Mathematics 31 Precalculus and Limits


 Dayna Lawrence
 1 years ago
 Views:
Transcription
1 Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals and radical epressions, limits of functions and limits of geometric sequences. General Outcomes: Perform operations on polynomials. Use eact values, arithmetic operations and algebraic operations on real numbers to solve problems. Solve coordinate geometry problems involving lines and line segments. Give eamples of the differences between intuitive and rigorous proofs in the contet of limits. Epress final algebraic and answers in a variety of equivalent forms, with the form chosen to be the most suitable form for the task at hand. Specific Outcomes Perform operations on irrational numbers of monomial and binomial form, using eact values. Factor polynomial epressions of the form a + b + c, a b y, difference and sum of squares. Determine the equation of a line, given information that uniquely determines the line Solve problems using slopes of: o parallel lines o perpendicular lines. Eplain the concept of a limit Solve of functions with limits, with lefthand or righthand limits, or with no limit Solve bounded and unbounded functions, and of bounded functions with no limit Eplain, and give eamples of, continuous and discontinuous functions Define the limit of an infinite sequence and an infinite series Eplain and illustrate the limit theorems for sum, difference, multiple, product, quotient and power Solve limits using intuitive and rigorous proofs in the contet of limits Find limits of functions and sequences, both at finite and infinite values of the independent variable Compute limits of functions, using definitions, limit theorems and calculator/ computer methods. Determine the limit of any algebraic function as the independent variable approaches finite or infinite values for continuous and discontinuous functions Calculate the sum of an infinite convergent geometric series Use definitions and limit theorems to determine the limit of any algebraic function as the independent variable approaches a fied value Using definitions and limit theorems to determine the limit of any algebraic function as the independent variable approaches ±. This unit is worth 10% of your mark in Mathematics 31.
2 The time line is: Lesson 1: Factoring and Rationalizing Review Lesson : Factoring More Comple Epressions Lesson 3: Linear Functions Lesson 4: The Tangent Problem Lesson 5: Solving Limits Using Intuitive Reasoning Lesson 6: The Limit theorems Lesson 7: Finding the Limits of Functions by Factoring Lesson 8: Finding the Limits of Rational and Radical Functions Lesson 9: One Sided Limits Lesson 10: Discontinuities Lesson 11: Using Limits to Find Tangents Lesson 1: Velocity as a Rate of Change Lesson 13: Infinite Sequences Precalculus and Limits Page of 37
3 Outcomes: Lesson 1 Factoring and Rationalizing Review Review skills factoring. Review: 1. Factor a) b) 5y 40y c) m 18m 0m d) a + 18a Factor a) + 3 b) 15a + 7a c) d) 3y + 13y 10 e) 6a + 17ab + 1b 4 3. Factor a) 5 16 b) 4 5m c) 7y 50 d) 16n 4 e) 16 1 f ) 1 p 4 6 Precalculus and Limits Page 3 of 37
4 4. Rationalize the following denominator. 7 a) b) c) d) Rationalize the following numerator. 8 7 a) b) c) 81 + Homework: Page 3 #1,a,g,h, Page 4 # 1ab,ab Precalculus and Limits Page 4 of 37
5 Lesson Factoring More Comple Epressions Outcomes: Factor epressions with fractional eponents and a sum and difference of cubes. Warm up: Multiply the following epressions a) ( a b)( a + ab + b ) b) ( a + b)( a ab + b ) From the above one can deduce the following formulas. Factoring a Difference and Sum of Cubes. 3 3 a b = ( a b)( a + ab + b ) 3 3 a + b = ( a + b)( a ab + b ) 1. Use the formula for sum and difference of cubes to factor the following. 3 a) 8 15 b) 7 6 8y 3 3 c) d) y 9 6 Precalculus and Limits Page 5 of 37
6 In pure mathematics 0 you worked with the factor theorem. Recall The Factor Theorem If polynomial P() is divided by b, the remainder P( b ) = 0. Then b is a factor of the polynomial. Factor the following epressions completely. 3 a) b) Factoring Epressions With Fractional Eponents When factoring epressions with fractional eponents factor out the smallest eponent. You will have to recall the eponent laws. 3. Factor the following epressions. a) b) 15a 8 + 7a 5 a Homework: Page 3 #1, b,c,d,e,f, 3,4 Precalculus and Limits Page 6 of 37
7 Outcomes: Lesson 3 Linear Functions Solve problems involving linear equations. Warm up: In mathematics 10 you found the slope of an equation through the use of the formula. y = y y 1 1 Find the slope between the following points. m = a) A( 1, 3), B( 8, 5 ) b) M( 6, 4), N(, 1 ) y y Use the formula to find the equation for the following situations. a) Find the equation of a line with a slope of 3 4 which passes through the point (,3) b) Determine the equation of the line that passes through the points Q(3,6) and R(1,). If we manipulate the formula we can arrive at y y = m( ) 1 1 a) Find the equation of a line with a slope of 4 which travels through the point C(5,7). 5 b) Find the equation of a line, which passes through the point (4,5) and (,7) Precalculus and Limits Page 7 of 37
8 Investigate: Since slope is defined as y the slope is the ratio of the change in y to the change in, it can be interpreted at the rate of change of y with respect to. Eamples: 1. A linear function is given by y = If increases by 3 how does y change?. A linear function is given by y = 6 5. If increases by 3 how does y change? 3. The deeper one enters the earth the warmer the temperature becomes. For every 5 m of depth the temperature increases about 1 0 C, up to 10km. If the ground temperature is 5 0 C, find an epression for the temperature T, as a function of the depth d. Homework: Page 9 # 1 6, 11, 1 Precalculus and Limits Page 8 of 37
9 Outcomes: Warm up: Eplain the concept of a limit Lesson 4 The Tangent Problem In pure mathematics 0 we found the tangent of a line to a circle. The tangent line was a line that touched the circle in only one place. In math 31 we will find tangent lines to curves at various positions. Investigate: At which points are the lines tangent to the given curve. Eamples: 1. Find the equation of a tangent line to the parabola y = at the point (1,1). Precalculus and Limits Page 9 of 37
10 . Find the equation of a tangent line to the cubic function y = 3 at the point (,8). The slopes in the above two eamples illustrate the concept of a limit. The rest of this unit will deal with the concept of limits. We will use limits to solve various problems in this unit. Homework: Page 9 # 8,9 Precalculus and Limits Page 10 of 37
11 Lesson 5 Solving Limits Using Intuitive Reasoning Outcomes: Give eamples of functions with limits, with lefthand or righthand limits, or with no limit Warm up: 1. Consider the function f ( ) = 1 a) Find the values of f( 1 ), f( 10 ), f(100 ), f( ) What is happening to the value of f( ), as is becoming smaller. From above we can intuitively conclude lim 1 = 0 This is stated: the limit of 1 as approaches negative infinity is 0. b) Find the values of f( 1 ), f( 10 ), f( 100 ), f( 1000 ) What is happening to the value of f( ), as is becoming larger. 1 From above we can intuitively conclude lim = 0 This is stated the limit of 1 as approaches infinity is 0.. Consider the function f ( ) = a) Find the values of f( 1 ), f( 10 ), f(100 ), f( ) What is happening to the value of f( ), as is becoming smaller. In the function f ( ) =, lim f ( ) = This is stated the limit of f () as approaches negative infinity is infinite. b) Find the values of f( 1 ), f( 10 ), f( 100 ), f( 1000 ) What is happening to the value of f( ), as is becoming larger. In the function f ( ) =, lim = This is stated the limit of f () as approaches infinity is infinite. Precalculus and Limits Page 11 of 37
12 Limits of continuous functions at a in f (a). Eamples: 1. Given f ( ) = + 4, Sketch the function. Find lim f ( ). 4 f ( ) f ( ) Find the lim Then draw a sketch of the function. 3. Find the lim Find the lim 9 Functions with the property lim f ( ) = f ( a ) are called continuous at a. a P Likewise in a rational function lim ( ) P( a) = is continuous at a. The function a Q( ) Q( a) is therefore defined at a. Precalculus and Limits Page 1 of 37
13 Limits of discontinuous functions at a in f (a) Investigate the function f ( ) = 4 a) State the nonpermissible values for f ( ) = 16 4 b) Determine the value of f ( 4 ) for the function c) Determine the lim 4 4. f ( ) f ( ) d) Graph the function f ( ) = 16 on your calculator. 4 e) Graph the function f ( ) = 16 4 manually. 16 f) What is the difference between the lim 4 4 and f ( 4 )? Eplain why this difference eists. Precalculus and Limits Page 13 of 37
14 . Investigate the function f ( ) = a) State the nonpermissible values for f ( ) = b) Determine the value of f ( 3 ) for the function f ( ) = c) Determine the lim f ( ) f ( ) d) Graph the function f ( ) = 3 on your calculator e) Graph the function f ( ) = 3 manually f) What is the difference between the lim 3 eists and f ( 3 )? Eplain why this difference Homework: Page 18 # 1,,4a,,7,8 Precalculus and Limits Page 14 of 37
15 Outcomes: Lesson 6 Limit Theorems Solve limit problems using the properties of limits. Investigate: Suppose that the limits lim f ( ) a 1. Find lim 5 4 and lim g ( ) a both eist and let c be a constant. The limit of a constant is the constant itself lim c = c a. Find lim( + ) and lim + lim The limits of a sum is the sum of the limits lim f ( ) + g( ) = lim f ( ) + lim g( ) a a a 3. Find lim( ) and lim lim The limits of a difference is the difference of the limits lim f ( ) g( ) = lim f ( ) lim g( ) a a a 4. Find lim( 5 ) and 5lim 5 a The limit of a product of a constant and a function is the product of the constant and the limit of the function lim cf ( ) = c lim f ( ) a a 5. Find lim( ) then lim lim The limit of a product is the product of the limits lim f ( ) g( ) = lim f ( ) lim g( ) a a a Precalculus and Limits Page 15 of 37
16 6. Find lim 5 3 and lim 5 lim3 5 The limit of a quotient is the quotient of the limits f ( ) lim f ( ) z lim = a g( ) lim g( ) a 7. Find lim( ) 5 3 and lim( ) 5 3 The limit of a power is the power of the limit lim f ( ) a n = lim f ( ) a n 8. Find lim ( ) 5 4 and lim( ) 5 4 The limit of a power is the power of the limit lim n f ( ) = n lim f ( ) a a Precalculus and Limits Page 16 of 37
17 Eamples: 1. Use the limit theorems to find lim( 3 ). Find lim( + 3 ) using the properties of limits 5 3. Find lim using the properties of limits 4. Find lim + 3 using the properties of limits Since all of the above functions were continuous at the value approaches. lim f ( ) = f ( a ) a The limit Theorems may seem rather useless at this time. That is because we chose equations in which the theorems were evident. Net class you will consider the eamples in which you will need to use the limit theorems to arrive at the correct solution. Practice the limit theorems you will need them. Homework: Page 19 # 3 Precalculus and Limits Page 17 of 37
18 Lesson 7 Finding the Limits of Functions by Factoring Outcomes: Solve limit problems which involve 0 0 Warm up: Find f ( ) of 4 0 Investigate: Since f ( ) = is meaningless you must simplify to obtain a function which is 0 continuous at, before you can find the limit of the function. Eamples: Find the following limits by factoring Find lim 16. Evaluate lim Find lim 8 4. Evaluate lim Precalculus and Limits Page 18 of 37
19 5. Find lim h 0 + h 4 h 6. Find lim h 0 1 h 1 h If a function cannot be written as another function which is continuous for the approached value of then the function does not have a limit, for the approached value of Show lim does not eist. Draw a graph to help with your understanding Find lim 4 4 Homework: Page 19 # 4(a f), 5 (a,b,d),6 (a,b,c,d,e,f,i) Precalculus and Limits Page 19 of 37
20 Lesson 8 Finding the Limits of Rational and Radical Functions Outcomes: Solve limit problems which involve 0 0 Warm up: 1. Rationalize the following denominators 3 a) b) Simplify the following epressions a) + b) ( + 1) 3 9 Eamples: 1. Find lim Find lim Precalculus and Limits Page 0 of 37
21 Find lim Find lim 0 5. Find lim Homework: Page 19 # 4(g,h), 5 (c,e,f),6 (g,h,j) Precalculus and Limits Page 1 of 37
22 Outcomes: Warm up: Lesson 9 One Sided  Limits Solve functions with limits, lefthand or righthand limits, or with no limit 5 Consider the function f ( ) = 3. Complete the table for the function. f ( ) f ( ) Sketch the above function. Investigate: approach. The limit as you approach 3 from the left is different than the limit when you The lefthand limit The righthand limit 5 lim = 5 lim = 3 3 the limit as approaches from the left is from the right is infinite negative infinite the limit as approaches 5 The lim does not eist. Since the left and right limits are not equal. 3 3 Precalculus and Limits Page of 37
23 Eamples: 1. Find the following using a table of values. a) lim b) lim c) lim 4 4. Find the following using a table of values. a) lim b) lim 4 ( ) ( 4 ) lim 4 ( ) 4 c 3. Find the following from the above graph a) lim f ( ) b) lim f ( ) c) lim f ( ) 0 d) lim f ( ) 5 e) lim f ( ) 5 + f) lim f ( ) 5 Precalculus and Limits Page 3 of 37
24 Some functions are not described by a single equation but rather multiple equations. Consider the follow. = if 1 4. f 3 if > 1 a) Find the values of f ( 1 ) f ( 0 ) f ( 1 ) f ( ) b) Sketch the function c) Find lim f ( ) 1 lim f ( ) 1 + lim f ( ) 1 5. Find a) lim 0 + b) lim 0 c) lim 0 6. Show that lim = 0 0 Precalculus and Limits Page 4 of 37
25 7. The Heaviside function H is defined by (details are in your book page 3) 0 if t < 0 H( t) = 1 if t 0 Find lim H( t) 0 lim H( t) 0 + lim H( t) 0 8. a) If if 1 f ( ) = if 1< < 1determine whether or not lim f ( ) and lim f ( ) eist. if Homework: Page 7 # 1,,4,5,6,7 Precalculus and Limits Page 5 of 37
26 Lesson 10 Discontinuous Functions Outcomes: Warm up: Distinguish between continuous and discontinuous functions. In lesson 5 we discussed continuous functions. Recall the following. Functions with the property lim f ( ) = f ( a ) are called continuous at a. a P Likewise in a rational function lim ( ) P( a) = is continuous at a. The function a Q( ) Q( a) is therefore defined at a. If a functions is not continuous at a, we say the function is discontinuous or the function has a discontinuity at a. If lim f ( ) does not eist then the function is discontinuous at a. a Eamples: Where are the following functions discontinuous? 3 if < 0 a) f ( ) = if 0 1 b) 1+ if > f ( ) = 0 1 < 0 = 0 > c) f ( ) = 1 = Homework: Page 7 # 3,8,9,10(a,b,c) Precalculus and Limits Page 6 of 37
27 Outcomes: Warm up: Lesson 11 Using Limits to Find Tangents Find Tangents to Curves In pure mathematics 0 we found the tangent of a line to a circle. The tangent line was a line that touched the circle in only one place. In math 31 we will find tangent lines to curves at various positions. Investigate: We can use the concept of limits to find the equation of a line tangent to a given curve. Consider the following eamples. Develop a formula for finding slope of a line tangent to a curve (, f()) (a,f(a)) Eamples: 1. Find the equation of a tangent line to the parabola y = at the point (1,1).. Find the equation of a tangent line to the cubic function y = 3 at the point (,8). Precalculus and Limits Page 7 of 37
28 Develop a second algebraic method to find the slope of a line tangent to a given curve by moving the point ( a + h) successively closer to a. This is often referred to as first principle of calculus. (a,f(a)) (a+h, f(a+h)) 3. Find the slope and the equation of the tangent line to the curve y = at the point (, 15). 4. Find the tangent line to the hyperbola y = 1 at the point ( ,  ½ ). Precalculus and Limits Page 8 of 37
29 5. Find the tangent line to the curve y = at the point where = 6. Graph the curve and the tangent line. 6. Find an equation to find the slope of a tangent line at any point tangent to the curve y = If f ( ) = 7, what is the value of where the slope is? Homework: Page 35 #1,,6,7(a,b),8,9 Precalculus and Limits Page 9 of 37
30 Lesson 1 Velocity as a Rate of Change Outcomes: Solve rate of change problems using limits Warm up: Find the slope of a line tangent to y = 3 5 through the point when = 1. distance travelled Average velocity = If you drive to Edmonton your average time elapsed velocity may be 100km/h. We all know that you would go slightly faster and slightly slower than that speed at any given moment. This speed at any moment is known as instantaneous velocity. Consider the speed at an instantaneous instance how much time is elapsed at that moment, how much distance is traveled. Consider the following investigation carefully. Investigate: Suppose that a ball is dropped from the upper observation deck of the CN tower, 450 m above the ground. How fast is the ball falling after 3 s? Approimate the speed using a table of values distance travelled Remember Average velocity = time elapsed Use time intervals closer and closer to 3 seconds to predict the speed. Time Distance Traveled Average Velocity s Average velocity = t f (3 + h) f (3) = h Precalculus and Limits Page 30 of 37
31 So the instantaneous velocity can be found using f lim ( 3 + h ) f ( 3 ) h 0 h Find the instantaneous velocity at 3 seconds. Eamples: 1. The displacement in metres, of a particle moving in a straight line is given by s = t + t, where t is measured in seconds. Find the velocity of the particle after 3 s. Homework: Page 43 # 1,,3 Precalculus and Limits Page 31 of 37
32 Lesson 13 Infinite Sequences Outcomes: Warm up: Investigate: Solve Geometric Sequence Problems Infinite sequences and their limits are basic to the understanding of Calculus. Even though sequences are more fully developed in pure mathematics 30, there are a number of key concepts you will need to know for mathematics 31. In a geometric sequence each term is multiplied by a constant to obtain the following term. In an arithmetic sequence each term is added to a constant to obtain the following term. Many other sequences can be generated using various methods. 1. The first two terms of a two geometric sequences are as follows: Sequence 1 Sequence t1 = 64 and t = 3 t = and t 8 = 4 a) Write the first 5 terms for each sequence. b) The first sequence is called a convergent sequence. What value does the sequence appear to be approaching. An infinite sequence is the range of a function which has the set of natural number as its domain. If the terms of an infinite sequence approach a unique finite value, that sequence is called a convergent sequence. A sequence which does not converge is called divergent. Eamples: 1. a) Determine the first five terms of the sequence defined by the function f ( n) = n 1, n N. b) Plot the points of sequence. c) What do you think lim f ( n) is? n Precalculus and Limits Page 3 of 37
33 . a) Determine the first five terms of the sequence defined by the function n t( n) = n N n + 1 b) Plot the points of sequence. c) What do you think lim f ( n) is? n 3. a) Determine the first five terms of the sequence defined by the function 1 t( n) = n N n b) Plot the points of sequence. 1 c) What do you think lim is? n n The following statements will prove very useful in your study of sequences 1 lim n n r = 0 if r > 0 n lim r = 0 if r < 1 n Precalculus and Limits Page 33 of 37
34 n 3 4. Find lim n n n n 5. Find lim n n + 1 3n 5n Find lim n n + 3n 7 7. Find the following limits if they eist. Which sequence is a convergent sequence. a) lim 1 n n 1 b) lim Homework: Page 50 #1 6, 8 Precalculus and Limits Page 34 of 37
35 Outcomes: Warm up: Investigate: Find S 1 Lesson 14 Infinite Series Define the limit of an infinite sequence. A series is the sum of a sequence. Consider the geometric series following geometric series S S 3 S 4 S 5 Plot above partial sums on the grid below. The value of n partial sums can be found using the formula. n a( 1 r ) Sn = r 1 1 r Find the value of the 6 th partial sum using the formula. Predict the value of S Precalculus and Limits Page 35 of 37
36 To find the sum of an infinite geometric series we would have to find lim ( n a 1 r ) 1 r To find a sum of a infinite geometric series S a =. 1 r Eamples: 4 1. Find the sum of the infinite geometric series ,.... Which infinite geometric series is convergent? What is the sum? a) b) Write each of the following as reduced fractions. a) 0. b) 0. 3 c) Precalculus and Limits Page 36 of 37
37 Note: r < 1 then the denominator would be less than 1 and the resulting series would be convergent. E: If r = 1 the common ratio is 1. So the series is a + a + a + a. therefore the series does not have a sum. E: If r =  1 the common ratio is 1. So the series is a  a + a  a. therefore the series does not have a sum. E: If r > 1, then the series grows indefinitely. Therefore lim ( n a 1 r ) 1 r 1 lim = 0 only if r < 1 n r n E: does not eist. Remember 4. For what values of are the following series convergent? In each case find the sum of the series for those values of a) ( + 1), ( + 1), ( + 1) 3,... b,,,,... c) 3,,,, Homework: Page 56#1,3,4,5 Precalculus and Limits Page 37 of 37
Prep for Calculus. Curriculum
Prep for Calculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationPrep for College Algebra
Prep for College Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More information25 Rational Functions
5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4
More informationCOURSE OUTLINE. MATHEMATICS 101 Intermediate Algebra
Degree Applicable I. Catalog Statement COURSE OUTLINE MATHEMATICS 101 Intermediate Algebra Glendale Community College October 2013 Mathematics 101 is an accelerated course of Intermediate Algebra. Topics
More informationIdentify examples of field properties: commutative, associative, identity, inverse, and distributive.
Topic: Expressions and Operations ALGEBRA II  STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers
More information3.1 Solving Quadratic Equations by Taking Square Roots
COMMON CORE 816 1 1 10 8 6 0 y Locker LESSON.1 Solving Quadratic Equations by Taking Square Roots Name Class Date.1 Solving Quadratic Equations by Taking Square Roots Essential Question: What is an imaginary
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationAdvanced Algebra 2. I. Equations and Inequalities
Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers
More informationMath 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =
Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationChapter 2 Limits Functions and Sequences sequence sequence Example
Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students
More informationMATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationAble Enrichment Centre  Prep Level Curriculum
Able Enrichment Centre  Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
More informationChapter R  Basic Algebra Operations (69 topics, due on 05/01/12)
Course Name: College Algebra 001 Course Code: R3RK6CTKHJ ALEKS Course: College Algebra with Trigonometry Instructor: Prof. Bozyk Course Dates: Begin: 01/17/2012 End: 05/04/2012 Course Content: 288 topics
More informationMATHEMATICS 31 A. COURSE OVERVIEW RATIONALE
MATHEMATICS 31 A. COURSE OVERVIEW RATIONALE To set goals and make informed choices, students need an array of thinking and problemsolving skills. Fundamental to this is an understanding of mathematical
More informationSAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?
SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red
More informationChapter 4A: Exponential Functions Outline
Chapter 4A: Eponential Functions Outline Lesson : Integer Eponents Lesson : Rational Eponents Lesson : Eponential Functions Basics Lesson 4: Finding Equations of Eponential Functions Lesson 5: The Method
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationLesson 6: Linear Functions and their Slope
Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation
More informationDraft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then
CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function,
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
More informationCommon Curriculum Map. Discipline: Math Course: College Algebra
Common Curriculum Map Discipline: Math Course: College Algebra August/September: 6A.5 Perform additions, subtraction and multiplication of complex numbers and graph the results in the complex plane 8a.4a
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationLyman Memorial High School. PreCalculus Prerequisite Packet. Name:
Lyman Memorial High School PreCalculus Prerequisite Packet Name: Dear PreCalculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These
More informationThe NotFormula Book for C1
Not The NotFormula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationHigher Education Math Placement
Higher Education Math Placement 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry (arith050) Subtraction with borrowing (arith006) Multiplication with carry
More informationCHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often
7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationpp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64
Semester 1 Text: Chapter 1: Tools of Algebra Lesson 11: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 12: Algebraic Expressions
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationAlgebra 12. A. Identify and translate variables and expressions.
St. Mary's College High School Algebra 12 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More information(x) = lim. x 0 x. (2.1)
Differentiation. Derivative of function Let us fi an arbitrarily chosen point in the domain of the function y = f(). Increasing this fied value by we obtain the value of independent variable +. The value
More informationMath Content
20122013 Math Content PATHWAY TO ALGEBRA I Unit Lesson Section Number and Operations in Base Ten Place Value with Whole Numbers Place Value and Rounding Addition and Subtraction Concepts Regrouping Concepts
More informationTaylor Polynomials. for each dollar that you invest, giving you an 11% profit.
Taylor Polynomials Question A broker offers you bonds at 90% of their face value. When you cash them in later at their full face value, what percentage profit will you make? Answer The answer is not 0%.
More informationMath Course: Algebra II Grade 10
MATH 401 Algebra II 1/2 credit 5 times per week (1 st Semester) Taught in English Math Course: Algebra II Grade 10 This is a required class for all 10 th grade students in the Mexican and/or U.S. Diploma
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationSkill Builders. (Extra Practice) Volume I
Skill Builders (Etra Practice) Volume I 1. Factoring Out Monomial Terms. Laws of Eponents 3. Function Notation 4. Properties of Lines 5. Multiplying Binomials 6. Special Triangles 7. Simplifying and Combining
More informationRational Functions ( )
Rational Functions A rational function is a function of the form r P Q where P and Q are polynomials. We assume that P() and Q() have no factors in common, and Q() is not the zero polynomial. The domain
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More information3.5 Summary of Curve Sketching
3.5 Summary of Curve Sketching Follow these steps to sketch the curve. 1. Domain of f() 2. and y intercepts (a) intercepts occur when f() = 0 (b) yintercept occurs when = 0 3. Symmetry: Is it even or
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationUnit 1: Polynomials. Expressions:  mathematical sentences with no equal sign. Example: 3x + 2
Pure Math 0 Notes Unit : Polynomials Unit : Polynomials : Reviewing Polynomials Epressions:  mathematical sentences with no equal sign. Eample: Equations:  mathematical sentences that are equated with
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationAlgebra II Pacing Guide First Nine Weeks
First Nine Weeks SOL Topic Blocks.4 Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole and natural. 7. Recognize that the
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More information1.3.1 Position, Distance and Displacement
In the previous section, you have come across many examples of motion. You have learnt that to describe the motion of an object we must know its position at different points of time. The position of an
More informationAlgebra 1 Course Objectives
Course Objectives The Duke TIP course corresponds to a high school course and is designed for gifted students in grades seven through nine who want to build their algebra skills before taking algebra in
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationAlgebra. Indiana Standards 1 ST 6 WEEKS
Chapter 1 Lessons Indiana Standards  11 Variables and Expressions  12 Order of Operations and Evaluating Expressions  13 Real Numbers and the Number Line  14 Properties of Real Numbers  15 Adding
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationMTH304: Honors Algebra II
MTH304: Honors Algebra II This course builds upon algebraic concepts covered in Algebra. Students extend their knowledge and understanding by solving openended problems and thinking critically. Topics
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More informationLIVE Online Math Algebra Scope and Sequence
LIVE Online Math Algebra Scope and Sequence The course is broken down into units. The units, and lessons that make up each unit, are below. Note: If there is a specific concept/technique that is not listed,
More informationALGEBRA 1/ALGEBRA 1 HONORS
ALGEBRA 1/ALGEBRA 1 HONORS CREDIT HOURS: 1.0 COURSE LENGTH: 2 Semesters COURSE DESCRIPTION The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 01 Sets There are no statemandated Precalculus 02 Operations
More informationMyMathLab ecourse for Developmental Mathematics
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationSenior Secondary Australian Curriculum
Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationMath 115 Spring 2014 Written Homework 3 Due Wednesday, February 19
Math 11 Spring 01 Written Homework 3 Due Wednesday, February 19 Instructions: Write complete solutions on separate paper (not spiral bound). If multiple pieces of paper are used, they must be stapled with
More informationLesson 71. Roots and Radicals Expressions
Lesson 71 Roots and Radicals Epressions Radical Sign inde Radical Sign n a Radicand Eample 1 Page 66 #6 Find all the real cube roots of 0.15 0.15 0.15 0.15 0.50 (0.50) 0.15 0.50 is the cube root of 0.15.
More informationChapter 1 Quadratic Equations in One Unknown (I)
Tin Ka Ping Secondary School 015016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real
More informationMath Analysis. A2. Explore the "menu" button on the graphing calculator in order to locate and use functions.
St. Mary's College High School Math Analysis Introducing the Graphing Calculator A. Using the Calculator for Order of Operations B. Using the Calculator for simplifying fractions, decimal notation and
More informationAlgebra I. Copyright 2014 Fuel Education LLC. All rights reserved.
Algebra I COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics, with an emphasis
More informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
More informationTo discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes.
INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3,... 10,...? Well, we could start creating sums of a finite number
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More information