Colegio del mundo IB. Programa Diploma REPASO The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

Size: px
Start display at page:

Download "Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t."

Transcription

1 REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f() = 6 +, for. Write f() in the form ( a) + b. Find the inverse function f. (c) State the domain of f.. The equation k + + = 0 has eactl one solution. Find the value of k.

2 REPASO 4. q The diagram shows part of the graph of the function f() =. p through the point A (, 0). The line (CD) is an asmptote. C 5 The curve passes 0 A D Find the value of (i) p; (ii) q.

3 REPASO The graph of f() is transformed as shown in the following diagram. The point A is transformed to A (, 0). 5 C A 5 D Give a full geometric description of the transformation.

4 4 REPASO 5. The diagram shows part of the graph of the curve = a ( h) + k, where a, h, k The verte is at the point (, ). Write down the value of h and of k. The point P(5, 9) is on the graph. Show that a =. () () (c) Hence show that the equation of the curve can be written as = + 9. () (d) (i) Find d. d A tangent is drawn to the curve at P (5, 9). (ii) (iii) Calculate the gradient of this tangent, Find the equation of this tangent. (4) (Total 0 marks) 4

5 5 REPASO 6. A famil of functions is given b f() = + + k, where k {,,, 4, 5, 6, 7}. One of these functions is chosen at random. Calculate the probabilit that the curve of this function crosses the -ais. 7. Let f() = e, and g() = +,. Find f () (g f)(). 8. The sketch shows part of the graph of = f() which passes through the points A(, ), B(0, ), C(l, 0), D(, ) and E(, 5) E 4 A B D C

6 6 REPASO A second function is defined b g() = f( ). Calculate g(0), g(), g() and g(). On the same aes, sketch the graph of the function g(). 9. Let g() = 4 +. Solve g() = 0. () Let f() = +. A part of the graph of f() is shown below. g( ) C A 0 B The graph has vertical asmptotes with equations = a and = b where a < b. Write down the values of (i) a; (c) (ii) b. The graph has a horizontal asmptote with equation = l. Eplain wh the value of f() approaches as becomes ver large. () () (d) The graph intersects the -ais at the points A and B. Write down the eact value of the - coordinate at (i) A; (ii) B. () 6

7 7 REPASO (e) The curve intersects the -ais at C. Use the graph to eplain wh the values of f () and f () are zero at C. () (Total 0 marks) 0. Given that f() = e, find the inverse function f ().. Consider the functions f : a 4 ( ) and g : a 6. Find g. Solve the equation (f g ) () = 4.. Solve the equation e = 5, giving our answer correct to four significant figures. á. Consider the function f() = Epress f() in the form a ( p) + q, where a, p, q. Find the minimum value of f(). 4. Consider functions of the form = e k Show that e d = Let k = k ( e k ). k () (i) Sketch the graph of = e 0.5, for, indicating the coordinates of the -intercept. (ii) Shade the region enclosed b this graph, the -ais, -ais and the line =. 7

8 8 REPASO (iii) Find the area of this region. (5) (c) (i) Find d in terms of k, where = e k. d The point P(, 0.8) lies on the graph of the function = e k. (ii) Find the value of k in this case. (iii) Find the gradient of the tangent to the curve at P. (5) (Total marks) 5. The diagram shows part of the graph of = a ( h) + k. The graph has its verte at P, and passes through the point A with coordinates (, 0). P A 0 Write down the value of (i) h; (ii) k. Calculate the value of a. 6. Let f() =, and g() =, ( ). 8

9 9 REPASO Find (g f) (); g (5). 7. The diagram below shows part of the graph of the function f : a A P 5 0 Q B The graph intercepts the -ais at A(,0), B(5,0) and the origin, O. There is a minimum point at P and a maimum point at Q. The function ma also be written in the form where a < b. Write down the value of (i) a; f : a ( a) ( b), (ii) b. () Find (i) f (); 9

10 0 REPASO (ii) the eact values of at which f '() = 0; (iii) the value of the function at Q. (7) (c) (i) Find the equation of the tangent to the graph of f at O. (ii) This tangent cuts the graph of f at another point. Give the -coordinate of this point. (4) (d) Determine the area of the shaded region. 8. In the diagram below, the points O(0, 0) and A(8, 6) are fied. The angle varies as the point P(, 0) moves along the horizontal line = 0. OPˆ A () (Total 5 marks) P(, 0) =0 A(8, 6) O(0, 0) (i) Show that AP = diagram to scale (ii) Write down a similar epression for OP in terms of. () Hence, show that cos OPˆA = {( ) (, + 00)} () (c) Find, in degrees, the angle OPˆ A when = 8. (d) Find the positive value of such that O PˆA = 60. () (4) 0

11 REPASO Let the function f be defined b f ( ) = cos OPˆA =, 0 5. {( ) ( + 00)} (e) Consider the equation f () =. (i) Eplain, in terms of the position of the points O, A, and P, wh this equation has a solution. (ii) Find the eact solution to the equation. 9. The diagram below shows the graph of = sin, for 0 < m, and 0 < n, where is in radians and m and n are integers. n (5) n 0 m m Find the value of m; n.

12 REPASO 0. The diagram shows parts of the graphs of = and = 5 ( 4). 8 = 6 = 5 ( 4) The graph of = ma be transformed into the graph of = 5 ( 4) b these transformations. A reflection in the line = 0 a vertical stretch with scale factor k a horizontal translation of p units a vertical translation of q units. followed b followed b followed b Write down the value of k; p; (c) q.. Factorise 0. Solve the equation 0 = 0.

13 REPASO. The diagram represents the graph of the function f : a ( p) ( q). C Write down the values of p and q. The function has a minimum value at the point C. Find the -coordinate of C. Total 4 marks). The diagrams show how the graph of = is transformed to the graph of = f() in three steps. For each diagram give the equation of the curve. 0 = 0 (c)

14 4 REPASO 4. On the following diagram, sketch the graphs of = e and = cos for. 0 The equation e = cos has a solution between and. Find this solution. 5. The function f is defined b f : a,. Evaluate f (5). 6. The following diagram shows the graph of = f (). It has minimum and maimum points at 4

15 5 REPASO (0, 0) and (, )..5 On the same diagram, draw the graph of = f ( ) +. What are the coordinates of the minimum and maimum points of = f ( ) +? 7. Michele invested 500 francs at an annual rate of interest of 5.5 percent, compounded annuall. Find the value of Michele s investment after ears. Give our answer to the nearest franc. () 5

16 6 REPASO How man complete ears will it take for Michele s initial investment to double in value? () (c) What should the interest rate be if Michele s initial investment were to double in value in 0 ears? (4) (Total 0 marks) 8. Note: Radians are used throughout this question. Let f () = sin ( + sin ). (i) Sketch the graph of = f (), for 0 6. (ii) Write down the -coordinates of all minimum and maimum points of f, for 0 6. Give our answers correct to four significant figures. (9) Let S be the region in the first quadrant completel enclosed b the graph of f and both coordinate aes. (i) Shade S on our diagram. (ii) Write down the integral which represents the area of S. (iii) Evaluate the area of S to four significant figures. (5) (c) Give reasons wh f () 0 for all values of. () 6

17 7 REPASO 9. f() = 4 sin + π. For what values of k will the equation f() = k have no solutions? 0. A group of ten leopards is introduced into a game park. After t ears the number of leopards, N, is modelled b N = 0 e 0.4t. How man leopards are there after ears? How long will it take for the number of leopards to reach 00? Give our answers to an appropriate degree of accurac. Give our answers to an appropriate degree of accurac.. Consider the function f : a +, Determine the inverse function f. What is the domain of f?. The diagram shows the graph of = f(), with the -ais as an asmptote. B(5, 4) A( 5, 4) On the same aes, draw the graph of =f( + ), indicating the coordinates of the 7

18 8 REPASO images of the points A and B. Write down the equation of the asmptote to the graph of = f( + ).. Sketch, on the given aes, the graphs of = and sin for Find the positive solution of the equation = sin, giving our answer correct to 6 significant figures. 8

19 9 REPASO 4. A ball is thrown verticall upwards into the air. The height, h metres, of the ball above the ground after t seconds is given b h = + 0t 5t, t 0 (c) Find the initial height above the ground of the ball (that is, its height at the instant when it is released). Show that the height of the ball after one second is 7 metres. At a later time the ball is again at a height of 7 metres. () () (i) (ii) Write down an equation that t must satisf when the ball is at a height of 7 metres. Solve the equation algebraicall. (4) (d) (i) Find d h. dt (ii) (iii) Find the initial velocit of the ball (that is, its velocit at the instant when it is released). Find when the ball reaches its maimum height. (iv) Find the maimum height of the ball. (7) (Total 5 marks) 5. The diagram shows part of the graph with equation = + p + q. The graph cuts the -ais at 9

20 0 REPASO and Find the value of p; q. 6. Each ear for the past five ears the population of a certain countr has increased at a stead rate of.7% per annum. The present population is 5. million. What was the population one ear ago? What was the population five ears ago? 7. The diagram shows the graph of the function = a + b + c. 0

21 REPASO Complete the table below to show whether each epression is positive, negative or zero. Epression positive negative zero a c b 4ac b 8. Initiall a tank contains litres of liquid. At the time t = 0 minutes a tap is opened, and liquid then flows out of the tank. The volume of liquid, V litres, which remains in the tank after t minutes is given b V = (0.9 t ). (c) Find the value of V after 5 minutes. Find how long, to the nearest second, it takes for half of the initial amount of liquid to flow out of the tank. The tank is regarded as effectivel empt when 95% of the liquid has flowed out. Show that it takes almost three-quarters of an hour for this to happen. () () () (d) (i) Find the value of V when t = 0.00 minutes. (ii) Hence or otherwise, estimate the initial flow rate of the liquid. Give our answer in litres per minute, correct to two significant figures. () (Total 0 marks) 9. Epress f() = in the form f() = ( h) + k, where h and k are to be determined.

22 REPASO Hence, or otherwise, write down the coordinates of the verte of the parabola with equation The diagram shows three graphs. B A C A is part of the graph of =. B is part of the graph of =. C is the reflection of graph B in line A. Write down the equation of C in the form =f(); the coordinates of the point where C cuts the -ais. 4. The quadratic equation 4 + 4k + 9 = 0, k > 0 has eactl one solution for. Find the value of k. 4. Two functions f, g are defined as follows: f : + 5

23 REPASO Find g : ( ) f (); (g o f)( 4). 4. Two functions f and g are defined as follows: f() = cos, 0 = π; g() = +,. Solve the equation (g o f) () = The function f is given b F() = +,,. (i) Show that = is an asmptote of the graph of = f(). () (ii) (iii) Find the vertical asmptote of the graph. Write down the coordinates of the point P at which the asmptotes intersect. () () (c) Find the points of intersection of the graph and the aes. Hence sketch the graph of = f(), showing the asmptotes b dotted lines. (4) (4) (d) Show that f () = 7 ( ) the point S where = 4. and hence find the equation of the tangent at (6) (e) The tangent at the point T on the graph is parallel to the tangent at S. Find the coordinates of T. (5)

24 4 REPASO (f) Show that P is the midpoint of [ST]. (l) 45. The diagram shows the parabola = (7 )(l + ). The points A and C are the -intercepts and the point B is the maimum point. B A 0 C Find the coordinates of A, B and C. 46. The function f is given b f() = n ( ). Find the domain of the function. 47. Let f() =, and g() =. Solve the equation (f o g)() = A population of bacteria is growing at the rate of. % per minute. How long will it take for the size of the population to double? Give our answer to the nearest minute. 49. Three of the following diagrams I, II, III, IV represent the graphs of = + cos = cos( + ) 4

25 5 REPASO (c) = cos +. Identif which diagram represents which graph. I II 4 π π π π π π π π π π III IV 5 4 π π π π π π π π π π 50. \$000 is invested at 5% per annum interest, compounded monthl. Calculate the minimum number of months required for the value of the investment to eceed \$000. 5

INVESTIGATIONS 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.

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

Polynomials Past Papers Unit 2 Outcome 1

PSf Polnomials Past Papers Unit 2 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor

AP Calculus AB 2005 Scoring Guidelines Form B

AP Calculus AB 5 coring Guidelines Form B The College Board: Connecting tudents to College uccess The College Board is a not-for-profit membership association whose mission is to connect students to college

Graphing Trigonometric Skills

Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE

M122 College Algebra Review for Final Exam

M122 College Algebra Review for Final Eam Revised Fall 2007 for College Algebra in Contet All answers should include our work (this could be a written eplanation of the result, a graph with the relevant

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

Core 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...

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

Circles - Past Edexcel Exam Questions

ircles - Past Edecel Eam Questions 1. The points A and B have coordinates (5,-1) and (13,11) respectivel. (a) find the coordinates of the mid-point of AB. [2] Given that AB is a diameter of the circle,

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

When I was 3.1 POLYNOMIAL FUNCTIONS

146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

C3: Functions. Learning objectives

CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

Identifying second degree equations

Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

y intercept Gradient Facts Lines that have the same gradient are PARALLEL

CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or

135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.

13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -

SAMPLE. Polynomial functions

Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

Section 5-9 Inverse Trigonometric Functions

46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

THE PARABOLA 13.2. section

698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.

Start Accuplacer. Elementary Algebra. Score 76 or higher in elementary algebra? YES

COLLEGE LEVEL MATHEMATICS PRETEST This pretest is designed to give ou the opportunit to practice the tpes of problems that appear on the college-level mathematics placement test An answer ke is provided

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

5.2 Inverse Functions

78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

AP Calculus AB 2004 Scoring Guidelines

AP Calculus AB 4 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from

D.3. Angles and Degree Measure. Review of Trigonometric Functions

APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

Straight Line. Paper 1 Section A. O xy

PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of

Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

SECTION 5-1 Exponential Functions

354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

Polynomial 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

MATH 185 CHAPTER 2 REVIEW

NAME MATH 18 CHAPTER REVIEW Use the slope and -intercept to graph the linear function. 1. F() = 4 - - Objective: (.1) Graph a Linear Function Determine whether the given function is linear or nonlinear..

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

COMPONENTS OF VECTORS

COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two

Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Algebra and coordinate geometry: Module 2. Coordinate geometry

1 Supporting Australian Mathematics Project 3 4 5 6 7 8 9 1 11 1 A guide for teachers Years 11 and 1 Algebra and coordinate geometr: Module Coordinate geometr Coordinate geometr A guide for teachers (Years

*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Non-calculator) NATIONAL QUALIFICATIONS 2014 TUESDAY, 6 MAY 1.00 PM 2.30 PM

X00//0 NTIONL QULIFITIONS 0 TUESY, 6 MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Non-calculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (0 marks) Instructions for completion

Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems

Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write

2.7 Applications of Derivatives to Business

80 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics Cost = C() In recent ears, economic decision making has become more and more mathematicall oriented.

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

5.3 Graphing Cubic Functions

Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

I think that starting

. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

STRAND: ALGEBRA Unit 3 Solving Equations

CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

MATHEMATICS: PAPER I. 5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.

NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 015 MATHEMATICS: PAPER I Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 1 pages and an Information

SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xy-plane. It is sketched in Figure 11.

SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651 SOLUTION The inequalities 1 2 2 2 4 can be rewritten as 2 FIGURE 11 1 0 1 s 2 2 2 2 so the represent the points,, whose distance from the origin is

Implicit Differentiation

Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some

In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)

Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and

REVIEW OF CONIC SECTIONS

REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

Rotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012

Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to

Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x

OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations

Lines and Planes 1. x(t) = at + b y(t) = ct + d

1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b

9.5 CALCULUS AND POLAR COORDINATES

smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS 379 Chapter 9 DIFFERENTIAL EQUATIONS He who seeks f methods without having a definite problem in mind seeks f the most part in vain. D. HILBERT 9. Introduction In Class XI and in

Connecting Transformational Geometry and Transformations of Functions

Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.

Equation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1

Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows :- gradient = vertical horizontal horizontal A B vertical

Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular

6706_PM10SB_C4_CO_pp192-193.qxd 5/8/09 9:53 AM Page 192 192 NEL

92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different

Core Maths C3. Revision Notes

Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...

Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

Polynomial and Rational Functions

Polnomial and Rational Functions 3 A LOOK BACK In Chapter, we began our discussion of functions. We defined domain and range and independent and dependent variables; we found the value of a function and

More Equations and Inequalities

Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

GRADE 1 TUTORIALS LO Topic Page 1 Number patterns and sequences 3 1 Functions and graphs 5 Algebra and equations 10 Finance 13 Calculus 15 Linear Programming 18 3 Analytical Geometry 4 3 Transformation

LINEAR FUNCTIONS OF 2 VARIABLES

CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for

Exponential equations will be written as, where a =. Example 1: Determine a formula for the exponential function whose graph is shown below.

.1 Eponential and Logistic Functions PreCalculus.1 EXPONENTIAL AND LOGISTIC FUNCTIONS 1. Recognize eponential growth and deca functions 2. Write an eponential function given the -intercept and another

2014 2015 Geometry B Exam Review

Semester Eam Review 014 015 Geometr B Eam Review Notes to the student: This review prepares ou for the semester B Geometr Eam. The eam will cover units 3, 4, and 5 of the Geometr curriculum. The eam consists

CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study

Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM

. Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,

2312 test 2 Fall 2010 Form B

2312 test 2 Fall 2010 Form B 1. Write the slope-intercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function

Core 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

Thnkwell 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

Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

Exponential Functions. Exponential Functions and Their Graphs. Example 2. Example 1. Example 3. Graphs of Exponential Functions 9/17/2014

Eponential Functions Eponential Functions and Their Graphs Precalculus.1 Eample 1 Use a calculator to evaluate each function at the indicated value of. a) f ( ) 8 = Eample In the same coordinate place,

1 Maximizing pro ts when marginal costs are increasing

BEE12 Basic Mathematical Economics Week 1, Lecture Tuesda 12.1. Pro t maimization 1 Maimizing pro ts when marginal costs are increasing We consider in this section a rm in a perfectl competitive market

PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

CHALLENGE PROBLEMS CHALLENGE PROBLEMS: CHAPTER 0 A Click here for answers. S Click here for solutions. m m m FIGURE FOR PROBLEM N W F FIGURE FOR PROBLEM 5. Each edge of a cubical bo has length m. The bo

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

Exponential and Logarithmic Functions

Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

o Graph an expression as a function of the chosen independent variable to determine the existence of a minimum or maximum

Two Parabolas Time required 90 minutes Teaching Goals:. Students interpret the given word problem and complete geometric constructions according to the condition of the problem.. Students choose an independent

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

1. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

Review 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

North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the

3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?

Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using