A11.1 Areas under curves
|
|
|
- Adelia Davis
- 5 years ago
- Views:
Transcription
1 Applications 11.1 Areas under curves A11.1 Areas under curves Before ou start You should be able to: calculate the value of given the value of in algebraic equations of curves calculate the area of a trapezium. bjectives You will be able to calculate an estimate for the area of the region bounded b a curve, the -ais and two lines parallel to the ais. You will be able to interpret the area under a velocit-time graph as a distance travelled. Wh do this? Working out areas under curves is ver important when studing the thermodnamics of engines. Get Read 1 In each case calculate the value of when i ii a b c 6 Calculate the area of the trapezium. 7 Ke Points An estimate of the area of the region between a curve and the -ais can be calculated b splitting the region into trapeziums and finding the total area of the trapeziums. The area of the region between a velocit-time (v/t) curve and the time (t)-ais is equal to the distance travelled. v t The area of this region is equal to the distance travelled. Eample 1 Here is a curve. Calculate an estimate of the area bounded b the curve, the -ais, and the lines and. 1
2 Chapter 11 Areas under curves A B C D Then add all these areas together. A: B: C: D: ( 8) (8 1) 11 (1 ) 19 ( ) 9 Split the area of the region into trapeziums and work out the area of each trapezium. A quicker wa to calculate the sum of the areas of the trapezium is to factorise the sum of the epressions: ( ) ( 8 1 ) [ (8 1 ) ] 6 square units Total area 6 square units Eample Here is a sketch of the graph of for values of from 0 to. 1 Calculate an estimate for the area of the region shown shaded in the diagram An estimate for the area is 8 square units. Split the region into trapeziums each of width 1 unit. Work out the lengths of the parallel sides of each trapezium b using the equation of the curve. (The first trapezium will be a triangle.) Eample A particle moves in one direction in a straight line. Its velocit v m/s after t seconds is given b v t t. Calculate an estimate of the distance it has travelled during the first seconds. t The distance travelled area between the curve and the time ais, from t 0 to t. Using the trapezium rule, an estimate for this area is 8 square units. So the distance travelled is 8 metres.
3 A Applications 11.1 Areas under curves Eercise 11A 1 Find the area of the regions bounded b the curve, the -ais and the lines shown parallel to the -ais. a b c B d e 1 f Draw the curve with equation for values of from to. Calculate an estimate for the area between the -ais, the curve and the lines and. a Cop and complete the table of values for the curve with equation b Draw the graph. c Calculate an estimate for the area between the curve, the -ais and the lines 1 and 6. The graph gives information about the velocit of a particle during the first seconds of its motion. a Calculate an estimate for the distance travelled during the first seconds. b State, with a reason, whether our answer to a is an overestimate or underestimate of the true distance travelled. Velocit m/s t
4 Chapter 11 Areas under curves A The graph shows the velocit, v m/s, with which a racing car has travelled during t seconds. v t a Calculate an estimate of the distance the racing car travelled during these seconds. b Calculate an estimate of the average speed of the car for these seconds. 6 The velocit, v m/s, of a particle at time t seconds is given b v t t. a Calculate an estimate of the distance it has travelled after seconds. b Calculate an estimate of the average speed of the particle during the first seconds. Review The approimate area under a graph can be found b approimating the region b a set of trapeziums and finding their total area. The area under a velocit-time graph is equal to the distance travelled.
5 Answers Answers Chapter 11 A11.1 Get Read answers 1 a i ii 18 b i ii 1. c i ii Eercise 11A 1 a 1 b 6 c 79 d 9 e 170 f 900 a metres b overestimate as the sloping edges of the trapeziums are alwas above the curve. a metres b 9.7 m/s 6 v 1 6 t a Distance travelled is approimatel metres. b Average speed is approimatel 16 m/s. Area is approimatel 6 square units b c 6 square units
Chapter 18 Quadratic Function 2
Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are
4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
5.6. Differential equations
5.6. Differential equations The relationship between cause and effect in phsical phenomena can often be formulated using differential equations which describe how a phsical measure () and its derivative
Regent College Maths Department. Core Mathematics 4 Trapezium Rule. C4 Integration Page 1
Regent College Maths Department Core Mathematics Trapezium Rule C Integration Page Integration It might appear to be a bit obvious but you must remember all of your C work on differentiation if you are
Speed (km/h) How can you determine the inverse of a function?
.7 Inverse of a Function Engineers have been able to determine the relationship between the speed of a car and its stopping distance. A tpical function describing this relationship is D.v, where D is the
Systems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes
Cartesian Components of Vectors 9.2 Introduction It is useful to be able to describe vectors with reference to specific coordinate sstems, such as the Cartesian coordinate sstem. So, in this Section, we
Higher. Differentiation 28
Higher Mathematics UNIT OUTCOME Differentiation Contents Differentiation 8 Introduction to Differentiation 8 Finding the Derivative 9 Differentiating with Respect to Other Variables 4 Rates of Change 4
UNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
Fitting Integrands to Basic Rules. x x 2 9 dx. Solution a. Use the Arctangent Rule and let u x and a dx arctan x 3 C. 2 du u.
58 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8 Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration rules Fitting Integrands
SM2H Average Rate of Change
Name Period Core Alignment: F.IF., F.IF., F.IF.9 SMH 5. - Average Rate of Change The average rate of change between two points is essentiall the slope of the line that connects the two points. For eample,
Practice Problem List II
Math 46 Practice Problem List II -------------------------------------------------------------------------------------------------------------------- Section 4.: 3, 3, 5, 9, 3, 9, 34, 39, 43, 53, 6-7 odd
Introducing Instantaneous Rate of Change
Introducing Instantaneous Rate of Change The diagram shows a door with an automatic closer. At time t = 0 seconds someone pushes the door. It swings open, slows down, stops, starts closing, then closes
UNIT 6 DESCRIBING DATA Lesson 2: Working with Two Variables. Instruction. Guided Practice Example 1
Guided Practice Eample 1 Andrew wants to estimate his gas mileage, or miles traveled per gallon of gas used. He records the number of gallons of gas he purchased and the total miles he traveled with that
QUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta
QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape
Fitting Integrands to Basic Rules
6_8.qd // : PM Page 8 8 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration
STRAND: GRAPHS Unit 1 Straight Lines
CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text STRAND: GRAPHS Unit Straight Lines TEXT Contents Section. Gradient. Gradients of Perpendicular Lines. Applications of Graphs. The Equation of
6 = 1 2. The right endpoints of the subintervals are then 2 5, 3, 7 2, 4, 2 9, 5, while the left endpoints are 2, 5 2, 3, 7 2, 4, 9 2.
5 THE ITEGRAL 5. Approimating and Computing Area Preliminar Questions. What are the right and left endpoints if [, 5] is divided into si subintervals? If the interval [, 5] is divided into si subintervals,
Algebra Skills Required for Entry to a Level Two Course in Mathematics
Algebra Skills Required for Entr to a Level Two Course in Mathematics This is a list of Level One skills ou will be required to demonstrate if ou are to gain entr to the Level Two Achievement Standard
= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
Section 4.1 Increasing and Decreasing Functions
Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates
means Name a function whose derivative is 2x. x 4, and so forth.
AP Slope Fields Worksheet Slope fields give us a great wa to visualize a famil of antiderivatives, solutions of differential equations. d Solving means Name a function whose derivative is. d Answers might
AP Calculus BC Chapter 4 (A) 12 (B) 40 (C) 46 (D) 55 (E) 66
AP Calculus BC Chapter 4 REVIEW 4.1 4.4 Name Date Period NO CALCULATOR IS ALLOWED FOR THIS PORTION OF THE REVIEW. 1. 4 d dt (3t 2 + 2t 1) dt = 2 (A) 12 (B) 4 (C) 46 (D) 55 (E) 66 2. The velocity of a particle
Section J Venn diagrams
Section J Venn diagrams A Venn diagram is a wa of using regions to represent sets. If the overlap it means that there are some items in both sets. Worked eample J. Draw a Venn diagram representing the
Derivatives of Multivariable Functions
Chapter 0 Derivatives of Multivariable Functions 0. Limits Motivating Questions In this section, we strive to understand the ideas generated b the following important questions: What do we mean b the limit
AQA Higher Practice paper (calculator 2)
AQA Higher Practice paper (calculator 2) Higher Tier The maimum mark for this paper is 8. The marks for each question are shown in brackets. Time: 1 hour 3 minutes 1 One billion in the UK is one thousand
MA123, Chapter 8: Idea of the Integral (pp , Gootman)
MA13, Chapter 8: Idea of the Integral (pp. 155-187, Gootman) Chapter Goals: Understand the relationship between the area under a curve and the definite integral. Understand the relationship between velocit
Slope as a Rate of Change
Find the slope of a line using two of its points. Interpret slope as a rate of change in real-life situations. FINDING THE SLOPE OF A LINE The slope m of the nonvertical line passing through the points
Slopes and Rates of Change
Slopes and Rates of Change If a particle is moving in a straight line at a constant velocity, then the graph of the function of distance versus time is as follows s s = f(t) t s s t t = average velocity
UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives
Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.
Quadratics NOTES.notebook November 02, 2017
1) Find y where y = 2-1 and a) = 2 b) = -1 c) = 0 2) Epand the brackets and simplify: (m + 4)(2m - 3) To find the equation of quadratic graphs using substitution of a point. 3) Fully factorise 4y 2-5y
Active Maths 2 Old Syllabus Strand 5
Junior certificate HIGHER LEVEL Active Maths Old Sllabus Strand 5 πr m = - - πr Oliver Murph Contents. Functions.... Functions and Graphs...5.3 Graphs...7.4 Linear Graphs...7.5 Quadratic Graphs...9.6 Real-Life
Lesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
CHAPTER 3 : QUADRARIC FUNCTIONS MODULE CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions Graphs of quadratic functions 4 Eercis
ADDITIONAL MATHEMATICS MODULE 5 QUADRATIC FUNCTIONS CHAPTER 3 : QUADRARIC FUNCTIONS MODULE 5 3.1 CONCEPT MAP Eercise 1 3. Recognizing the quadratic functions 3 3.3 Graphs of quadratic functions 4 Eercise
The area under a graph can be estimated by counting squares.
A15 CALCULATE OR ESTIMATE GRADIENTS OF GRAPHS AND AREAS UNDER GRAPHS (INCLUDING QUADRATIC AND OTHER NON-LINEAR GRAPHS), AND INTERPRET RESULTS IN CASES SUCH AS DISTANCE- TIME GRAPHS, VELOCITY-TIME GRAPHS
CHAPTER 72 AREAS UNDER AND BETWEEN CURVES
CHAPTER 7 AREAS UNDER AND BETWEEN CURVES EXERCISE 8 Page 77. Show by integration that the area of the triangle formed by the line y, the ordinates and and the -ais is 6 square units. A sketch of y is shown
Precalculus Honors - AP Calculus A Information and Summer Assignment
Precalculus Honors - AP Calculus A Information and Summer Assignment General Information: Competenc in Algebra and Trigonometr is absolutel essential. The calculator will not alwas be available for ou
Derivatives 2: The Derivative at a Point
Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned
Answers to Some Sample Problems
Answers to Some Sample Problems. Use rules of differentiation to evaluate the derivatives of the following functions of : cos( 3 ) ln(5 7 sin(3)) 3 5 +9 8 3 e 3 h 3 e i sin( 3 )3 +[ ln ] cos( 3 ) [ln(5)
Learning Outcomes and Assessment Standards
Lesson 5 CALCULUS (8) Rate of change Learning Outcomes and Assessment Standards Learning Outcome : Functions and Algebra Assessment standard 1..7(e) Solve practical problems involving optimisation and
How can you write an equation of a line when you are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines
.7 Writing Equations in Point-Slope Form How can ou write an equation of a line when ou are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines Work with a partner. Sketch the
AMB111F Notes 3 Quadratic Equations, Inequalities and their Graphs
AMB111F Notes 3 Quadratic Equations, Inequalities and their Graphs The eqn y = a +b+c is a quadratic eqn and its graph is called a parabola. If a > 0, the parabola is concave up, while if a < 0, the parabola
recognise quadratics of the form y = kx 2 and y = (x + a) 2 + b ; a, b Î Z, from their graphs solve quadratic equations by factorisation
QUADRATIC FUNCTIONS B the end of this unit, ou should be able to: (a) (b) (c) (d) (e) recognise quadratics of the form = k 2 and = ( + a) 2 + b ; a, b Î Z, from their graphs identif the nature and coordinates
4.2 Mean Value Theorem Calculus
4. MEAN VALUE THEOREM The Mean Value Theorem is considered b some to be the most important theorem in all of calculus. It is used to prove man of the theorems in calculus that we use in this course as
14.3. Volumes of Revolution. Introduction. Prerequisites. Learning Outcomes
Volumes of Revolution 14.3 Introduction In this Section we show how the concept of integration as the limit of a sum, introduced in Section 14.1, can be used to find volumes of solids formed when curves
REVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES
Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.
P.4 Lines in the Plane
28 CHAPTER P Prerequisites P.4 Lines in the Plane What ou ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables
10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
2 MOTION ALONG A STRAIGHT LINE
MOTION ALONG A STRAIGHT LINE Download full Solution manual for Universit phsics with modern phsics 14t http://testbankcollection.com/download/solution-manual-for-universit-phsics-withmodern-phsics-14th.1.
Limits and the derivative function. Limits and the derivative function
The Velocity Problem A particle is moving in a straight line. t is the time that has passed from the start of motion (which corresponds to t = 0) s(t) is the distance from the particle to the initial position
Graph and Write Equations of Ellipses. You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses.
TEKS 9.4 a.5, A.5.B, A.5.C Before Now Graph and Write Equations of Ellipses You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses. Wh? So ou can model
(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
![(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)](/thumbs/85/92620692.jpg "(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)")
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
Displacement, Velocity and Acceleration in one dimension
Displacement, Velocity and Acceleration in one dimension In this document we consider the general relationship between displacement, velocity and acceleration. Displacement, velocity and acceleration are
Vertex form of a quadratic equation
Verte form of a quadratic equation Nikos Apostolakis Spring 017 Recall 1. Last time we looked at the graphs of quadratic equations in two variables. The upshot was that the graph of the equation: k = a(
CALCULUS AB SECTION II, Part A
CALCULUS AB SECTION II, Part A Time 45 minutes Number of problems 3 A graphing calculator is required for some problems or parts of problems. pt 1. The rate at which raw sewage enters a treatment tank
STRAIGHT LINE GRAPHS. Lesson. Overview. Learning Outcomes and Assessment Standards
STRAIGHT LINE GRAPHS Learning Outcomes and Assessment Standards Lesson 15 Learning Outcome : Functions and Algebra The learner is able to investigate, analse, describe and represent a wide range o unctions
13. x 2 = x 2 = x 2 = x 2 = x 3 = x 3 = x 4 = x 4 = x 5 = x 5 =
Section 8. Eponents and Roots 76 8. Eercises In Eercises -, compute the eact value... 4. (/) 4. (/). 6 6. 4 7. (/) 8. (/) 9. 7 0. (/) 4. (/6). In Eercises -4, perform each of the following tasks for the
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the graph to evaluate the limit. ) lim 0 f() ) - - - - - - - - A) - B) 0 C)does not eist
y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions
AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)
Section Differential Equations: Modeling, Slope Fields, and Euler s Method
Section.. Differential Equations: Modeling, Slope Fields, and Euler s Method Preliminar Eample. Phsical Situation Modeling Differential Equation An object is taken out of an oven and placed in a room where
Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.
Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of
Conic Section: Circles
Conic Section: Circles Circle, Center, Radius A circle is defined as the set of all points that are the same distance awa from a specific point called the center of the circle. Note that the circle consists
Mini-Lecture 8.1 Solving Quadratic Equations by Completing the Square
Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.
A. Simplifying Polynomial Expressions
A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: 5-7 + 10 +
APPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
Limits 4: Continuity
Limits 4: Continuit 55 Limits 4: Continuit Model : Continuit I. II. III. IV. z V. VI. z a VII. VIII. IX. Construct Your Understanding Questions (to do in class). Which is the correct value of f (a) in
y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions
AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)
5 Linear Graphs and Equations
Linear Graphs and Equations. Coordinates Firstl, we recap the concept of (, ) coordinates, illustrated in the following eamples. Eample On a set of coordinate aes, plot the points A (, ), B (0, ), C (,
Solve Quadratic Equations by Graphing
0.3 Solve Quadratic Equations b Graphing Before You solved quadratic equations b factoring. Now You will solve quadratic equations b graphing. Wh? So ou can solve a problem about sports, as in Eample 6.
Lesson 5.1 Exercises, pages
Lesson 5.1 Eercises, pages 346 352 A 4. Use the given graphs to write the solutions of the corresponding quadratic inequalities. a) 2 2-8 - 10 < 0 The solution is the values of for which y
5. Zeros. We deduce that the graph crosses the x-axis at the points x = 0, 1, 2 and 4, and nowhere else. And that s exactly what we see in the graph.
. Zeros Eample 1. At the right we have drawn the graph of the polnomial = ( 1) ( 2) ( 4). Argue that the form of the algebraic formula allows ou to see right awa where the graph is above the -ais, where
Unit 10 - Graphing Quadratic Functions
Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif
(b) Find the difference quotient. Interpret your result. 3. Find the average rate of change of ƒ(x) = x 2-3x from
6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 67 CHAPTER Summar 67 69. ƒ() = 3 70. ƒ() = -5 (b) Find the difference quotient. Interpret our result. 7. ƒ() = - 7. ƒ() = 0 73. ƒ() = + 74. ƒ() = -3 + 4 75. ƒ() =
3.1 Graph Quadratic Functions
3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
INVESTIGATE the Math
. Graphs of Reciprocal Functions YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software f() = GOAL Sketch the graphs of reciprocals of linear and quadratic functions.
HOMEWORK SOLUTIONS MATH 1910 Sections 5.1, 5.2 Fall 2016
HOMEWORK SOLUTIOS MATH 9 Sections.,. Fall 6 Problem.. An ostrich runs with velocit km/h for minutes min), km/h for min, and km/h for another minute. Compute the total distance traveled and indicate with
May 27, QUADRATICS.notebook. Apr 26 17:43. Apr 26 18:27. Apr 26 18:40. Apr 28 10:22. Apr 28 10:34. Apr 28 10:33. Starter
1. Factorise: 2 - - 6 2. Solve for : 2( + 1) = - 1 3. Factorise: 2-25 To solve quadratic equations.. Factorise: 2 2-8 5. State the gradient of the line: + 12 = 2 Apr 26 17:3 Apr 26 18:27 Solving Quadratic
For use after the chapter Graphing Linear Equations and Functions 3 D. 7. 4y 2 3x 5 4; (0, 1) x-intercept: 6 y-intercept: 3.
Chapter Test A Write the coordinates of the point.. A. B. D. C. A. D C B.... Tell whether the ordered pair is a solution of the equation.. ; (, ) 7.. ; (, ). 7. ; (, ). Draw the line that has the given
Lesson 8.2 Exercises, pages
Lesson 8. Eercises, pages 38 A Students should verif the solutions to all equations.. Which values of are not roots of each equation? a) ƒ - 3 ƒ = 7 = 5 or =- Use mental math. 5: L.S. 7 R.S. 7 : L.S. 7
Algebra I Notes Direct Variation Unit 04e
OBJECTIVES: F.IF.B.4 Interpret functions that arise in applications in terms of the contet. For a function that models a relationship between two quantities, interpret ke features of graphs and tables
y sin n x dx cos x sin n 1 x n 1 y sin n 2 x cos 2 x dx y sin n xdx cos x sin n 1 x n 1 y sin n 2 x dx n 1 y sin n x dx
SECTION 7. INTEGRATION BY PARTS 57 EXAPLE 6 Prove the reduction formula N Equation 7 is called a reduction formula because the eponent n has been reduced to n and n. 7 sin n n cos sinn n n sin n where
5.2 Solving Linear-Quadratic Systems
Name Class Date 5. Solving Linear-Quadratic Sstems Essential Question: How can ou solve a sstem composed of a linear equation in two variables and a quadratic equation in two variables? Resource Locker
QUADRATIC FUNCTION REVIEW
Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important
Cartesian coordinates in space (Sect. 12.1).
Cartesian coordinates in space (Sect..). Overview of Multivariable Calculus. Cartesian coordinates in space. Right-handed, left-handed Cartesian coordinates. Distance formula between two points in space.
Tangent Line Approximations
60_009.qd //0 :8 PM Page SECTION.9 Dierentials Section.9 EXPLORATION Tangent Line Approimation Use a graphing utilit to graph. In the same viewing window, graph the tangent line to the graph o at the point,.
Chapter 3: Three faces of the derivative. Overview
Overview We alread saw an algebraic wa of thinking about a derivative. Geometric: zooming into a line Analtic: continuit and rational functions Computational: approimations with computers 3. The geometric
3.7 InveRSe FUnCTIOnS
CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
Interpret Linear Graphs
Interpret Linear Graphs Objectives: -Interpret the meaning of the and intercepts, slope, and points on and off the line of a graph, in the contet of a real world situation. Common Core Standards: N.Q.1
Lesson 6.2 Exercises, pages
Lesson 6.2 Eercises, pages 448 48 A. Sketch each angle in standard position. a) 7 b) 40 Since the angle is between Since the angle is between 0 and 90, the terminal 90 and 80, the terminal arm is in Quadrant.
5.3 Modelling Periodic Behaviour
5.3 Modelling Periodic Behaviour There are man eamples of periodic behaviour in nature. Familiar eamples include the rising and setting of the sun, and the rise and fall of tides. The rhthm of the human
All work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. What is the definition of a derivative?. What is the alternative definition of a
Number Plane Graphs and Coordinate Geometry
Numer Plane Graphs and Coordinate Geometr Now this is m kind of paraola! Chapter Contents :0 The paraola PS, PS, PS Investigation: The graphs of paraolas :0 Paraolas of the form = a + + c PS Fun Spot:
One of the most common applications of Calculus involves determining maximum or minimum values.
8 LESSON 5- MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..
LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
Linear Equation Theory - 2
Algebra Module A46 Linear Equation Theor - Copright This publication The Northern Alberta Institute of Technolog 00. All Rights Reserved. LAST REVISED June., 009 Linear Equation Theor - Statement of Prerequisite
CHAPTER 2 : LINEAR LAW Contents Page 3.0 CONCEPT MAP UNDERSTAND AND USE THE CONCEPT OF LINES OF BEST FIT Draw lines of best fit b inspecti
ADDITIONAL ATHEMATICS FORM 5 MODULE 3 LINEAR LAW CHAPTER 2 : LINEAR LAW Contents Page 3.0 CONCEPT MAP 2 3.1 UNDERSTAND AND USE THE CONCEPT OF LINES OF BEST FIT 3.1.1 Draw lines of best fit b inspection
Sections 5.1: Areas and Distances
Sections.: Areas and Distances In this section we shall consider problems closely related to the problems we considered at the beginning of the semester (the tangent and velocity problems). Specifically,
5A Exponential functions
Chapter 5 5 Eponential and logarithmic functions bjectives To graph eponential and logarithmic functions and transformations of these functions. To introduce Euler s number e. To revise the inde and logarithm
Review Sheet for Exam 1 SOLUTIONS
Math b Review Sheet for Eam SOLUTIONS The first Math b midterm will be Tuesday, February 8th, 7 9 p.m. Location: Schwartz Auditorium Room ) The eam will cover: Section 3.6: Inverse Trig Appendi F: Sigma