e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)
|
|
- May Brown
- 5 years ago
- Views:
Transcription
1 Chapter 0 Application of differential calculus 014 GDC required 1. Consider the curve with equation f () = e for 0. Find the coordinates of the point of infleion and justify that it is a point of infleion. (Total 7 marks). A television screen, BC, of height one metre, is built into a wall. The bottom of the television screen at B is one metre above an observer s eye level. The angles of elevation ( Aˆ OC, Aˆ OB ) from the observer s eye at O to the top and bottom of the television screen are and radians respectively. The horizontal distance from the observer s eye to the wall containing the television screen is metres. The observer s angle of vision ( Bˆ OC ) is radians, as shown below. 1 (i) Show that = arctan arctan. Hence, or otherwise, find the eact value of for which is a maimum and justify that this value of gives the maimum value of. (iii) Find the maimum value of. Find where the observer should stand so that the angle of vision is 15. (17) (Total marks). Sketch and label the curves y = for, and y = 1 ln for 0 <. () Find the -coordinate of P, the point of intersection of the two curves. If the tangents to the curves at P meet the y-ais at Q and R, calculate the area of the triangle PQR. () (6) (d) Prove that the two tangents at the points where = a, a > 0, on each curve are always perpendicular. (Total 14 marks) 4. Let f () = ln 5, 0. 5 < <, a, b; (a, b are values of for which f () is not defined). (i) Sketch the graph of f (), indicating on your sketch the number of zeros of f (). Show also the position of any asymptotes. Find all the zeros of f (), (that is, solve f () = 0). () Find the eact values of a and b. 1
2 Chapter 0 Application of differential calculus 014 Find f (), and indicate clearly where f () is not defined. (d) Find the eact value of the -coordinate of the local maimum of f (), for 0 < < 1.5. (You may assume that there is no point of infleion.) (e) Write down the definite integral that represents the area of the region enclosed by f () and the -ais. (Do not evaluate the integral.) () (Total 16 marks) 5. Let f ( ) ( 1), Sketch the graph of f (). (An eact scale diagram is not required.) On your graph indicate the approimate position of (i) (iii) each zero; each maimum point; each minimum point. (i) Find f (), clearly stating its domain. Find the -coordinates of the maimum and minimum points of f (), for 1 < < 1. (7) Find the -coordinate of the point of infleion of f (), where > 0, giving your answer correct to four decimal places. () (Total 1 marks) 6. The point B(a, b) is on the curve f () = such that B is the point which is closest to A(6, 0). Calculate the value of a. 7. A rectangle is drawn so that its lower vertices are on the -ais and its upper vertices are on the No GDC! curve y = e. The area of this rectangle is denoted by A. Write down an epression for A in terms of. Find the maimum value of A. 8. A particle moves along a straight line. When it is a distance s from a fied point, where s > 1,
3 Chapter 0 Application of differential calculus 014 (s ) the velocity v is given by v =. (s 1) Find the acceleration when s =. (Total 4 marks) 9. Given that y = e find d y ; d the eact values of the -coordinates of the points of infleion on the graph of y = justifying that they are points of infleion. e, 10. A closed cylindrical can has a volume of 500 cm. The height of the can is h cm and the radius of the base is r cm. Find an epression for the total surface area A of the can, in terms of r. Given that there is a minimum value of A for r > 0, find this value of r. 11. A normal to the graph of y = arctan ( 1), for 0, has equation y = + c, where c. Find the value of c. 1. A family of cubic functions is defined as f k () = k k +, k +. Epress in terms of k (i) f k () and f k (); the coordinates of the points of infleion P k on the graphs of f k. Show that all P k lie on a straight line and state its equation. (6) () Show that for all values of k, the tangents to the graphs of f k at P k are parallel, and find the equation of the tangent lines. (Total 1 marks) 1. André wants to get from point A located in the sea to point Y located on a straight stretch of beach. P is the point on the beach nearest to A such that AP = km and PY = km. He does this by swimming in a straight line to a point Q located on the beach and then running to Y.
4 Chapter 0 Application of differential calculus 014 When André swims he covers 1 km in 5 5 minutes. When he runs he covers 1 km in 5 minutes. If PQ = km, 0, find an epression for the time T minutes taken by André to reach point Y. dt 5 5 Show that 5. d 4 dt (i) Solve 0. d Use the value of found in part (i) to determine the time, T minutes, taken for André to reach point Y. (iii) Show that minimum. d T 0 5 d 4 and hence show that the time found in part is a (11) (Total 18 marks) 14. The radius and height of a cylinder are both equal to cm. The curved surface area of the cylinder is increasing at a constant rate of 10 cm /sec. When =, find the rate of change of the radius of the cylinder; the volume of the cylinder. 15. Car A is travelling on a straight east-west road in a westerly direction at 60 km h 1. Car B is travelling on a straight north-south road in a northerly direction at 70 km h 1. The roads intersect at the point O. When Car A is km east of O, and Car B is y km south of O, the distance between the cars is z km. Find the rate of change of z when Car A is 0.8 km east of O and Car B is 0.6 km south of O. 16. The following diagram shows the points A and B on the circumference of a circle, centre O, and radius 4 cm, where A ÔB =. Points A and B are moving on the circumference so that is increasing at a constant rate. 4
5 Chapter 0 Application of differential calculus 014 O A B Given that the rate of change of the length of the minor arc AB is numerically equal to the rate of change of the area of the shaded segment, find the acute value of. 17. The volume of a solid is given by 4 V = πr πr h. At the time when the radius is cm, the volume is 1 cm, the radius is changing at a rate of cm/min and the volume is changing at a rate of 04 cm /min. Find the rate of change of the height at this time. 18. A man PF is standing on horizontal ground at F at a distance from the bottom of a vertical wall GE. He looks at the picture AB on the wall. The angle BPA is. E B A P D F G Let DA = a, DB = b, where angle Pˆ DE is a right angle. Find the value of for which tan is a maimum, giving your answer in terms of a and b. Justify that this value of does give a maimum value of tan. (Total 9 marks) 19. The curve y = + 4 has a local maimum point at P and a local minimum point at 5
6 Chapter 0 Application of differential calculus 014 Q. Determine the equation of the straight line passing through P and Q, in the form a + by + c = 0, where a, b, c. 0. The diagram shows a trapezium OABC in which OA is parallel to CB. O is the centre of a circle radius r cm. A, B and C are on its circumference. Angle O ĈB = θ. O r A C B Let T denote the area of the trapezium OABC. Show that T = r (sin θ + sin θ). For a fied value of r, the value of T varies as the value of θ varies. Show that T takes its maimum value when θ satisfies the equation 4 cos θ + cos θ = 0, and verify that this value of T is a maimum. Given that the perimeter of the trapezium is 75 cm, find the maimum value of T. (6) (Total 15 marks) 1. Give eact answers in this part of the question. The temperature g (t) at time t of a given point of a heated iron rod is given by g (t) = ln t, where t > 0. t Find the interval where g (t) > 0. Find the interval where g (t) > 0 and the interval where g (t) < 0. (d) Find the value of t where the graph of g (t) has a point of infleion. Let t* be a value of t for which g (t*) = 0 and g (t*) < 0. Find t*. 6
7 Chapter 0 Application of differential calculus 014 (e) Find the point where the normal to the graph of g (t) at the point (t*, g (t*)) meets the t-ais. (Total 18 marks). A curve has equation y = 8. Find the equation of the normal to the curve at the point (, 1).. The following diagram shows an isosceles triangle ABC with AB = 10 cm and AC = BC. The verte C is moving in a direction perpendicular to (AB) with speed cm per second. C Calculate the rate of increase of the angle A B C ÂB at the moment the triangle is equilateral. 4. An airplane is flying at a constant speed at a constant altitude of km in a straight line that will take it directly over an observer at ground level. At a given instant the observer notes that the 1 1 angle is radians and is increasing at radians per second. Find the speed, in 60 kilometres per hour, at which the airplane is moving towards the observer. Airplane km Observer 5. The function f is defined by f () =, for > 0. (i) Show that f () = ln Obtain an epression for f (), simplifying your answer as far as possible. (i) Find the eact value of satisfying the equation f () = 0 Show that this value gives a maimum value for f (). 7
8 Chapter 0 Application of differential calculus 014 Find the -coordinates of the two points of infleion on the graph of f. (Total 1 marks) 6. Air is pumped into a spherical ball which epands at a rate of 8 cm per second (8 cm s 1 ). Find the eact rate of increase of the radius of the ball when the radius is cm. 7. Find the -coordinate of the point of infleion on the graph of y = e, A rectangle is drawn so that its lower vertices are on the -ais and its upper vertices are on the curve y = sin, where 0 n. Write down an epression for the area of the rectangle. Find the maimum area of the rectangle. (Total marks) 9. Let f : e sin. Find f (). There is a point of infleion on the graph of f, for 0 < < 1. Write down, but do not solve, an equation in terms of, that would allow you to find the value of at this point of infleion. (Total marks) 0. Consider the function f ( ), where n Show that the derivative f ( ) f ( ). Sketch the function f (), showing clearly the local maimum of the function and its horizontal asymptote. You may use the fact that 1n lim 0. Find the Taylor epansion of f () about = e, up to the second degree term, and show that this polynomial has the same maimum value as f () itself. (Total 1 marks) 1. A curve has equation y + y =. Find the equation of the tangent to this curve at the point (1, 1). 8
IB Practice - Calculus - Differentiation Applications (V2 Legacy)
IB Math High Level Year - Calc Practice: Differentiation Applications IB Practice - Calculus - Differentiation Applications (V Legacy). A particle moves along a straight line. When it is a distance s from
More informationThe region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.
Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.
More information1. Find the area enclosed by the curve y = arctan x, the x-axis and the line x = 3. (Total 6 marks)
1. Find the area enclosed by the curve y = arctan, the -ais and the line = 3. (Total 6 marks). Show that the points (0, 0) and ( π, π) on the curve e ( + y) = cos (y) have a common tangent. 3. Consider
More information2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW
FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the values of k for which the line y 6 is a tangent to the curve k 7 y. Find also the coordinates of the point at which this tangent touches the curve.
More informationSolutionbank C2 Edexcel Modular Mathematics for AS and A-Level
file://c:\users\buba\kaz\ouba\c_rev_a_.html Eercise A, Question Epand and simplify ( ) 5. ( ) 5 = + 5 ( ) + 0 ( ) + 0 ( ) + 5 ( ) + ( ) 5 = 5 + 0 0 + 5 5 Compare ( + ) n with ( ) n. Replace n by 5 and
More informationIB Math High Level Year 2 Calc Differentiation Practice IB Practice - Calculus - Differentiation (V2 Legacy)
IB Math High Level Year Calc Differentiation Practice IB Practice - Calculus - Differentiation (V Legac). If =, fin the two values of when = 5. Answer:.. (Total marks). Differentiate = arccos ( ) with
More information(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
More informationCalculus 1 - Lab ) f(x) = 1 x. 3.8) f(x) = arcsin( x+1., prove the equality cosh 2 x sinh 2 x = 1. Calculus 1 - Lab ) lim. 2.
) Solve the following inequalities.) ++.) 4 >.) Calculus - Lab { + > + 5 + < +. ) Graph the functions f() =, g() = + +, h() = cos( ), r() = +. ) Find the domain of the following functions.) f() = +.) f()
More informationPossible C4 questions from past papers P1 P3
Possible C4 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P January 001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More information1993 AP Calculus AB: Section I
99 AP Calculus AB: Section I 90 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among
More informationy=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions
AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)
More information1993 AP Calculus AB: Section I
99 AP Calculus AB: Section I 9 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among
More informationg y = (x 2 + 3)(x 3) h y = 2x 6x i y = 6 Find the coordinates of any stationary points on each curve. By evaluating
C Worksheet A In each case, find any values of for which d y d = 0. a y = + 6 b y = 4 + + c y = d y = 4 + 9 e y = 5 + f y = + 9 g y = ( + )( ) h y = Find the set of values of for which f() is increasing
More informationAP Calculus AB/BC ilearnmath.net
CALCULUS AB AP CHAPTER 1 TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-8: Calculator, 5 minutes. Choose the letter that best completes the statement or
More informationAPPLICATIONS OF DERIVATIVES OBJECTIVES. The approimate increase in the area of a square plane when each side epands from c m to.0 cm is () 0.00 sq. cm () 0.006 sq. cm () 0.06 sq. cm () None. If y log then
More informationAdd Math (4047) Paper 2
1. Solve the simultaneous equations 5 and 1. [5]. (i) Sketch the graph of, showing the coordinates of the points where our graph meets the coordinate aes. [] Solve the equation 10, giving our answer correct
More information1985 AP Calculus AB: Section I
985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of
More informationName: Index Number: Class: CATHOLIC HIGH SCHOOL Preliminary Examination 3 Secondary 4
Name: Inde Number: Class: CATHOLIC HIGH SCHOOL Preliminary Eamination 3 Secondary 4 ADDITIONAL MATHEMATICS 4047/1 READ THESE INSTRUCTIONS FIRST Write your name, register number and class on all the work
More informationAlgebra y funciones [219 marks]
Algebra y funciones [219 marks] Let f() = 3 ln and g() = ln5 3. 1a. Epress g() in the form f() + lna, where a Z +. 1b. The graph of g is a transformation of the graph of f. Give a full geometric description
More information1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.
Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient
More information(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N.
Calculus Review Packet 1. Consider the function f() = 3 3 2 24 + 30. Write down f(0). Find f (). Find the gradient of the graph of f() at the point where = 1. The graph of f() has a local maimum point,
More informationCHAPTER 72 AREAS UNDER AND BETWEEN CURVES
CHAPTER 7 AREAS UNDER AND BETWEEN CURVES EXERCISE 8 Page 77. Show by integration that the area of the triangle formed by the line y, the ordinates and and the -ais is 6 square units. A sketch of y is shown
More informationAP Calculus (BC) Summer Assignment (169 points)
AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationPure Core 2. Revision Notes
Pure Core Revision Notes June 06 Pure Core Algebra... Polynomials... Factorising... Standard results... Long division... Remainder theorem... 4 Factor theorem... 5 Choosing a suitable factor... 6 Cubic
More informationy x is symmetric with respect to which of the following?
AP Calculus Summer Assignment Name: Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number. Part : Multiple Choice Solve
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 000 MATHEMATICS /3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal
More informationAlgebra y funciones [219 marks]
Algebra y funciones [9 marks] Let f() = 3 ln and g() = ln5 3. a. Epress g() in the form f() + lna, where a Z +. attempt to apply rules of logarithms e.g. ln a b = b lna, lnab = lna + lnb correct application
More information1. Given the function f (x) = x 2 3bx + (c + 2), determine the values of b and c such that f (1) = 0 and f (3) = 0.
Chapter Review IB Questions 1. Given the function f () = 3b + (c + ), determine the values of b and c such that f = 0 and f = 0. (Total 4 marks). Consider the function ƒ : 3 5 + k. (a) Write down ƒ ().
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)
N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS /3 UNIT (COMMON) Time allowed Three hours (Plus minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL
More informationabc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS
More informationNATIONAL QUALIFICATIONS
Mathematics Higher Prelim Eamination 04/05 Paper Assessing Units & + Vectors NATIONAL QUALIFICATIONS Time allowed - hour 0 minutes Read carefully Calculators may NOT be used in this paper. Section A -
More informationQuestion. [The volume of a cone of radius r and height h is 1 3 πr2 h and the curved surface area is πrl where l is the slant height of the cone.
Q1 An experiment is conducted using the conical filter which is held with its axis vertical as shown. The filter has a radius of 10cm and semi-vertical angle 30. Chemical solution flows from the filter
More informationMARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION 2
Class Inde Number Name MRIS STELL HIGH SCHOOL PRELIMINRY EXMINTION DDITIONL MTHEMTICS 406/0 8 September 008 Paper hours 0minutes dditional Materials: nswer Paper (6 Sheets RED THESE INSTRUCTIONS FIRST
More informationFinal Exam Review / AP Calculus AB
Chapter : Final Eam Review / AP Calculus AB Use the graph to find each limit. 1) lim f(), lim f(), and lim π - π + π f 5 4 1 y - - -1 - - -4-5 ) lim f(), - lim f(), and + lim f 8 6 4 y -4 - - -1-1 4 5-4
More informationMathematics Extension 1
Teacher Student Number 008 TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Extension 1 General Instructions o Reading Time- 5 minutes o Working Time hours o Write using a blue or black pen o Approved
More informationTotal marks 70. Section I. 10 marks. Section II. 60 marks
THE KING S SCHOOL 03 Higher School Certificate Trial Eamination Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Board-approved calculators
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
More informationDecember 2012 Maths HL Holiday Pack. Paper 1.2 Paper 1 from TZ2 Paper 2.2 Paper 2 from TZ2. Paper 1.1 Paper 1 from TZ1 Paper 2.
December 2012 Maths HL Holiday Pack This pack contains 4 past papers from May 2011 in the following order: Paper 1.2 Paper 1 from TZ2 Paper 2.2 Paper 2 from TZ2 Paper 1.1 Paper 1 from TZ1 Paper 2.1 Paper
More informationMathematics Extension 1
013 HIGHER SCHL CERTIFICATE EXAMINATIN Mathematics Etension 1 General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators
More informationAnswers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16)
Answers for NSSH eam paper type of questions, based on the syllabus part (includes 6) Section Integration dy 6 6. (a) Integrate with respect to : d y c ( )d or d The curve passes through P(,) so = 6/ +
More informationAP Calculus I and Calculus I Summer 2013 Summer Assignment Review Packet ARE YOU READY FOR CALCULUS?
Name: AP Calculus I and Calculus I Summer 0 Summer Assignment Review Packet ARE YOU READY FOR CALCULUS? Calculus is a VERY RIGOROUS course and completing this packet with your best effort will help you
More informationANOTHER FIVE QUESTIONS:
No peaking!!!!! See if you can do the following: f 5 tan 6 sin 7 cos 8 sin 9 cos 5 e e ln ln @ @ Epress sin Power Series Epansion: d as a Power Series: Estimate sin Estimate MACLAURIN SERIES ANOTHER FIVE
More informationPhysicsAndMathsTutor.com
1. The diagram above shows the sector OA of a circle with centre O, radius 9 cm and angle 0.7 radians. Find the length of the arc A. Find the area of the sector OA. The line AC shown in the diagram above
More informationUNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level and Advanced Level
UNIVERSITY F CMBRIDGE INTERNTINL EXMINTINS General Certificate of Education dvanced Subsidiary Level and dvanced Level *336370434* MTHEMTICS 9709/11 Paper 1 Pure Mathematics 1 (P1) ctober/november 013
More informationy=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions
AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)
More informationCalculus-Lab ) f(x) = 1 x. 3.8) f(x) = arcsin( x+1., prove the equality cosh 2 x sinh 2 x = 1. Calculus-Lab ) lim. 2.7) lim. 2.
) Solve the following inequalities.) ++.) 4 > 3.3) Calculus-Lab { + > + 5 + < 3 +. ) Graph the functions f() = 3, g() = + +, h() = 3 cos( ), r() = 3 +. 3) Find the domain of the following functions 3.)
More informationCHAPTER 9. Trigonometry. The concept upon which differentiation of trigonometric functions depends is based
49 CHAPTER 9 Trigonometry The concept upon which differentiation of trigonometric functions depends is based on the fact that sin = 1. " 0 It should be remembered that, at the Calculus level, when we talk
More informationAdd Math (4047/02) Year t years $P
Add Math (4047/0) Requirement : Answer all questions Total marks : 100 Duration : hour 30 minutes 1. The price, $P, of a company share on 1 st January has been increasing each year from 1995 to 015. The
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More informationCreated by T. Madas. Candidates may use any calculator allowed by the regulations of this examination.
IYGB GCE Mathematics SYN Advanced Level Snoptic Paper C Difficult Rating: 3.895 Time: 3 hours Candidates ma use an calculator allowed b the regulations of this eamination. Information for Candidates This
More informationHere is a link to the formula booklet:
IB MATH SL2 SUMMER ASSIGNMENT review of topics from year 1. We will be quizzing on this when you return to school. This review is optional but you will earn bonus points if you complete it. Questions?
More informationI K J are two points on the graph given by y = 2 sin x + cos 2x. Prove that there exists
LEVEL I. A circular metal plate epands under heating so that its radius increase by %. Find the approimate increase in the area of the plate, if the radius of the plate before heating is 0cm.. The length
More informationMathematics 2001 HIGHER SCHOOL CERTIFICATE EXAMINATION
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time 3 hours Write using black or blue pen Board-approved calculators may be used A table of standard
More informationCreated by T. Madas. Candidates may use any calculator allowed by the Regulations of the Joint Council for Qualifications.
IYGB Special Paper Q Time: 3 hours 30 minutes Candidates may use any calculator allowed by the Regulations of the Joint Council for Qualifications. Information for Candidates This practice paper follows
More informationMATHEMATICS EXTENSION 2
Sydney Grammar School Mathematics Department Trial Eaminations 008 FORM VI MATHEMATICS EXTENSION Eamination date Tuesday 5th August 008 Time allowed hours (plus 5 minutes reading time) Instructions All
More informationAP Calculus BC Chapter 4 AP Exam Problems A) 4 B) 2 C) 1 D) 0 E) 2 A) 9 B) 12 C) 14 D) 21 E) 40
Extreme Values in an Interval AP Calculus BC 1. The absolute maximum value of x = f ( x) x x 1 on the closed interval, 4 occurs at A) 4 B) C) 1 D) 0 E). The maximum acceleration attained on the interval
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS / UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal value.
More information2 and v! = 3 i! + 5 j! are given.
1. ABCD is a rectangle and O is the midpoint of [AB]. D C 2. The vectors i!, j! are unit vectors along the x-axis and y-axis respectively. The vectors u! = i! + j! 2 and v! = 3 i! + 5 j! are given. (a)
More informationAP Calculus AB Information and Summer Assignment
AP Calculus AB Information and Summer Assignment General Information: Competency in Algebra and Trigonometry is absolutely essential. The calculator will not always be available for you to use. Knowing
More information1. Find A and B so that f x Axe Bx. has a local minimum of 6 when. x 2.
. Find A and B so that f Ae B has a local minimum of 6 when.. The graph below is the graph of f, the derivative of f; The domain of the derivative is 5 6. Note there is a cusp when =, a horizontal tangent
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)
N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions.
More informationReview Sheet for Second Midterm Mathematics 1300, Calculus 1
Review Sheet for Second Midterm Mathematics 300, Calculus. For what values of is the graph of y = 5 5 both increasing and concave up? >. 2. Where does the tangent line to y = 2 through (0, ) intersect
More information6675/01 Edexcel GCE Pure Mathematics P5 Further Mathematics FP2 Advanced/Advanced Subsidiary
6675/1 Edecel GCE Pure Mathematics P5 Further Mathematics FP Advanced/Advanced Subsidiary Monday June 5 Morning Time: 1 hour 3 minutes 1 1. (a) Find d. (1 4 ) (b) Find, to 3 decimal places, the value of.3
More informationMathematics 2003 HIGHER SCHOOL CERTIFICATE EXAMINATION
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Board-approved calculators may be used A table of standard
More informationx n+1 = ( x n + ) converges, then it converges to α. [2]
1 A Level - Mathematics P 3 ITERATION ( With references and answers) [ Numerical Solution of Equation] Q1. The equation x 3 - x 2 6 = 0 has one real root, denoted by α. i) Find by calculation the pair
More informationTrigonometric ratios:
0 Trigonometric ratios: The six trigonometric ratios of A are: Sine Cosine Tangent sin A = opposite leg hypotenuse adjacent leg cos A = hypotenuse tan A = opposite adjacent leg leg and their inverses:
More informationTime: 1 hour 30 minutes
Paper Reference (complete below) Centre No. Surname Initial(s) Candidate No. Signature Paper Reference(s) 6663 Edexcel GCE Pure Mathematics C Advanced Subsidiary Specimen Paper Time: hour 30 minutes Examiner
More informationSolutions to O Level Add Math paper
Solutions to O Level Add Math paper 4. Bab food is heated in a microwave to a temperature of C. It subsequentl cools in such a wa that its temperature, T C, t minutes after removal from the microwave,
More informationName Date Period. AP Calculus AB/BC Practice TEST: Curve Sketch, Optimization, & Related Rates. 1. If f is the function whose graph is given at right
Name Date Period AP Calculus AB/BC Practice TEST: Curve Sketch, Optimization, & Related Rates. If f is the function whose graph is given at right Which of the following properties does f NOT have? (A)
More informationTopic 3 Part 1 [449 marks]
Topic 3 Part [449 marks] a. Find all values of x for 0. x such that sin( x ) = 0. b. Find n n+ x sin( x )dx, showing that it takes different integer values when n is even and when n is odd. c. Evaluate
More information43603H. (MAR H01) WMP/Mar13/43603H. General Certificate of Secondary Education Higher Tier March Unit H
Centre Number Surname Candidate Number For Examiner s Use Other Names Candidate Signature Examiner s Initials General Certificate of Secondary Education Higher Tier March 2013 Pages 3 4 5 Mark Mathematics
More information1. (a) B, D A1A1 N2 2. A1A1 N2 Note: Award A1 for. 2xe. e and A1 for 2x.
1. (a) B, D N (b) (i) f () = e N Note: Award for e and for. (ii) finding the derivative of, i.e. () evidence of choosing the product rule e.g. e e e 4 e f () = (4 ) e AG N0 5 (c) valid reasoning R1 e.g.
More informationBE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: Unlimited and Continuous! (21 points)
BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: United and Continuous! ( points) For #- below, find the its, if they eist.(#- are pt each) ) 7 ) 9 9 ) 5 ) 8 For #5-7, eplain why
More information2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3
. Find an equation for the line that contains the points (, -) and (6, 9).. Find the value of y for which the line through A and B has the given slope m: A(-, ), B(4, y), m.. Find an equation for the line
More informationPractice Papers Set D Higher Tier A*
Practice Papers Set D Higher Tier A* 1380 / 2381 Instructions Information Use black ink or ball-point pen. Fill in the boxes at the top of this page with your name, centre number and candidate number.
More informationCALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt
CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental
More informationSolutionbank C2 Edexcel Modular Mathematics for AS and A-Level
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Eercise A, Question Find the values of for which f() is an increasing function, given that f() equals: (a) + 8 + (b) (c) 5 8 (d) 5 + 6 (e) +
More informationMathematics. Knox Grammar School 2012 Year 11 Yearly Examination. Student Number. Teacher s Name. General Instructions.
Teacher s Name Student Number Kno Grammar School 0 Year Yearly Eamination Mathematics General Instructions Reading Time 5 minutes Working Time 3 hours Write using black or blue pen Board approved calculators
More informationSolutionbank C1 Edexcel Modular Mathematics for AS and A-Level
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Find the values of x for which f ( x ) = x x is a decreasing function. f ( x ) = x x f ( x ) = x x Find f ( x ) and put
More informationARE YOU READY FOR CALCULUS?? Name: Date: Period:
ARE YOU READY FOR CALCULUS?? Name: Date: Period: Directions: Complete the following problems. **You MUST show all work to receive credit.**(use separate sheets of paper.) Problems with an asterisk (*)
More informationCALCULUS AB SECTION II, Part A
CALCULUS AB SECTION II, Part A Time 45 minutes Number of problems 3 A graphing calculator is required for some problems or parts of problems. pt 1. The rate at which raw sewage enters a treatment tank
More informationPROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS - APPLICATIONS OF DERIVATIVES Page ( ) Water seeps out of a conical filter at the constant rate of 5 cc / sec. When the height of water level in the cone is 5 cm, find the rate at which the height
More informationName Date Class. Logarithmic/Exponential Differentiation and Related Rates Review AP Calculus. Find dy. dx. 1. y 4 x. y 6. 3e x.
Name Date Class Find dy d. Logarithmic/Eponential Differentiation and Related Rates Review AP Calculus 1. y 4. 1 y ln. y ln 1 4. y log9 1 5. e y 6. y log 7. y e 8. e y e 4 1 1 9. y e e 10. 1 y ln 1 e 11.
More information1. Which of the following defines a function f for which f ( x) = f( x) 2. ln(4 2 x) < 0 if and only if
. Which of the following defines a function f for which f ( ) = f( )? a. f ( ) = + 4 b. f ( ) = sin( ) f ( ) = cos( ) f ( ) = e f ( ) = log. ln(4 ) < 0 if and only if a. < b. < < < < > >. If f ( ) = (
More informationPossible C2 questions from past papers P1 P3
Possible C2 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P1 January 2001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More informationSolutions to O Level Add Math paper
Solutions to O Level Add Math paper 04. Find the value of k for which the coefficient of x in the expansion of 6 kx x is 860. [] The question is looking for the x term in the expansion of kx and x 6 r
More information2008 CALCULUS AB SECTION I, Part A Time 55 minutes Number of Questions 28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION
8 CALCULUS AB SECTION I, Part A Time 55 minutes Number of Questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems. After eamining the form
More informationCore Mathematics C12
Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C12 Advanced Subsidiary Monday 19 May 2014 Morning Time: 2 hours 30
More informationUnit Circle: The unit circle has radius 1 unit and is centred at the origin on the Cartesian plane. POA
The Unit Circle Unit Circle: The unit circle has radius 1 unit and is centred at the origin on the Cartesian plane THE EQUATION OF THE UNIT CIRCLE Consider any point P on the unit circle with coordinates
More informationAP Calculus Chapter 4 Testbank (Mr. Surowski)
AP Calculus Chapter 4 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions 1. Let f(x) = x 3 + 3x 2 45x + 4. Then the local extrema of f are (A) a local minimum of 179 at x = 5 and a local maximum
More informationObjective Mathematics
. A tangent to the ellipse is intersected by a b the tangents at the etremities of the major ais at 'P' and 'Q' circle on PQ as diameter always passes through : (a) one fied point two fied points (c) four
More informationMathematics Department
Mathematics Department Grade 9 Placement Test Name This assessment will help the maths department to make a provisional placement. The final placement will be determined after a suitable period of in-class
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION. Mathematics
009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Board-approved calculators may be used A table of standard
More informationMathematics Extension 2
0 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators
More information1. Peter cuts a square out of a rectangular piece of metal. accurately drawn. x + 2. x + 4. x + 2
1. Peter cuts a square out of a rectangular piece of metal. 2 x + 3 Diagram NOT accurately drawn x + 2 x + 4 x + 2 The length of the rectangle is 2x + 3. The width of the rectangle is x + 4. The length
More informationNational Quali cations
National Quali cations AH08 X70/77/ Mathematics of Mechanics TUESDAY, 9 MAY :00 PM :00 PM Total marks 00 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which
More informationMATHEMATICS LEVEL 2 TEST FORM B Continued
Mathematics Level Test Form B For each of the following problems, decide which is the BEST of the choices given. If the eact numerical value is not one of the choices, select the choice that best approimates
More information1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3
[STRAIGHT OBJECTIVE TYPE] Q. Point 'A' lies on the curve y e and has the coordinate (, ) where > 0. Point B has the coordinates (, 0). If 'O' is the origin then the maimum area of the triangle AOB is (A)
More informationAP Calculus (BC) Summer Assignment (104 points)
AP Calculus (BC) Summer Assignment (0 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationAP CALCULUS BC SUMMER ASSIGNMENT
AP CALCULUS BC SUMMER ASSIGNMENT Work these problems on notebook paper. All work must be shown. Use your graphing calculator only on problems -55, 80-8, and 7. Find the - and y-intercepts and the domain
More information