Click here for answers.


 Junior Gregory
 2 years ago
 Views:
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) ,theintercept 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,andtheintercept 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) Sotheintercept N for the normal line is given b 0 0 = 9 ( N 0) N = = 5 0,andtheintercept 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 radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive 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()
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationREVIEW 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
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.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
More informationSection 14.6 Tangent planes and differentials
Section 14.6 Tangent planes and differentials (3/23/08) Overview: In this section we stud linear functions of two variables and equations of tangent planes to the graphs of functions of two variables.
More information8 FURTHER CALCULUS. 8.0 Introduction. 8.1 Implicit functions. Objectives. Activity 1
8 FURTHER CALCULUS Chapter 8 Further Calculus Objectives After studing this chapter ou should be able to differentiate epressions defined implicitl; be able to use approimate methods for integration such
More informationYears 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 Straightline depreciation. The Circle Definition Anone who has drawn a circle using a compass
More informationSECTION 115 Parametric Equations. Parametric Equations and Plane Curves. Parametric Equations and Plane Curves Projectile Motion Cycloid
 Parametric Equations 9. A hperbola with foci (, ) and (6, ) and vertices (, ) and (, ).. An ellipse with foci (, ) and (, 6) and vertices (, ) and (, ).. A parabola with ais the ais and passing through
More informationD.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
More informationAngles and Degree Measure. Figure D.25 Figure D.26
APPENDIX D. Review of Trigonometric Functions D7 APPE N DIX D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D. Review of Trigonometric Functions D7 APPENDIX D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving
More information1. 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,
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationTHE 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.
More information10.2. Introduction to Conics: Parabolas. Conics. What you should learn. Why you should learn it
3330_00.qd /8/05 9:00 AM Page 735 Section 0. Introduction to Conics: Parabolas 735 0. Introduction to Conics: Parabolas What ou should learn Recognize a conic as the intersection of a plane and a doublenapped
More information1.1 Rectangular Coordinates; Graphing Utilities
CHAPTER 1 Graphs 1.1 Rectangular Coordinates; Graphing Utilities PREPARING FOR THIS SECTION Before getting started, review the following: Algebra Review (Appendi, Section A.1, pp. 951 959) Geometr Review
More informationSolutions 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
More informationIdentifying 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 +
More informationHigher. Differentiation 1
Higher Mathematics Contents Introduction to RC Finding the Derivative RC Differentiating with Respect to Other Variables RC 4 Rates of Change RC 7 5 Equations of Tangents RC 8 Increasing and Decreasing
More information2.1 The Tangent and Velocity Problems
. The Tangent and Velocit Problems Suppose we want to find the slope of the line between (, ) and (, ). We could do this easil with the slope formula: below. m = = =. Now consider the function =. Suppose
More informationLesson 4.6 Exercises, pages
Lesson 4.6 Eercises, pages 306 311 A 3. Identif the intercepts of the graph of each quadratic function. a = ( + 3(  b = (  4(  5 The intercepts are: 3 and The intercepts are: 4 and 5 c = ( + 1( +
More informationQUADRATIC RELATIONS AND FUNCTIONS
CHAPTER 3 CHAPTER TABLE OF CONTENTS 3 Solving Quadratic Equations 32 The Graph of a Quadratic Function 33 Finding Roots from a Graph 34 Graphic Solution of a QuadraticLinear Sstem 35 Algebraic Solution
More informationQuadratic 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
More information6.3 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
More informationGCSE Further Mathematics
THOMS WHITHM SIXTH FORM GCSE Further Mathematics Revision Guide S J Cooper The book contains a number of worked eamples covering the topics needed in the Further Mathematics Specifications. This includes
More informationInvestigate Slopes of Parallel and Perpendicular Lines
7. Parallel and Perpendicular Lines Focus on identifing whether two lines are parallel, perpendicular, or neither writing the equation of a line using the coordinates of a point on the line and the equation
More informationAlgebra 2 with Trigonometry Final Exam Review
Algebra with Trigonometr Final Exam Review Find the value of the given expression. 1. [ + (0( ))]. Evaluate the given expression if x =, = 10, w = 11, and z = 18. (x ) + 10wz Solve the given equation.
More informationFunctions 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
More informationHIRES STILL TO BE SUPPLIED
1 MRE GRAPHS AND EQUATINS HIRES STILL T BE SUPPLIED Differentshaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object
More informationHONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 2001 MATHEMATICS PAPER 2
00CE 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
More information13 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,
More informationHigher. Differentiation 33
hsn uknet Higher Mathematics UNIT UTCME Differentiation Contents Differentiation Introduction to Differentiation Finding the Derivative 4 Differentiating with Respect to ther Variables 8 4 Rates of Change
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationAPPLICATION 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
More informationAP Calculus Testbank (Chapter 7) (Mr. Surowski)
AP Calculus Testbank (Chapter 7) (Mr. Surowski) Part I. MultipleChoice Questions. Suppose that a function = f() is given with f() for 4. If the area bounded b the curves = f(), =, =, and = 4 is revolved
More informationMathematics. 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 Boardapproved calculators ma be
More informationFocusing 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()
More information1.3 The Geometry of FirstOrder DIfferential Equations. main 2007/2/16 page CHAPTER 1 FirstOrder Differential Equations
page 20 20 CHAPTER 1 FirstOrder Differential Equations 35. = cos, (0) = 2, (0) = 1. 36. = 6, (0) = 1, (0) = 1, (0) = 4. 37. = e, (0) = 3, (0) = 4. 38. Prove that the general solution to = 0onan interval
More informationCHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often
7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.
More informationD.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
More informationSection 105 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
More informationConics and Polar Coordinates
Contents 17 Conics and Polar Coordinates 17.1 Conic Sections 2 17.2 Polar Coordinates 23 17.3 Parametric Curves 33 Learning outcomes In this Workbook ou will learn about some of the most important curves
More informationChapter 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,
More informationAngle Measure sin θ cos θ tan θ s 2
Aim #27: What is the relationship between the sine and cosine of complementar angles and special angles? 25 Do Now: Given the diagram of the right triangle, complete the following table. Calculate the
More informationMessiah College Calculus 1 Placement Exam Topics and Review: Key
Messiah College Calculus Placement Eam Topics and Review: Key Answers to problems in the tet are listed in the back of the course tetbook: Calculus, 9th edition by Larson, Hostetler, and Edwards. Solutions
More informationSTRAIGHT 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 xaxis in anticlockwise direction, then the value of tan θ is called the slope
More informationArchimedes 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
More informationSL 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
More informationPure Core 3. Revision Notes
Pure Core Revision Notes June 06 Pure Core Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions... 5 Notation...
More information3 Rectangular Coordinate System and Graphs
060_CH03_13154.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
More informationCoordinate Geometry. Positive gradients: Negative gradients:
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
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree 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 seconddegree
More informationCALCULUS 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
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationIIT ASHRAM. Common Practice Test9 JEE MAINS PAPER. PART III : Mathematics. then the value of A is. (a) 2 (b) cos a (c) sin a (d) none of these
UG &, Concorde Comple, Above OBC Bank. R.C. Dutt Road., Alkapuri Baroda. Ph. 665979, 655 Cell 965086 5 IIT ASHRAM W here w orking hard is a habit Class  XII PART III : Mathematics d A. If sin = sin (
More informationChapter Test A For use after the chapter Quadratic Relations and Conic Sections
Test A. Find the distance between the points (, ) and (, 7). What is the midpoint of the line segment joining the two points?. The vertices of a triangle are (, ), (, ), and (, ). Classif the triangle
More informationHigher. 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
More information4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant
SECTION 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant 6 What ou ll learn about The Tangent Function The Cotangent Function The Secant Function The Cosecant Function... and wh This will give us
More informationSLOPES AND EQUATIONS OF LINES CHAPTER
CHAPTER 90 8 CHAPTER TABLE OF CONTENTS 8 The Slope of a Line 8 The Equation of a Line 83 Midpoint of a Line Segment 84 The Slopes of Perpendicular Lines 85 Coordinate Proof 86 Concurrence of the
More informationChange 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
More informationπ 2 + h 1 is A) 1 B) 1 C) 0 D) E) none of these
Part A (NO CALCULATORS). lim [] (where [] is the greatest integer in ) is A) B) C) D) E) noneistent lim. h n. = n+ 4 n= sin π + h is A) B) C) D) E) none of these h A) 4 B) C) D) 4 E) diverges 4. The equation
More informationTrigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric
Trigonometric equations 6 sllabusref eferenceence Topic: Periodic functions and applications In this cha 6A 6B 6C 6D 6E chapter Simple trigonometric equations Equations using radians Further trigonometric
More informationDISTANCE, 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
More informationSolutionbank C2 Edexcel Modular Mathematics for AS and ALevel
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question Epand and simplify ( ) 5. ( ) 5 = + 5 ( ) + 0 ( ) + 0 ( ) + 5 ( ) + ( ) 5 = 5 + 0 0 + 5 5 Compare ( + )
More informationTrigonometric 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
More informationTHE 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
More informationD.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
More information10.2 The Unit Circle: Cosine and Sine
Foundations of Trigonometr 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 a circular path
More informationSeparation of Variables and the Logistic Equation. Separation of Variables. M x N y dy. x 2 3y dy. sin x y cos x. dy cot x dx 1 e y 1 dy 2 x dx
SECTION 63 Separation of Variables and the Logistic Equation 2 Section 63 Separation of Variables and the Logistic Equation Recognize and solve differential equations that can be solved b separation of
More informationHoover High School Math League
Hoover High School Math League Coordinate Geometry Solutions 1. (MH 111 1997) Find the distance between (, 1) and (7,4). (a) 6 (b) 50 (c) 10 (d) 1 Solution. Using the distance formula d = we have d =
More informationGraphing 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
More information10 cm. 6 m. In the diagram, angle ABC = angle ABD = 90, AC = 6 m, BD = 5 m and angle ACB = angle DAB =.
5 (i) Sketch, on the same diagram and for 0 x 2, the graphs of y = 1 4 + sin x and y = 1 2 cos 2x. [4] (ii) The xcoordinates of the points of intersection of the two graphs referred to in part (i) satisfy
More informationFinding Related Rates
SECTION.6 Related Rates 9 Section.6 r r h h Related Rates Find a related rate. Use related rates to solve reallife problems. Finding Related Rates You have seen how the Chain Rule can be used to find
More informationREVIEW 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.
More informationChapter 13 Functions of Several Variables
Chapter 13 Functions of Several Variables Chapter Summar Section Topics 13.1 Introduction to Functions of Several Variables Understand the notation for a function of several variables. Sketch the graph
More informationSAMPLE. 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
More informationName Class Date. Deriving the Equation of a Parabola
Name Class Date Parabolas Going Deeper Essential question: What are the defining features of a parabola? Like the circle, the ellipse, and the hperbola, the parabola can be defined in terms of distance.
More informationSolving 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
More informationONLY FOR PERSONAL USE. This digital version of the DictaatRekenvaardigheden  Algebraic Skills is for personal use because of copyright.
ONLY FOR PERSONAL USE This digital version of the DictaatRekenvaardigheden  Algebraic Skills is for personal use because of copyright. c Algebraic Skills Department of Mathematics and Computer Science
More informationConnecting 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.
More informationDERIVATIVES: RULES. Find the derivative of the constant function f(x) = c using the definition of derivative.
Part : Derivatives of Polynomial Functions DERIVATIVES: RULES We can use the definition of the derivative in order to generalize solutions and develop rules to find derivatives. The simplest derivatives
More information, 6, 7 A
Additional Topics in Analtic Geometr Outline 7 Conic Sections; Parabola 72 Ellipse 73 Hperbola 74 Translation of Aes 75 Parametric Equations Chapter 7 Group Activit: ocal Chords Chapter 7 Review Cumulative
More informationActuarial Exam Questions. f(x) = x 2 (x + 1) 1/3
Actuarial Exam Questions 6 7 8 5.) The point (5, 8, q) is on the line passing through the points (,, ) and (, 5, ). What is the value of q?.) What is the maximum value of on the interval (, ]? f(x) = x
More informationExploration 81a: NumberLine Graphs for Derivatives
Eploration 8a: NumberLine Graphs for Derivatives Objective: Relate the signs of the first and second derivatives to the graph of a function. The graph shows the quartic function f() H 3 D 52 3 C 3 2
More information2010 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
More informationAP Calculus Sample Examination I Calculators NOT allowed. You have 45 minutes to complete questions 127.
AP Calculus Sample Eamination I Calculators NOT allowed. You have 45 minutes to complete questions 7.. If f is a continuous function defined by f ( ) 6 (B) 5 (C) 4 4 5 + b, 5 = π 5sin, > 5 then b =.
More informationSECTION 9.1 THREEDIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xyplane. It is sketched in Figure 11.
SECTION 9.1 THREEDIMENSIONAL 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
More informationClass X. Mathematics. Course Structure 2 nd Term
Class X Mathematics Course Structure 2 nd Term S.No Units Marks 1. Algebra 23 2. Geometry 17 3. Trigonometry 08 4. Probability 08 5. Coordinate Geometry 11 6. Mensuration 23 Total 90 Mathematics X Blue
More informationFUNCTIONS Trigonometric graphs
LESSON FUNCTIONS Trigonometric graphs In grade 0, ou studied the graphs of = sin, = cos AND = tan Let us recall these. The graph of = sin 60º 70º 80º 90º 90º 80º 70º 60º The graph of = sin has a period
More informationIB Mathematics HL Year 1 Unit 8: Integral Calculus II (Core Topic 7)
IB Mathematics HL Year 1 Unit 8: Integral Calculus II (Core Topic 7) Homework for Unit 8 Lesson 47 Miscelleneous review (integration) 6E: 4 7. Review Set 6A:, 8, 9, 10. Review Set 6B: 3, 5, 7. 7A: 1,,
More informationAP Calculus AB AP Calculus BC
AP Calculus AB AP Calculus BC FreeResponse Questions and Solutions 989 997 Copyright 3 College Entrance Eamination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Vertical
More informationChapter 5 Graphing Linear Equations and Inequalities
.1 The Rectangular Coordinate Sstem (Page 1 of 28) Chapter Graphing Linear Equations and Inequalities.1 The Rectangular Coordinate Sstem The rectangular coordinate sstem (figure 1) has four quadrants created
More informationReview for Final Exam on Saturday December 8, Section 12.1 ThreeDimensional Coordinate System
Mat 7 Calculus III Updated on /0/07 Dr. Firoz Review for Final Eam on Saturda December 8, 007 Chapter Vectors and Geometr of Space Section. ThreeDimensional Coordinate Sstem. Find the equation of a sphere
More informationTrigonometric Functions
Chapter 4 Trigonometric Functions Course Number Section 4.1 Radian and Degree Measure Objective: In this lesson ou learned how to describe an angle and to convert between degree and radian measure. Instructor
More informationMATH PLACEMENT TEST STUDY GUIDE
MATH PLACEMENT TEST STUDY GUIDE The study guide is a review of the topics covered by the Columbia College Math Placement Test The guide includes a sample test question for each topic The answers are given
More informationCOORDINATE 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
More informationSLOPE A MEASURE OF STEEPNESS through 7.1.5
SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in form, the m is the
More informationOptimization Practice [83 marks]
Optimization Practice [83 marks] The following diagram shows a semicircle centre O, diameter [AB], with radius. Let P be a point on the circumference, with POB = θ radians. 1a. Find the area of the triangle
More informationWrite equations of parabolas in standard form. Graph parabolas. are parabolas used in manufacturing?
Parabolas Vocabular parabola conic section focus directri latus rectum Write equations of parabolas in standard form. Graph parabolas. are parabolas used in manufacturing? A mirror or other reflective
More information