + y + 0) 5. x )x )x + ) 5. x + )x )x )
6 .. Answers for: Complete te Square 5. x + ) 9 5. ) t y ) x ) x ) x ).. Answers for: Perform te Indicated Operations 59. 5x + 0 x x x 0 6. x + 6 x 8 6. y + 5y 6. x + x + 9 xx 9) 6. x 0x + x + )x )x 5) 65. 8x + 0x 6 x x + ) 66. x xx + ) x + 5 x sr r s x x x + + )x + ), x +, xx + ) x 5x + ) x xx + ) 79. x 5)8x + ) x + ) 80. x + ) 5 9x 89) x 5) 8. x + 9) x + x 9) x + 6) 8. xx + ) x ) x x + ) x 5) 5 8. x + x + ) 85. xx ) x ) 6x x + ) x + )..5 Answers for: Rationalize te Numerator 87. x + y) x + y), x y 88. x + ) + + x +, 0
7 ..6 Answers for: Solve te Equation 89. x = or x = 90. x = or x = 9. No solutions - false 9. All R reals) 9. x = 9. x = 5 or x = 95. x =, x = 0, or x = x = 0, x = 7, or x = 97. x = ± x = ± No real solutions 00. x = ± x = or x = 0. x = or x = 0. z = or z = 0. x =, x =, x = 0, or x = 05. x = 06. x = x = 08. t = 09. No solution x ± ) 0. q = p S) S p). q = fp p f RR R. R = R R RR RR. x = x=-0 is extraneous). x = x=-6 is extraneous) 5. x = 9 6. x = 8 7. x = or x = ± 5 8. x =, x =, or x =..7 Answers for: Solve te Absolute Value Inequality 9., ) 0. 7, 9)., 8) or 6, )., ). [0, 8]., 5] or [, )..8 Answers for: Polynomial Division 5. x x x + 6. x + x + x 7. 5x x + x 7 x 8. x + 5 x x x ) 9. x + x + 0. x. x x + + x x +. x + x x + 0 x + x + 5
8 ..9 Answers for: Function Notation. a) 0 b) c) 0 7t d) 7x a) 0 b) c) 5 d) a) b) Undefined c) d) x + x x x a) b) c) 0 d) 8 7. a) b) 0 c) d) 8. a) x + 8x b) x + c) x x 0 d) x +, x + +, 0 0. x, 0. x x x + ), 0. x + x, 0..0 Answers for: Function Operations. a) fgx)) = 6x 8x + b) gfx)) = x + c) fg)) =. a) fgx)) = x + b) gfx)) = x + c) fg)) = 5. a) fgx)) = x b) gfx)) = x c) fg)) = 6. a) fgx)) = 5x + 0x 7 b) gfx)) = 5x + 0x c) fg)) = 8 7. f x) = x + 8. f x) = x 9. f x) = x f x) = x x, x 5. [ 5, ) 5., ], ) 5., ) S [, ] S, ) 5., ), 5) algebra review.dvi
Math 110 Midterm 1 Study Guide October 14, 2013
Name: For more practice exercises, do the study set problems in sections: 3.4 3.7, 4.1, and 4.2. 1. Find the domain of f, and express the solution in interval notation. (a) f(x) = x 6 D = (, ) or D = R
Combining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)
Precalculus Test 2 Practice Questions Page 1. Note: You can expect other types of questions on the test than the ones presented here!
Precalculus Test 2 Practice Questions Page Note: You can expect oter types of questions on te test tan te ones presented ere! Questions Example. Find te vertex of te quadratic f(x) = 4x 2 x. Example 2.
Chapter 7 Algebra 2 Honors 1 Polynomials
Chapter 7 Algebra 2 Honors 1 Polynomials Polynomial: - - Polynomials in one variable Degree Leading coefficient f(x) = 3x 3 2x + 4 f(2) = f(t) = f(y -1) = 3f(x) = Using your graphing calculator sketch/graph
Note: The actual exam will consist of 20 multiple choice questions and 6 show-your-work questions. Extra questions are provided for practice.
College Algebra - Unit 2 Exam - Practice Test Note: The actual exam will consist of 20 multiple choice questions and 6 show-your-work questions. Extra questions are provided for practice. MULTIPLE CHOICE.
Chapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010
Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression
MAT 1339-S14 Class 2
MAT 1339-S14 Class 2 July 07, 2014 Contents 1 Rate of Cange 1 1.5 Introduction to Derivatives....................... 1 2 Derivatives 5 2.1 Derivative of Polynomial function.................... 5 2.2 Te
Chapter 8. Exploring Polynomial Functions. Jennifer Huss
Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial
Ron Paul Curriculum 9th Grade Mathematics Problem Set #2
Problem Set # Find the value of each expression below, and then name the sets of numbers to which each value belongs:.. + 6.8. 7. 8 (.6). 8+ 5 5. 9/ Name the property illustrated by each equation: 6. z
PreCalculus: Semester 1 Final Exam Review
Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain
Function Notation We use the f(x) (read f of x) notation to represent a function. E.g. f(x) = 3x 1 Here, f is the name of the function, x is the
Functions Informal definition of a function: A function between two sets is a rule that assigns to each member in the first set (called the domain) one and only one member in the second set (called the
MATH 150 TOPIC 3 FUNCTIONS: COMPOSITION AND DIFFERENCE QUOTIENTS
Mat 50 T3-Functions and Difference Quotients Review Page MATH 50 TOPIC 3 FUNCTIONS: COMPOSITION AND DIFFERENCE QUOTIENTS I. Composition of Functions II. Difference Quotients Practice Problems Mat 50 T3-Functions
Solutions Manual for Precalculus An Investigation of Functions
Solutions Manual for Precalculus An Investigation of Functions David Lippman, Melonie Rasmussen 1 st Edition Solutions created at Te Evergreen State College and Soreline Community College 1.1 Solutions
MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3
MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 JAMIE HADDOCK 1. Agenda Functions Composition Graphs Average Rate of Change..............................................................................................................
MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points
MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim
Skills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).
8 Further theory of function limits and continuity
8 Further theory of function limits and continuity 8.1 Algebra of limits and sandwich theorem for real-valued function limits The following results give versions of the algebra of limits and sandwich theorem
Intermediate Algebra Study Guide
Chapter 1 Intermediate Algebra Study Guide 1. Simplify the following. (a) ( 6) + ( 4) ( 9) (b) ( 7) ( 6)( )( ) (c) 8 5 9 (d) 6x(xy x ) x (y 6x ) (e) 7x {6 [8 (x ) (6 x)]} (f) Evaluate x y for x =, y =.
( u. Algebra Review Day 1. PreCalculus y _2z0l1W7] LKOuGtzaE lsmotfbtowtacrbec elqloci.y H VAvlUle LrfiKgphUthsA qrdedsmemrmvpebdw. Name.
![( u. Algebra Review Day 1. PreCalculus y _2z0l1W7] LKOuGtzaE lsmotfbtowtacrbec elqloci.y H VAvlUle LrfiKgphUthsA qrdedsmemrmvpebdw. Name.](/thumbs/96/126601907.jpg "( u. Algebra Review Day 1. PreCalculus y _2z0l1W7] LKOuGtzaE lsmotfbtowtacrbec elqloci.y H VAvlUle LrfiKgphUthsA qrdedsmemrmvpebdw. Name.")
PreCalculus y _z0l1w7] LKOuGtzaE lsmotfbtowtacrbec elqloci.y H VAvlUle LrfiKgphUthsA qrdedsmemrmvpebdw. Algebra Review Day 1 Simplify. Name Date Period 1) 5 (64xy ) x y Simplify. Your answer should contain
FUNCTIONS - PART 2. If you are not familiar with any of the material below you need to spend time studying these concepts and doing some exercises.
Introduction FUNCTIONS - PART 2 This handout is a summary of the basic concepts you should understand and be comfortable working with for the second math review module on functions. This is intended as
Higher Portfolio Quadratics and Polynomials
Higher Portfolio Quadratics and Polynomials Higher 5. Quadratics and Polynomials Section A - Revision Section This section will help you revise previous learning which is required in this topic R1 I have
Continuity. Example 1
Continuity MATH 1003 Calculus and Linear Algebra (Lecture 13.5) Maoseng Xiong Department of Matematics, HKUST A function f : (a, b) R is continuous at a point c (a, b) if 1. x c f (x) exists, 2. f (c)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) x 8. C) y = x + 3 2
Precalculus Fall Final Exam Review Name Date Period MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplify the expression. Assume that the variables
for every x in the gomain of g
Section.7 Definition of Inverse Function Let f and g be two functions such that f(g(x)) = x for every x in the gomain of g and g(f(x)) = x for every x in the gomain of f Under these conditions, the function
1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x)
Calculus. Gradients and te Derivative Q f(x+) δy P T δx R f(x) 0 x x+ Let P (x, f(x)) and Q(x+, f(x+)) denote two points on te curve of te function y = f(x) and let R denote te point of intersection of
ALGEBRA AND TRIGONOMETRY REVIEW by Dr TEBOU, FIU. A. Fundamental identities Throughout this section, a and b denotes arbitrary real numbers.
ALGEBRA AND TRIGONOMETRY REVIEW by Dr TEBOU, FIU A. Fundamental identities Trougout tis section, a and b denotes arbitrary real numbers. i) Square of a sum: (a+b) =a +ab+b ii) Square of a difference: (a-b)
Pre-Calculus Summer Math Packet 2018 Multiple Choice
Pre-Calculus Summer Math Packet 208 Multiple Choice Page A Complete all work on separate loose-leaf or graph paper. Solve problems without using a calculator. Write the answers to multiple choice questions
CHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
Spring Nikos Apostolakis
Spring 07 Nikos Apostolakis Review of fractions Rational expressions are fractions with numerator and denominator polynomials. We need to remember how we work with fractions (a.k.a. rational numbers) before
Solution: f( 1) = 3 1)
Gateway Questions How to Evaluate Functions at a Value Using the Rules Identify the independent variable in the rule of function. Replace the independent variable with big parenthesis. Plug in the input
Partial Derivatives October 2013
Partial Derivatives 14.3 02 October 2013 Derivative in one variable. Recall for a function of one variable, f (a) = lim h 0 f (a + h) f (a) h slope f (a + h) f (a) h a a + h Partial derivatives. For a
( ) y 2! 4. ( )( y! 2)
1. Dividing: 4x3! 8x 2 + 6x 2x 5.7 Division of Polynomials = 4x3 2x! 8x2 2x + 6x 2x = 2x2! 4 3. Dividing: 1x4 + 15x 3! 2x 2!5x 2 = 1x4!5x 2 + 15x3!5x 2! 2x2!5x 2 =!2x2! 3x + 4 5. Dividing: 8y5 + 1y 3!
CHAPTER 4: Polynomial and Rational Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).
MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and
Linear equations are equations involving only polynomials of degree one.
Chapter 2A Solving Equations Solving Linear Equations Linear equations are equations involving only polynomials of degree one. Examples include 2t +1 = 7 and 25x +16 = 9x 4 A solution is a value or a set
1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point
MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note
MATH 1A Midterm Practice September 29, 2014
MATH A Midterm Practice September 9, 04 Name: Problem. (True/False) If a function f : R R is injective, ten f as an inverse. Solution: True. If f is injective, ten it as an inverse since tere does not
DIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3. The Product and Quotient Rules In this section, we will learn about: Formulas that enable us to differentiate new functions formed from old functions by
Example: Limit definition. Geometric meaning. Geometric meaning, y. Notes. Notes. Notes. f (x, y) = x 2 y 3 :
Partial Derivatives 14.3 02 October 2013 Derivative in one variable. Recall for a function of one variable, f (a) = lim h 0 f (a + h) f (a) h slope f (a + h) f (a) h a a + h Partial derivatives. For a
3. Use absolute value notation to write an inequality that represents the statement: x is within 3 units of 2 on the real line.
PreCalculus Review Review Questions 1 The following transformations are applied in the given order) to the graph of y = x I Vertical Stretch by a factor of II Horizontal shift to the right by units III
Unit 7 Inverse and Radical Functions
Unit 7 Inverse and Radical Functions Test date: Name: By the end of this unit, you will be able to Find the inverse of a function and identify its domain and range Compose two functions and identify the
Chapter 1: Precalculus Review
: Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,
1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math 0320 Final Exam Review
Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:
Remember, you may not use a calculator when you take the assessment test.
Elementary Algebra problems you can use for practice. Remember, you may not use a calculator when you take the assessment test. Use these problems to help you get up to speed. Perform the indicated operation.
Section Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
Algebra 2 Honors: Final Exam Review
Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt
1 FUNCTIONS _ 5 _ 1.0 RELATIONS
1 FUNCTIONS 1.0 RELATIONS Notes : (i) Four types of relations : one-to-one many-to-one one-to-many many-to-many. (ii) Three ways to represent relations : arrowed diagram set of ordered pairs graph. (iii)
Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006
Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2
11.5. The Chain Rule. Introduction. Prerequisites. Learning Outcomes
The Chain Rule 11.5 Introduction In this Section we will see how to obtain the derivative of a composite function (often referred to as a function of a function ). To do this we use the chain rule. This
AFM Review Test Review
Name: Class: Date: AFM Review Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. What are the solutions of the inequality?. q + (q ) > 0 q < 3 q
Math 421, Homework #9 Solutions
Math 41, Homework #9 Solutions (1) (a) A set E R n is said to be path connected if for any pair of points x E and y E there exists a continuous function γ : [0, 1] R n satisfying γ(0) = x, γ(1) = y, and
Section 5.1 Composite Functions
Section 5. Composite Functions Objective #: Form a Composite Function. In many cases, we can create a new function by taking the composition of two functions. For example, suppose f(x) x and g(x) x +.
x y More precisely, this equation means that given any ε > 0, there exists some δ > 0 such that
Chapter 2 Limits and continuity 21 The definition of a it Definition 21 (ε-δ definition) Let f be a function and y R a fixed number Take x to be a point which approaches y without being equal to y If there
2.3 Product and Quotient Rules
.3. PRODUCT AND QUOTIENT RULES 75.3 Product and Quotient Rules.3.1 Product rule Suppose tat f and g are two di erentiable functions. Ten ( g (x)) 0 = f 0 (x) g (x) + g 0 (x) See.3.5 on page 77 for a proof.
Further Topics in Functions
Chapter 5 Further Topics in Functions 5. Function Composition Before we embark upon any further adventures with functions, we need to take some time to gather our thoughts and gain some perspective. Chapter
Section 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
171S2.2q The Algebra of Functions. February 13, MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3
Chapter 4: Radicals and Complex Numbers
Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y
Homework 5 Solutions
Homework 5 Solutions ECS 0 (Fall 17) Patrice Koehl koehl@cs.ucdavis.edu February 8, 019 Exercise 1 a) Show that the following statement is true: If there exist two integers n and m such that n + n + 1
3. Solve the following inequalities and express your answer in interval notation.
Youngstown State University College Algebra Final Exam Review (Math 50). Find all Real solutions for the following: a) x 2 + 5x = 6 b) 9 x2 x 8 = 0 c) (x 2) 2 = 6 d) 4x = 8 x 2 e) x 2 + 4x = 5 f) 36x 3
Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i
Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add
Summer Work for students entering PreCalculus
Summer Work for students entering PreCalculus Name Directions: The following packet represent a review of topics you learned in Algebra 1, Geometry, and Algebra 2. Complete your summer packet on separate
NAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer.
2-1 Power and Radical Functions What You ll Learn Scan Lesson 2-1. Predict two things that you expect to learn based on the headings and Key Concept box. 1. 2. Lesson 2-1 Active Vocabulary extraneous solution
171S4.3 Polynomial Division; The Remainder and Factor Theorems. March 24, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
171S4.3 Polynomial Division; The Remainder and Factor Theorems. October 26, Polynomial Division; The Remainder and Factor Theorems
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
Precalculus 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
MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE. A rational expression is just a fraction involving polynomials, for example 3x2 2
MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE SECTION 1.2: PRECALCULUS REVIEW II Practice: 3, 7, 13, 17, 19, 23, 29, 33, 43, 45, 51, 57, 69, 81, 89 1. Rational Expressions and Other Algebraic Fractions A rational
Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions
Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions Date: Objectives: SWBAT (Simplify Rational Expressions) Main Ideas: Assignment: Rational Expression is an expression that can be written
Rational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions
Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W
Definition of the Derivative
Te Limit Definition of te Derivative Tis Handout will: Define te limit grapically and algebraically Discuss, in detail, specific features of te definition of te derivative Provide a general strategy of
Preface. The version of the textbook that has been modified specifically for Math 1100 at MU is available at:
Preface This manual of notes and worksheets was developed by Teri E. Christiansen at the University of Missouri- Columbia. The goal was to provide students in Math 1100 (College Algebra) a resource to
Advanced Algebra 2 - Assignment Sheet Chapter 1
Advanced Algebra - Assignment Sheet Chapter #: Real Numbers & Number Operations (.) p. 7 0: 5- odd, 9-55 odd, 69-8 odd. #: Algebraic Expressions & Models (.) p. 4 7: 5-6, 7-55 odd, 59, 6-67, 69-7 odd,
Limit Theorems. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Limit Theorems
Limit s MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Bounded Functions Definition Let A R, let f : A R, and let c R be a cluster point of A. We say that f is bounded
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational Functions; Task 5.3.1
TASK 5.3.: RATIONAL FUNCTIONS AND THEIR RECIPROCALS Solutions Rational functions appear frequently in business, science, engineering, and medical applications. This activity explores some aspects of rational
Derivatives. if such a limit exists. In this case when such a limit exists, we say that the function f is differentiable.
Derivatives 3. Derivatives Definition 3. Let f be a function an a < b be numbers. Te average rate of cange of f from a to b is f(b) f(a). b a Remark 3. Te average rate of cange of a function f from a to
Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!
1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3 x 9 D) 27. y 4 D) -8x 3 y 6.
Precalculus Review - Spring 018 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplify the exponential expression. Assume that variables represent
CALCULUS 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
PMI Unit 2 Working With Functions
Vertical Shifts Class Work 1. a) 2. a) 3. i) y = x 2 ii) Move down 2 6. i) y = x ii) Move down 1 4. i) y = 1 x ii) Move up 3 7. i) y = e x ii) Move down 4 5. i) y = x ii) Move up 1 Vertical Shifts Homework
Student: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
Algebra II Through Competitions Chapter 7 Function Composition and Operations
. FUNCTIONS. Definition A function is a relationship between the independent variable x and dependent variable y. Each value of x corresponds exactly one value of y. Note two different values of x can
Numerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
Math Placement Test Review Sheet Louisburg College _ Summer = c = d. 5
1. Preform indicated operations with fractions and decimals: a. 7 14 15 = b. 2 = c. 5 + 1 = d. 5 20 4 5 18 12 18 27 = 2. What is the least common denominator of fractions: 8 21 and 9 14. The fraction 9
Examples 2: Composite Functions, Piecewise Functions, Partial Fractions
Examples 2: Composite Functions, Piecewise Functions, Partial Fractions September 26, 206 The following are a set of examples to designed to complement a first-year calculus course. objectives are listed
Sect Polynomial and Rational Inequalities
158 Sect 10.2 - Polynomial and Rational Inequalities Concept #1 Solving Inequalities Graphically Definition A Quadratic Inequality is an inequality that can be written in one of the following forms: ax
4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.
Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra
Exam 1 Review Solutions
Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),
8. Limit Laws. lim(f g)(x) = lim f(x) lim g(x), (x) = lim x a f(x) g lim x a g(x)
8. Limit Laws 8.1. Basic Limit Laws. If f and g are two functions and we know the it of each of them at a given point a, then we can easily compute the it at a of their sum, difference, product, constant
8 Building New Functions from Old Ones
Arkansas Tech University MATH 2243: Business Calculus Dr. Marcel B. Finan 8 Building New Functions from Old Ones In this section we discuss various ways for building new functions from old ones. New functions
PreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College
PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions
The most factored form is usually accomplished by common factoring the expression. But, any type of factoring may come into play.
MOST FACTORED FORM The most factored form is the most factored version of a rational expression. Being able to find the most factored form is an essential skill when simplifying the derivatives found using
Princeton High School
Princeton High School Mathematics Department PreCalculus Summer Assignment Summer assignment vision and purpose: The Mathematics Department of Princeton Public Schools looks to build both confidence and
x 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
Algebra II Final Examination Mr. Pleacher Name (A) - 4 (B) 2 (C) 3 (D) What is the product of the polynomials (4c 1) and (3c + 5)?
Algebra II Final Examination Mr. Pleacher Name I. Multiple Choice 1. If f( x) = x 1, then f ( 3) = (A) - 4 (B) (C) 3 (D) 4. What is the product of the polynomials (4c 1) and (3c + 5)? A) 7c 4 B) 1c + 17c
Inverse Variation. y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0.
Inverse Variation y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0. Inverse variation xy = k or y = k where k is not equal to 0. x Identify whether the following functions
Math Boot Camp Functions and Algebra
Fall 017 Math Boot Camp Functions and Algebra FUNCTIONS Much of mathematics relies on functions, the pairing (relation) of one object (typically a real number) with another object (typically a real number).
Chapter 2 Prerequisite Skills BLM Evaluate Functions 1. Given P(x) = x 4 3x 2 + 5x 11, evaluate.
Chapter Prerequisite Skills BLM 1.. Evaluate Functions 1. Given P(x) = x 4 x + 5x 11, evaluate. a) P( ) b) P() c) P( 1) 1 d) P 4 Simplify Expressions. Expand and simplify. a) (x x x + 4)(x 1) + b) (x +
CALCULUS II - Self Test
175 2- CALCULUS II - Self Test Instructor: Andrés E. Caicedo November 9, 2009 Name These questions are divided into four groups. Ideally, you would answer YES to all questions in group A, to most questions
Chapter 2. Polynomial and Rational Functions. 2.6 Rational Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter Polynomial and Rational Functions.6 Rational Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Find the domains of rational functions. Use arrow notation.