Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 lines at P and Q is an equilateral triangle ; Find the point where the curves 4 and are tangent to each other, that is, have a common tangent line Illustrate b sketching both curves and the common tangent Suppose f is a function that satisfies the equation P Q f f f B 0 C for all real numbers and Suppose also that FIGURE FOR PROBLEM 1 lim l0 f 1 FIGURE FOR PROBLEM 4 (a) Find f 0 (b) Find f 0 (c) Find f 4 A car is traveling at night along a highwa shaped like a parabola with its verte at the origin (see the figure) The car starts at a point 100 m west and 100 m north of the origin and travels in an easterl direction There is a statue located 100 m east and 50 m north of the origin At what point on the highwa will the car s headlights illuminate the statue? d n 5 Prove that d n sin4 cos 4 4 n1 cos4 n 6 Find the nth derivative of the function f n 1 7 The figure shows a circle with radius 1 inscribed in the parabola Find the center of the circle = If f is differentiable at a, where a 0, evaluate the following limit in terms of f a: 9 The figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length 1 m The pin P slides back and forth along the -ais as the wheel rotates counterclockwise at a rate of 60 revolutions per minute (a) Find the angular velocit of the connecting rod, ddt, in radians per second, when (b) Epress the distance OP in terms of (c) Find an epression for the velocit of the pin P in terms of lim l a f f a s sa A O å P (, 0)

2 CHALLENGE PROBLEMS 10 Tangent lines and are drawn at two points and on the parabola and the intersect at a point P Another tangent line T is drawn at a point between P 1 and P ; it intersects T 1 at Q 1 and T at Show that Q T 1 T P 1 P PQ1 PQ 1 PP1 PP T 0 N T N N P T 11 Let T and N be the tangent and normal lines to the ellipse at an point P on the ellipse in the first quadrant Let T and T be the - and -intercepts of T and N and N be the intercepts of N As P moves along the ellipse in the first quadrant (but not on the aes), what values can T, T, N, and N take on? First tr to guess the answers just b looking at the figure Then use calculus to solve the problem and see how good our intuition is sin sin 9 1 Evaluate lim l 0 1 (a) Use the identit for tan (see Equation 14b in Appendi A) to show that if two lines L and intersect at an angle, then L 1 FIGURE FOR PROBLEM 11 m m1 tan 1 m 1m where m 1 and m are the slopes of L 1 and L, respectivel (b) The angle between the curves C 1 and C at a point of intersection P is defined to be the angle between the tangent lines to C 1 and C at P (if these tangent lines eist) Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection (i) and (ii) and Let P 1, 1 be a point on the parabola 4p with focus F p, 0 Let be the angle between the parabola and the line segment FP, and let be the angle between the horizontal line 1 and the parabola as in the figure Prove that (Thus, b a principle of geometrical optics, light from a source placed at F will be reflected along a line parallel to the -ais This eplains wh paraboloids, the surfaces obtained b rotating parabolas about their aes, are used as the shape of some automobile headlights and mirrors for telescopes) = P(, ) å 0 F( p, 0) =4p A Q C R O P 15 Suppose that we replace the parabolic mirror of Problem 14 b a spherical mirror Although the mirror has no focus, we can show the eistence of an approimate focus In the figure, C is a semicircle with center O A ra of light coming in toward the mirror parallel to the ais along the line PQ will be reflected to the point R on the ais so that PQO OQR (the angle of incidence is equal to the angle of reflection) What happens to the point R as P is taken closer and closer to the ais? 16 If f and t are differentiable functions with f 0 t0 0 and t0 0, show that lim l 0 f t f 0 t0 FIGURE FOR PROBLEM 15 sina sina sin a 17 Evaluate lim l 0 18 Given an ellipse a b 1, where a b, find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals

3 CHALLENGE PROBLEMS 19 Find the two points on the curve 4 that have a common tangent line 0 Suppose that three points on the parabola have the propert that their normal lines intersect at a common point Show that the sum of their -coordinates is 0 1 A lattice point in the plane is a point with integer coordinates Suppose that circles with radius r are drawn using all lattice points as centers Find the smallest value of r such that an line with slope intersects some of these circles 5 A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1 cms into a tall clinder of radius R centimeters that is partiall filled with water How fast is the water level rising at the instant the cone is completel submerged? A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top It is partiall filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid (The surface area of a cone is rl, where r is the radius and l is the slant height) If we pour the liquid into the container at a rate of cm min, then the height of the liquid decreases at a rate of 0 cmmin when the height is 10 cm If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour the liquid into the container? CAS 4 (a) The cubic function f 6 has three distinct zeros: 0,, and 6 Graph f and its tangent lines at the average of each pair of zeros What do ou notice? (b) Suppose the cubic function f a b c has three distinct zeros: a, b, and c Prove, with the help of a computer algebra sstem, that a tangent line drawn at the average of the zeros a and b intersects the graph of f at the third zero

4 4 CHALLENGE PROBLEMS ANSWERS S Solutions 1 (s, 1 4 ) (a) 0 (b) 1 (c) f 1 7 (0, 5 4 ) 9 (a) 4ss11 rads (b) 40(cos s8 cos ) cm (c) 11 T,, T,, N (0, 5 ), N ( 5, 0) 1 (b) (i) 5 (or 17) (ii) 6 (or 117) 15 R approaches the midpoint of the radius AO 17 sin a 19 1,, 1, 0 1 s sin (1 cos s8 cos ) cms cm min

5 CHALLENGE PROBLEMS 5 SOLUTIONS E Eercises 1 Let a be the -coordinate of Q Since the derivative of =1 is 0 =, the slope at Q is a But since the triangle is equilateral, AO/OC = /1, so the slope at Q is Therefore, we must have that a = a = Thus, the point Q has coordinates, 1 =, 1 andbsmmetr,p has coordinates 4, 1 4 (a) Put =0and =0in the equation: f(0 + 0) = f(0) + f(0) f(0) = f(0) Subtracting f(0) from each side of this equation gives f(0) = 0 (b) f 0 f(0 + h) f(0) f(0) + f(h)+0 h +0h f(0) f(h) (0) =lim h 0 h =lim f() 0 =1 (c) f 0 f( + h) f() f()+f(h)+ h + h f() () =lim h 0 h f(h)+ h + h f(h) + + h =1+ 5 We use mathematical induction Let S n be the statement that S 1 is true because d n d n sin 4 +cos 4 =4 n 1 cos(4 + nπ/) d d sin 4 +cos 4 =4sin cos 4cos sin =4sin cos sin cos Now assume S k is true, that is, = 4sin cos cos = sin cos = sin 4 =sin( 4) =cos π ( 4) =cos π +4 =4 n 1 cos 4 + n π when n =1 d k sin 4 +cos 4 =4 k 1 cos 4 + k π d k Then d k+1 d sin 4 +cos 4 = d d k k+1 d d sin 4 +cos 4 = d k d 4 k 1 cos 4 + k π = 4 k 1 sin 4 + k π d d 4 + k π = 4 k sin 4 + k π =4 k sin 4 k π =4 k cos π 4 k π =4 k cos 4 +(k +1) π which shows that S k+1 is true Therefore, d n sin 4 +cos 4 =4 n 1 cos 4 + n π d n for ever positive integer n, b mathematical induction Another proof: First write sin 4 +cos 4 = sin +cos sin cos =1 1 sin =1 1 (1 cos 4) = cos 4 Then we have dn sin 4 +cos 4 = dn d n d + 1 cos 4 n 4 4 = 1 4 4n cos 4 + n π =4 n 1 cos 4 + n π 7 We must find a value 0 such that the normal lines to the parabola = at = ± 0 intersect at a point one unit from the points ± 0, 0 The normals to = at = ± 0 have slopes 1 and pass through ± 0, 0 ± 0 respectivel, so the normals have the equations 0 = 1 0 ( 0 ) and 0 = 1 0 ( + 0 ) The common

6 6 CHALLENGE PROBLEMS -intercept is Wewanttofind the value of 0 for which the distance from 0, to 0, 0 equals 1 The square of the distance is ( 0 0) ,the-intercept is = 5 4, so the center of the circle is at 0, 5 4 = =1 4 0 = ± For these values of Another solution: Let the center of the circle be (0,a) Then the equation of the circle is +( a) =1 Solving with the equation of the parabola, =,weget + a =1 + 4 a + a =1 4 +(1 a) + a 1=0 The parabola and the circle will be tangent to each other when this quadratic equation in hasequalroots;thatis,whenthediscriminantis0 Thus,(1 a) 4 a 1 =0 1 4a +4a 4a +4=0 4a =5,soa = 5 4 The center of the circle is 0, We can assume without loss of generalit that θ =0at time t =0,sothatθ =1πt rad [The angular velocit of the wheel is 60 rpm =60 (π rad)/(60 s) =1π rad/s] Then the position of A as a function of time is A =(40cosθ, 40 sin θ) = (40 cos 1πt, 40 sin 1πt),sosin α = 40 sin θ = = sin θ = 1 sin 1πt 1 m 10 (a) Differentiating the epression for sin α, wegetcos α dα dt = 1 1π cos 1πt =4π cos θ When θ = π,wehavesin α = 1 sin θ = 6,socos α = 1 dα dt = 4π cos π cos α = π 11/1 = 4π rad/s 6 = 11 1 and (b) B the Law of Cosines, AP = OA + OP OA OP cos θ 10 =40 + OP 40 OP cos θ OP (80 cos θ) OP 1,800 = 0 OP = 1 80 cos θ ± 6400 cos θ +51,00 =40cosθ ± 40 cos θ +8 =40 cos θ + 8+cos θ cm [since OP > 0] Asacheck,notethat OP = 160 cm when θ =0and OP =80 cm when θ = π (c) B part (b), the -coordinate of P is given b =40 cos θ + 8+cos θ,so d dt = d dθ dθ dt =40 cosθ sin θ sin θ cos θ 1π = 480π sin θ 1+ cm/s 8+cos θ 8+cos θ In particular, d/dt =0cm/swhenθ =0and d/dt = 480π cm/swhenθ = π 11 It seems from the figure that as P approaches the point (0, ) from the right, T and T + AsP approaches the point (, 0) from the left, it appears that T + and T Soweguessthat T (, ) and T (, ) Itismoredifficult to estimate the range of values for N and N We might perhaps guess that N (0, ),and N (, 0) or (, 0) In order to actuall solve the problem, we implicitl differentiate the equation of the ellipse to find the equation of the tangent line: = =0,so 0 = 4 9 So at the point ( 0, 0 ) on the ellipse, an equation of the tangent line is 0 = 4 0 ( 0 ) or = This can be written as 0

7 CHALLENGE PROBLEMS = =1 =1, because (0,0) lies on the ellipse So an equation of the tangent line is Therefore, the -intercept T for the tangent line is given b 0T 9 =1 T = 9 0,andthe-intercept T is given b 0T 4 =1 T = 4 0 So as 0 takes on all values in (0, ), T takes on all values in (, ),andas 0 takes on all values in (0, ), T takes on all values in (, ) At the point ( 0, 0 ) on the ellipse, the slope of the normal line is 1 0 ( 0, 0 ) = 9 0, and its equation is 0 = 9 0 ( 0) Sothe-intercept N for the normal line is given b 0 0 = 9 ( N 0) N = = 5 0,andthe-intercept N is given b 9 0 N 0 = 9 (0 0) N = = 50 4 So as 0 takes on all values in (0, ), N takes on all values in 0, 5,andas0 takes on all values in (0, ), N takes on all values in 5, 0 1 (a) If the two lines L 1 and L have slopes m 1 and m and angles of inclination φ 1 and φ,thenm 1 =tanφ 1 and m =tanφ The triangle in the figure shows that φ 1 + α + (180 φ )=180 and so α = φ φ 1 Therefore, using the identit for tan( ), we have tan α =tan(φ φ 1 )= tan φ tan φ 1 1+tanφ tan φ 1 and so tan α = m m1 1+m 1m (b) (i) The parabolas intersect when =( ) =1If =,then 0 =, so the slope of the tangent to = at (1, 1) is m 1 =(1)=If =( ),then 0 =( ), so the slope of the tangent to =( ) at (1, 1) is m =(1 ) = Therefore, tan α = m m1 = 1+m 1m 1+( ) = 4 and so α =tan (or 17 ) (ii) =and 4 + +=0intersect when 4 + +=0 ( ) = 0 =0or,but0 is etraneous If =,then = ±1If =then 0 =0 0 = / and 4 + += =0 0 = At (, 1) the slopes are m 1 =and m =0,sotan α = = α 117 At(, 1) the slopes are m 1 = and m =0, so tan α = 0 ( ) 1+( )(0) = α 6 (or 117 )

8 8 CHALLENGE PROBLEMS 15 Since ROQ = OQP = θ, the triangle QOR is isosceles, so QR = RO = BtheLawofCosines, = + r r cos θ Hence, r cos θ = r,so = (since sin θ = /r), and hence r r cos θ = r cosθ Note that as 0+, θ 0 + r cos0 = r Thus,asP is taken closer and closer to the -ais, the point R approaches the midpoint of the radius AO 17 lim 0 sin(a +) sin(a + )+sina 0 sin a cos +cosa sin sina cos cosa sin +sina 0 sin a (cos cos +1)+cosa (sin sin) 0 sin a ( cos 1 cos +1)+cosa ( sin cos sin) 0 sin a ( cos )(cos 1) + cos a ( sin )(cos 1) (cos 1)[sin a cos +cosa sin ](cos +1) 0 (cos +1) sin [sin(a + )] sin sin(a + ) = lim 0 (cos +1) 0 cos +1 = sin(a +0) (1) cos = sin a 19 = 4 0 =4 4 1 The equation of the tangent line at = a is (a 4 a a) =(4a 4a 1)( a) or =(4a 4a 1) +( a 4 +a ) and similarl for = b So if at = a and = b we have the same tangent line, then 4a 4a 1=4b 4b 1 and a 4 +a = b 4 +b Thefirst equation gives a b = a b (a b)(a + ab + b )=(a b) Assuming a 6=b, wehave1=a + ab + b The second equation gives (a 4 b 4 )=(a b ) (a b )(a + b )=(a b ) which is true if a = b Substituting into 1=a + ab + b gives 1=a a + a a = ±1 so that a =1and b = 1 or vice versa Thus, the points (1, ) and ( 1, 0) have a common tangent line As long as there are onl two such points, we are done So we show that these are in fact the onl two such points Suppose that a b 6=0Then(a b )(a + b )=(a b ) gives (a + b )= or a + b = Thus, ab = a + ab + b a + b =1 = 1,sob = 1 a Hence, a + 1 9a =,so9a4 +1=6a 0=9a 4 6a +1= a 1 Soa 1=0 a = 1 b = 1 9a = 1 = a, contradicting our assumption that a 6=b 1 Because of the periodic nature of the lattice points, it suffices to consider the points in the 5 grid shown We can

9 CHALLENGE PROBLEMS 9 see that the minimum value of r occurs when there is a line with slope 5 which touches the circle centered at (, 1) and the circles centered at (0, 0) and (5, ) Tofind P, the point at which the line is tangent to the circle at (0, 0), we simultaneousl solve + = r and = = r = 4 r = r, = 5 rtofind Q, we either use smmetr or solve ( ) +( 1) = r and 1= 5 ( ) Asabove,weget = r, =1+ 5 r Now the slope of the line PQis,so 5 m PQ = 1+ 5 r 5 r r r = r 4 r = + 10r 4r = r =6 8r 58r = r = So the minimum value of r for which an line with slope 58 5 intersects circles with radius r centered at the lattice points on the plane is r = B similar triangles, r 5 = h 16 Equating the two epressions for dv dt l = 5 8 gives us k = r = 5h The volume of the cone is 16 5h V = 1 πr h = 1 π h = 5π h,so dv dt = 5π dh 56 h dt Nowtherateof change of the volume is also equal to the difference of what is being added ( cm /min) and what is oozing out (kπrl,whereπrl is the area of the cone and k is a proportionalit constant) Thus, dv dt and substituting h =10, dh dt = kπrl = 0, r = 5(10) 16 81,weget 5π 56 (10) ( 0) = kπ kπ = 5 8,and l 81 = =+ 750π Solving for k π 50π To maintain a certain height, the rate of oozing, kπrl, must equal the rate of the liquid 81 beingpouredin;thatis, dv dt π =0 kπrl = 50π 81 π π = 1104 cm /min 8 18

### 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.

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()

### 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

### C1: Coordinate geometry of straight lines

B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

### Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304

Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

### 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...

### 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

### 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,

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

### 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.

### Quadratic Functions. MathsStart. Topic 3

MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE

### 6.3 Parametric Equations and Motion

SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh

### 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 +

### HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 2001 MATHEMATICS PAPER 2

00-CE MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 00 MATHEMATICS PAPER 5 am 45 pm (½ hours) Subject Code 80 Read carefull the instructions on the Answer Sheet

### Functions and Graphs

PSf Functions and Graphs Paper 1 Section B 1. The points A and B have coordinates (a, a 2 ) and (2b, 4b 2 ) respectivel. Determine the gradient of AB in its simplest form. 2 2. hsn.uk.net Page 1 Questions

### Mathematics. Total marks 100. Section I Pages marks Attempt Questions 1 10 Allow about 15 minutes for this section

04 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators ma be

### AP Calculus Testbank (Chapter 7) (Mr. Surowski)

AP Calculus Testbank (Chapter 7) (Mr. Surowski) Part I. Multiple-Choice Questions. Suppose that a function = f() is given with f() for 4. If the area bounded b the curves = f(), =, =, and = 4 is revolved

### Archimedes quadrature of the parabola and the method of exhaustion

JOHN OTT COLLEGE rchimedes quadrature of the parabola and the method of ehaustion CLCULUS II SCIENCE) Carefull stud the tet below and attempt the eercises at the end. You will be evaluated on this material

### APPLICATION OF DERIVATIVES

6. Overview 6.. Rate of change of quantities For the function y f (x), d (f (x)) represents the rate of change of y with respect to x. dx Thus if s represents the distance and t the time, then ds represents

### HI-RES STILL TO BE SUPPLIED

1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

### CHAPTER 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.

### SL Calculus Practice Problems

Alei - Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram

### CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

### Focusing properties of spherical and parabolic mirrors

Physics 5B Winter 2009 Focusing properties of spherical and parabolic mirrors 1 General considerations Consider a curved mirror surface that is constructed as follows Start with a curve, denoted by y()

### 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.

### Section 10-5 Parametric Equations

88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### SLOPES AND EQUATIONS OF LINES CHAPTER

CHAPTER 90 8 CHAPTER TABLE OF CONTENTS 8- The Slope of a Line 8- The Equation of a Line 8-3 Midpoint of a Line Segment 8-4 The Slopes of Perpendicular Lines 8-5 Coordinate Proof 8-6 Concurrence of the

### 3 Rectangular Coordinate System and Graphs

060_CH03_13-154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter

### 13 Graphs, Equations and Inequalities

13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,

8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight

### STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

### 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

### 116 Chapter 6 Transformations and the Coordinate Plane

116 Chapter 6 Transformations and the Coordinate Plane Chapter 6-1 The Coordinates of a Point in a Plane Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive

### Change of Coordinates in Two Dimensions

CHAPTER 5 Change of Coordinates in Two Dimensions Suppose that E is an ellipse centered at the origin. If the major and minor aes are horiontal and vertical, as in figure 5., then the equation of the ellipse

### DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

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

### 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 P(X, Y) X

Rotation of Aes For a discussion of conic sections, see Appendi. In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the general second-degree

### Chapter 12. The Straight Line

302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

### Trigonometric Functions

LIALMC5_1768.QXP /6/4 1:4 AM Page 47 5 Trigonometric Functions Highwa transportation is critical to the econom of the United States. In 197 there were 115 billion miles traveled, and b the ear this increased

### Higher. Polynomials and Quadratics 64

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

### INTEGRATION FINDING AREAS

INTEGRTIN FINDING RES Created b T. Madas Question 1 (**) = 4 + 10 3 The figure above shows the curve with equation = 4 + 10, R. Find the area of the region, bounded b the curve the coordinate aes and the

### THE PARABOLA section. Developing the Equation

80 (-0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or

### REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

### 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

### Math 21a Old Exam One Fall 2003 Solutions Spring, 2009

1 (a) Find the curvature κ(t) of the curve r(t) = cos t, sin t, t at the point corresponding to t = Hint: You ma use the two formulas for the curvature κ(t) = T (t) r (t) = r (t) r (t) r (t) 3 Solution:

### 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

### 2.1 Three Dimensional Curves and Surfaces

. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

### Lines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line

Lines and planes in space (Sect. 2.5) Lines in space (Toda). Review: Lines on a plane. The equations of lines in space: Vector equation. arametric equation. Distance from a point to a line. lanes in space

### 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.

### 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

### COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS

G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs

### Polynomial and Rational Functions

Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

### 4.1 Radian and Degree Measure

Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position

### Mathematics Paper 1 (Non-Calculator)

H National Qualifications CFE Higher Mathematics - Specimen Paper A Duration hour and 0 minutes Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full

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

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### 17.1. Conic Sections. Introduction. Prerequisites. Learning Outcomes. Learning Style

Conic Sections 17.1 Introduction The conic sections (or conics) - the ellipse, the parabola and the hperbola - pla an important role both in mathematics and in the application of mathematics to engineering.

### Solving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots

### If (a)(b) 5 0, then a 5 0 or b 5 0.

chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic

### Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom

Free Response Questions 1969-005 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions

### Graphing Nonlinear Systems

10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities

### CHAPTER 54 SOME APPLICATIONS OF DIFFERENTIATION

CHAPTER 5 SOME APPLICATIONS OF DIFFERENTIATION EXERCISE 0 Page 65. An alternating current, i amperes, is given b i = 0 sin πft, where f is the frequenc in hertz and t the time in seconds. Determine the

.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

### REVIEW, pages

REVIEW, pages 69 697 8.. Sketch a graph of each absolute function. Identif the intercepts, domain, and range. a) = ƒ - + ƒ b) = ƒ ( + )( - ) ƒ 8 ( )( ) Draw the graph of. It has -intercept.. Reflect, in

### 27.2. Multiple Integrals over Non-rectangular Regions. Introduction. Prerequisites. Learning Outcomes

Multiple Integrals over Non-rectangular Regions 7. Introduction In the previous Section we saw how to evaluate double integrals over simple rectangular regions. We now see how to etend this to non-rectangular

### SECTION 9-1 Conic Sections; Parabola

66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships

### Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

### Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

### COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS

a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus 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

### FLC Ch 1 & 3.1. A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A.

Math 335 Trigonometry Sec 1.1: Angles Definitions A line is an infinite set of points where between any two points, there is another point on the line that lies between them. Line AB, A line segment is

### x 2 would be a solution to d y

CATHOLIC JUNIOR COLLEGE H MATHEMATICS JC PRELIMINARY EXAMINATION PAPER I 0 System of Linear Equations Assessment Objectives Solution Feedback To use a system of linear c equations to model and solve y

### 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

### 2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2

00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

### y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx

Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

### 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

### Section 7-2 Ellipse. Definition of an Ellipse The following is a coordinate-free definition of an ellipse: DEFINITION

7- Ellipse 3. Signal Light. A signal light on a ship is a spotlight with parallel reflected light ras (see the figure). Suppose the parabolic reflector is 1 inches in diameter and the light source is located

### 2 Analysis of Graphs of

ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

### 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

### Solution for Final Review Problems 1

Solution for Final Review Problems 1 (1) Compute the following its. (a) ( 2 + 1 2 1) ( 2 + 1 2 1) ( 2 + 1 2 1)( 2 + 1 + 2 1) 2 + 1 + 2 1 2 2 + 1 + 2 1 = (b) 1 3 3 1 (c) 3 1 3 1 ( 1)( 2 + ) 1 ( 1)( 2 +

### Functions. Chapter A B C D E F G H I J. The reciprocal function x

Chapter 1 Functions Contents: A B C D E F G H I J Relations and functions Function notation, domain and range Composite functions, f± g Sign diagrams Inequalities (inequations) The modulus function 1 The

### Catholic Schools Trial Examination 2004 Mathematics

0 Catholic Trial HSC Examination Mathematics Page Catholic Schools Trial Examination 0 Mathematics a If x 5 = 5000, find x correct to significant figures. b Express 0. + 0.. in the form b a, where a and

### 3 Unit Circle Trigonometry

0606_CH0_-78.QXP //0 :6 AM Page Unit Circle Trigonometr In This Chapter. The Circular Functions. Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions. Special Identities.5 Inverse

### Warm-Up y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.

CST/CAHSEE: Warm-Up Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and

### Chapter 3A - Rectangular Coordinate System

- Chapter A Chapter A - Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been

### Mathematics 1. Lecture 5. Pattarawit Polpinit

Mathematics 1 Lecture 5 Pattarawit Polpinit Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance

### 2-5. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math

Chapter 2 Lesson 2-5 The Graph of = k 2 BIG IDEA The graph of the set of points (, ) satisfing = k 2, with k constant, is a parabola with verte at the origin and containing the point (1, k). Vocabular

### 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

### Analyzing the Graph of a Function

SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the

### 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

### 1. The Six Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1.2 The Rectangular Coordinate System 1.3 Definition I: Trigonometric

1. The Si Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1. The Rectangular Coordinate Sstem 1.3 Definition I: Trigonometric Functions 1.4 Introduction to Identities 1.5 More on Identities

### Graphing and transforming functions

Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6

A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which

### 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

### 1. By how much does 1 3 of 5 2 exceed 1 2 of 1 3? 2. What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3.

1 By how much does 1 3 of 5 exceed 1 of 1 3? What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3 A ticket fee was \$10, but then it was reduced The number of customers