SL Calculus Practice Problems


 Alexina Snow
 2 years ago
 Views:
Transcription
1 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 below shows part of the graph of the function f : α A P 5 Q B The graph intercepts the ais at A(, ), B(5, ) and the origin, O. There is a minimum point at P and a maimum point at Q. (a) The function ma also be written in the form f : α ( a) ( b), where a < b. Write down the value of (i) a; (ii) b. Find (i) f (); (ii) the eact values of at which f '() = ; (iii) the value of the function at Q. (7) (c) (i) Find the equation of the tangent to the graph of f at O. (ii) This tangent cuts the graph of f at another point. Give the coordinate of this point. () (d) Determine the area of the shaded region. (Total 5 marks). A ball is dropped verticall from a great height. Its velocit v is given b v = 5 5e.t, t where v is in metres per second and t is in seconds. (a) Find the value of v when (i) t = ; (ii) t =. \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
2 (i) Find an epression for the acceleration, a, as a function of t. (ii) What is the value of a when t =? (c) (i) As t becomes large, what value does v approach? (ii) As t becomes large, what value does a approach? (iii) Eplain the relationship between the answers to parts (i) and (ii). Alei  Desert Academ (d) Let metres be the distance fallen after t seconds. (i) Show that = 5t + 5e.t + k, where k is a constant. (ii) Given that = when t =, find the value of k. (iii) Find the time required to fall 5 m, giving our answer correct to four significant figures. (7) (Total 5 marks). The graph of = + + has a maimum point between = and =. Find the coordinates of this maimum point In this question, s represents displacement in metres, and t represents time in seconds. (a) The velocit v m s d s of a moving bod ma be written as v = = at, where a is a constant. dt Given that s = when t =, find an epression for s in terms of a and t. Trains approaching a station start to slow down when the pass a signal which is m from the station. The velocit of Train t seconds after passing the signal is given b v = 5t. (i) Write down its velocit as it passes the signal. (ii) Show that it will stop before reaching the station. (c) Train slows down so that it stops at the station. Its velocit is given b v = d s = at, where a is a constant. dt (i) Find, in terms of a, the time taken to stop. (ii) Use our solutions to parts (a) and (c)(i) to find the value of a. \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of (Total 5 marks) 6. Consider the function h () = 5. (i) Find the equation of the tangent to the graph of h at the point where = a, (a ). Write the equation in the form = m + c. (ii) Show that this tangent intersects the ais at the point ( a, ). (Total 5 marks) 7. An aircraft lands on a runwa. Its velocit v m s at time t seconds after landing is given b the equation v = 5 + 5e.5t, where t. (a) Find the velocit of the aircraft (i) when it lands; (ii) when t =. () Write down an integral which represents the distance travelled in the first four seconds. (c) Calculate the distance travelled in the first four seconds. After four seconds, the aircraft slows down (decelerates) at a constant rate and comes to rest when t =. (d) Sketch a graph of velocit against time for t. Clearl label the aes and mark on the graph the point where t =.
3 Alei  Desert Academ (e) Find the constant rate at which the aircraft is slowing down (decelerating) between t = and t =. (f) Calculate the distance travelled b the aircraft between t = and t =. (Total 8 marks) 8. The diagram on the left shows the graph of = f (). On the right hand grid sketch the graph of = f (). 9. Consider the function f () = + e. (a) (i) Find f (). (ii) Eplain briefl how this shows that f () is a decreasing function for all values of (ie that f () alwas decreases in value as increases). Let P be the point on the graph of f where =. Find an epression in terms of e for (i) the coordinate of P; (ii) the gradient of the tangent to the curve at P. (c) Find the equation of the tangent to the curve at P, giving our answer in the form = a + b. (d) (i) Sketch the curve of f for. (ii) Draw the tangent at =. (iii) (iv) Shade the area enclosed b the curve, the tangent and the ais. Find this area.. Consider the function f given b f () = +,. ( ) A part of the graph of f is given at right. The graph has a vertical asmptote and a horizontal asmptote, as shown. (a) Write down the equation of the vertical asmptote. () f () =.9 f ( ) =.9 f () =.99 (i) Evaluate f ( ). (ii) Write down the equation of the horizontal asmptote. (c) 9 7 Show that f () =, ( ). (7) (Total marks) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
4 Alei  Desert Academ 7 8 The second derivative is given b f () =,. ( ) (d) Using values of f () and f () eplain wh a minimum must occur at =. (e) There is a point of infleion on the graph of f. Write down the coordinates of this point. (Total marks). A car starts b moving from a fied point A. Its velocit, v m s after t seconds is given b v = t + 5 5e t. Let d be the displacement from A when t =. (a) Write down an integral which represents d. Calculate the value of d. Working: (a)..... The displacement s metres of a car, t seconds after leaving a fied point A, is given b s = t.5t. (a) Calculate the velocit when t =. Calculate the value of t when the velocit is zero. (c) Calculate the displacement of the car from A when the velocit is zero. Working: (a).... (c)... The function f () is defined as f () = ( h) + k. The diagram at right shows part of the graph of f (). The maimum point on the curve is P (, ). (a) Write down the value of (i) h; (ii) k. Show that f () can be written as f () = () 8 (c) Find f (). The point Q lies on the curve and has coordinates (, ). A straight line L, through Q, is perpendicular to the tangent at Q. (d) (i) Calculate the gradient of L. (ii) Find the equation of L. (iii) The line L intersects the curve again at R. Find the coordinate of R. (8) (Total marks). Let = g () be a function of for 7. The graph of g has an infleion point at P, and a minimum point at M. Partial sketches of the curves of g and g are shown below P(, ) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
5 g ( ) g ( ) Alei  Desert Academ = g ( ) = g ( ) Use the above information to answer the following. (a) Write down the coordinate of P, and justif our answer. Write down the coordinate of M, and justif our answer. (c) Given that g () =, sketch the graph of g. On the sketch, mark the points P and M. 5. The velocit v m s of a moving bod at time t seconds is given b v = 5 t. (a) Find its acceleration in m s. The initial displacement s is metres. Find an epression for s in terms of t. 6. The function f is defined b f : α (a) Write down (i) f (); (ii) f (). 5 () (Total 8 marks) Let N be the normal to the curve at the point where the graph intercepts the ais. Show that the equation of N ma be written as = Let g : α (c) (i) Find the solutions of f () = g (). (ii) Hence find the coordinates of the other point of intersection of the normal and the curve. (d) Let R be the region enclosed between the curve and N. (i) Write down an epression for the area of R. (ii) Hence write down the area of R. 7. The diagram below shows the graphs of f () = + e, g () = +,.5. 6 (6) (Total 6 marks) f g (a) (i) Write down an epression for the vertical distance p between the graphs of f and g. (ii) Given that p has a maimum value for.5, find the value of at which this occurs. (6) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 5 of 8 p.5.5
6 The graph of = f () onl is shown in the diagram at right. When = a, = 5. Alei  Desert Academ 6 (i) Find f (). (ii) Hence show that a = ln. 8 (c) The region shaded in the diagram is rotated through 6 about the  ais. Write down an epression for the volume obtained. (Total marks) 5.5 a.5 8. The equation of a curve ma be written in the form = a( p)( q). The curve intersects the ais at A(, ) and B(, ). The curve of = f () is shown in the diagram at right. (a) (i) Write down the value of p and of q. (ii) Given that the point (6, 8) is on the curve, find the value of a. (iii) Write the equation of the curve in the form = a + b + c. (i) d Find. 6 (ii) A tangent is drawn to the curve at a point P. The gradient of this tangent is 7. Find the coordinates of P. (c) The line L passes through B(, ), and is perpendicular to the tangent to the curve at point B. (i) Find the equation of L. (ii) Find the coordinate of the point where L intersects the curve again. 9. The diagram shows a rectangular area ABCD enclosed on three sides b 6 m of fencing, and on the fourth b a wall AB. Find the width of the rectangle that gives its maimum area. A B 6 () (6) (Total 5 marks). A particle moves with a velocit v m s given b v = 5 t where t. (a) The displacement, s metres, is when t is. Find an epression for s in terms of t. (6) Find t when s reaches its maimum value. (c) The particle has a positive displacement for m t n. Find the value of m and the value of n. (Total marks). If f () = cos, and f π =, find f (). Working:... (Total marks) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 6 of
7 Alei  Desert Academ. (a) Sketch the graph of = π sin,, on millimetre square paper, using a scale of cm per unit on each ais. Label and number both aes and indicate clearl the approimate positions of the intercepts and the local maimum and minimum points. Find the solution of the equation π sin =, >. () (c) Find the indefinite integral ( π sin ) and hence, or otherwise, calculate the area of the region enclosed b the graph, the ais and the line =. () (Total marks). A curve with equation =f () passes through the point (, ). Its gradient function is f () = +. Find the equation of the curve.... (Total marks). Given that f () = ( + 5) find (a) f (); f ( ). 5. The diagram shows the graph of the function = +, <. Find the eact value of the area of the shaded region.... (Total marks) (Total marks) 6. In this question ou should note that radians are used throughout. (a) (i) Sketch the graph of = cos, for making clear the approimate positions of the positive intercept, the maimum point and the endpoints. (ii) Write down the approimate coordinates of the positive intercept, the maimum point and the endpoints. (7) Find the eact value of the positive intercept for. Let R be the region in the first quadrant enclosed b the graph and the ais. (c) (i) Shade R on our diagram. (ii) Write down an integral which represents the area of R. (d) Evaluate the integral in part (c)(ii), either b using a graphic displa calculator, or b using the following information. d ( sin + cos sin ) = cos. (Total 5 marks) 7. In this part of the question, radians are used throughout. = + \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 7 of
8 The function f is given b f () = (sin ) cos. The diagram shows part of the graph of = f (). The point A is a maimum point, the point B lies on the  ais, and the point C is a point of infleion. Alei  Desert Academ A C (a) Give the period of f. () O B From consideration of the graph of = f (), find to an accurac of one significant figure the range of f. () (c) (i) Find f (). (ii) Hence show that at the point A, cos = (iii) Find the eact maimum value.. (d) Find the eact value of the coordinate at the point B. (e) (i) Find f (). (ii) Find the area of the shaded region in the diagram. (f) Given that f () = 9(cos ) 7 cos, find the coordinate at the point C. 8. Let f () =. Given that f =, find f (). Working: (9) () () () (Total marks)... (Total marks) 9. Note: Radians are used throughout this question. (a) Draw the graph of = π + cos, 5, on millimetre square graph paper, using a scale of cm per unit. Make clear (i) the integer values of and on each ais; (ii) the approimate positions of the intercepts and the turning points. Without the use of a calculator, show that π is a solution of the equation π + cos =. (c) (d) (e) Find another solution of the equation π + cos = for 5, giving our answer to si significant figures. Let R be the region enclosed b the graph and the aes for π. Shade R on our diagram, and write down an integral which represents the area of R. Evaluate the integral in part (d) to an accurac of si significant figures. (If ou consider it necessar, d ou can make use of the result ( sin + cos ) = cos.) (Total 5 marks) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 8 of
9 . Find (a) sin ( + 7); Alei  Desert Academ d e.... (Total marks). The derivative of the function f is given b f () =.5 sin, for. + The graph of f passes through the point (, ). Find an epression for f ()..... Consider functions of the form = e k (a) Show that e k = k ( e k ). Let k =.5 (i) Sketch the graph of = e.5, for, indicating the coordinates of the intercept. (ii) Shade the region enclosed b this graph, the ais, ais and the line =. (iii) Find the area of this region. (c) (i) d Find in terms of k, where = e k. The point P(,.8) lies on the graph of the function = e k. (ii) Find the value of k in this case. (iii) Find the gradient of the tangent to the curve at P.. Let f () =. Find (a) f (); f ( ). (Total marks).... The diagram shows part of the curve = sin. The shaded region π is bounded b the curve and the lines = and =. π π Given that sin = and cos eact area of the shaded region. =, calculate the... π π \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 9 of
10 5. The diagram below shows a sketch of the graph of the function = sin (e ) where, and is in radians. The graph cuts the ais at A, and the ais at C and D. It has a maimum point at B. (a) Find the coordinates of A. The coordinates of C ma be written as (ln k, ). Find the eact value of k. (c) (i) Write down the coordinate of B. (ii) d Find. Alei  Desert Academ B A C D (6) (d) (i) Write down the integral which represents the shaded area. (ii) Evaluate this integral. (e) (i) Cop the above diagram into our answer booklet. (There is no need to cop the shading.) On our diagram, sketch the graph of =. (ii) The two graphs intersect at the point P. Find the coordinate of P. (Total 8 marks) 6. Given that g ( ) =, deduce the value of (a) g ( ); ( g ( ) + ). 7. Consider the function f () = cos + sin. (a) (i) Show that f ( π ) =. (ii) Find in terms of π, the smallest positive value of which satisfies f () =. The diagram shows the graph of = e (cos + sin ),. The graph has a maimum turning point at C(a, b) and a point of infleion at D. d Find.... (c) Find the eact value of a and of b. () (d) Show that at D, = π e. (e) Find the area of the shaded region. (Total 7 marks) 6 D C(a, b) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
11 d 8. It is given that = + and that = when =. Find in terms of (a) Find ( + sin ( + )). The diagram shows part of the graph of the function f () = + sin ( + ). The area of the shaded region is given b a f ( ). Alei  Desert Academ Find the value of a..... (a) Consider the function f () = +. The diagram below is a sketch of part of the graph of = f (). 5 Cop and complete the sketch of f (). (i) Write down the intercepts and intercepts of f (). (ii) Write down the equations of the asmptotes of f (). () (c) (i) Find f (). (ii) There are no maimum or minimum points on the graph of f (). Use our epression for f () to eplain wh. The region enclosed b the graph of f (), the ais and the lines = and =, is labelled A, as shown at right. (d) (i) Find f (). (ii) Write down an epression that represents the area labelled A. (iii) Find the area of A. (7) (Total 6 marks). The derivative of the function f is given b f () = e +, <. The graph of = f () passes through the point (, ). Find an epression for f (). Working: A \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
12 Alei  Desert Academ. Let f be a function such that f ( ) = 8. (a) Deduce the value of (i) f ( ) ; + (ii) ( f ( ) ). d If f ( ) = 8, write down the value of c and of d. c (a) (i)... (ii)... c =..., d =.... Let h () = ( ) sin ( ) for 5 5. The curve of h () is shown at right. There is a minimum point at R and a maimum point at S. The curve intersects the ais at the points (a, ) (, ) (, ) and (b, ). (a) Find the eact value of (i) a; (ii) b. The regions between the curve and the ais are shaded for a as shown. (i) Write down an epression which represents the total area of the shaded regions. (ii) Calculate this total area. (c) (i) The coordinate of R is.. Find the coordinate of S. (ii) Hence or otherwise, find the range of values of k for which the equation ( ) sin ( ) = k has four distinct solutions.. Let f () =. + (a) Write down the equation of the horizontal asmptote of the graph of f. Find f (). 6 (c) The second derivative is given b f () =. (+ ) Let A be the point on the curve of f where the gradient of the tangent is a maimum. Find the coordinate of A. (d) Let R be the region under the graph of f, between = and = () ( a, ) 5 5 R S ( b, ) () (Total marks) R (), as shaded in the diagram at right Write down the definite integral which represents the area of R. (Total marks) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
13 5. The function f is given b f () = sin (5 ). (a) Find f " (). Write down f ( ). 6. Let f () = ( + ) 5. Find (a) f (); Alei  Desert Academ f () The curve = f () passes through the point (, 6). d Given that = 5, find in terms of The table below shows some values of two functions, f and g, and of their derivatives f and g. f () 5 g () 5 f () g () 6 Calculate the following. (a) d (f () + g ()), when = ; ( g' ( ) + 6). k 9. Given = ln 7, find the value of k. 5. The graph of = sin from π is shown at right. The area of the shaded region is.85. Find the value of k.... \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
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 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 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 informationINTEGRATION 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
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationAP Calculus AB 2005 Scoring Guidelines Form B
AP Calculus AB 5 coring Guidelines Form B The College Board: Connecting tudents to College uccess The College Board is a notforprofit membership association whose mission is to connect students to college
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 informationAP Calculus AB 2004 Scoring Guidelines
AP Calculus AB 4 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from
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 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 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 informationx = y + 2, and the line
WS 8.: Areas between Curves Name Date Period Worksheet 8. Areas between Curves Show all work on a separate sheet of paper. No calculator unless stated. Multiple Choice. Let R be the region in the first
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 information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
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 information14.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes. Learning Style
The Area Bounded b a Curve 14.3 Introduction One of the important applications of integration is to find the area bounded b a curve. Often such an area can have a phsical significance like the work done
More informationHigher. Functions and Graphs. Functions and Graphs 18
hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms
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 informationt hours This is the distance in miles travelled in 2 hours when the speed is 70mph. = 22 yards per second. = 110 yards.
The area under a graph often gives useful information. Velocittime graphs Constant velocit The sketch shows the velocittime graph for a car that is travelling along a motorwa at a stead 7 mph. 7 The
More informationAP Calculus BC 2008 Scoring Guidelines
AP Calculus BC 8 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationCHAPTER 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
More informationGraphing 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
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 informationChapter 3. Curve Sketching. By the end of this chapter, you will
Chapter 3 Curve Sketching How much metal would be required to make a ml soup can? What is the least amount of cardboard needed to build a bo that holds 3 cm 3 of cereal? The answers to questions like
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 information4 NonLinear relationships
NUMBER AND ALGEBRA NonLinear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationSection C Non Linear Graphs
1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions  4, use the graph at the right.. Eplain wh the graph
More informationTrigonometric functions
Chapter0 Trigonometric functions Sllabus reference: 3. Contents: A B C D E F Periodic behaviour The sine function Modelling using sine functions The cosine function The tangent function General trigonometric
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationCatholic 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
More informationSection 14.5 Directional derivatives and gradient vectors
Section 4.5 Directional derivatives and gradient vectors (3/3/08) Overview: The partial derivatives f ( 0, 0 ) and f ( 0, 0 ) are the rates of change of z = f(,) at ( 0, 0 ) in the positive  and directions.
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 informationDIFFERENTIATION OPTIMIZATION PROBLEMS
DIFFERENTIATION OPTIMIZATION PROBLEMS Question 1 (***) 4cm 64cm figure 1 figure An open bo is to be made out of a rectangular piece of card measuring 64 cm by 4 cm. Figure 1 shows how a square of side
More informationFurther Calculus Past Papers Unit 3 Outcome 2
PSf Further Calculus Past Papers Unit 3 utcome 2 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Differentiate 3 cos ( 2 π ) 6 with respect to. A. 3 sin(2) B. 3 sin(2
More informationHigher Mathematics Homework A
Non calcuator section: Higher Mathematics Homework A 1. Find the equation of the perpendicular bisector of the line joining the points A(3,1) and B(5,3) 2. Find the equation of the tangent to the circle
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More informationPractice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths.
Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths. Your formal test will be of a similar standard. Read the description of each assessment standard carefully to
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationAP Calculus AB 2010 FreeResponse Questions Form B
AP Calculus AB 2010 FreeResponse Questions Form B The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity.
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More informationIf (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
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 information3 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
More information*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Noncalculator) NATIONAL QUALIFICATIONS 2014 TUESDAY, 6 MAY 1.00 PM 2.30 PM
X00//0 NTIONL QULIFITIONS 0 TUESY, 6 MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Noncalculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (0 marks) Instructions for completion
More informationAP Calculus AB 2007 Scoring Guidelines Form B
AP Calculus AB 7 Scoring Guidelines Form B The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationMath 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:
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationSolution 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 +
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 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 informationAnalyzing 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
More informationSECTION 25 Combining Functions
2 Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
More informationFree Response Questions Compiled by Kaye Autrey for facetoface student instruction in the AP Calculus classroom
Free Response Questions 1969005 Compiled by Kaye Autrey for facetoface student instruction in the AP Calculus classroom 1 AP Calculus FreeResponse Questions 1969 AB 1 Consider the following functions
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 informationA Summary of Curve Sketching. Analyzing the Graph of a Function
0_00.qd //0 :5 PM Page 09 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
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 informationx 2. 4x x 4x 1 x² = 0 x = 0 There is only one xintercept. (There can never be more than one yintercept; do you know why?)
Math Learning Centre Curve Sketching A good graphing calculator can show ou the shape of a graph, but it doesn t alwas give ou all the useful information about a function, such as its critical points and
More informationClick here for answers.
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
More informationGive a formula for the velocity as a function of the displacement given that when s = 1 metre, v = 2 m s 1. (7)
. The acceleration of a bod is gien in terms of the displacement s metres as s a =. s (a) Gie a formula for the elocit as a function of the displacement gien that when s = metre, = m s. (7) (b) Hence find
More informationQuadratic Equations in One Unknown
1 Quadratic Equations in One Unknown 1A 1. Solving Quadratic Equations Using the Factor Method Name : Date : Mark : Ke Concepts and Formulae 1. An equation in the form a + b + c, where a, b and c are real
More informationContents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...
Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima
More informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
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 informationIntegral Calculus  Exercises
Integral Calculus  Eercises 6. Antidifferentiation. The Indefinite Integral In problems through 7, find the indicated integral.. Solution. = = + C = + C.. e Solution. e =. ( 5 +) Solution. ( 5 +) = e
More information4.4 Logarithmic Functions
SECTION 4.4 Logarithmic Functions 87 4.4 Logarithmic Functions PREPARING FOR THIS SECTION Before getting started, review the following: Solving Inequalities (Appendi, Section A.8, pp. 04 05) Polnomial
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
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 informationNATIONAL QUALIFICATIONS
H Mathematics Higher Paper 1 Practice Paper A Time allowed 1 hour 0 minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions 1 0 (40 marks) Instructions
More information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
More informationMATH Area Between Curves
MATH  Area Between Curves Philippe Laval September, 8 Abstract This handout discusses techniques used to nd the area of regions which lie between two curves. Area Between Curves. Theor Given two functions
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Eam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice eam contributors: Benita Albert Oak Ridge High School,
More informationImplicit Differentiation
Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some
More informationScholarship 2014 Calculus
93202Q 932022 S Scholarship 2014 Calculus 9.30 am Wednesda 19 November 2014 Time allowed: Three hours Total marks: 40 QUESTION BOOKLET There are five questions in this booklet. Answer ALL FIVE questions,
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
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 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 information27.2. Multiple Integrals over Nonrectangular Regions. Introduction. Prerequisites. Learning Outcomes
Multiple Integrals over Nonrectangular 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 nonrectangular
More informationHSC Mathematics  Extension 1. Workshop E4
HSC Mathematics  Extension 1 Workshop E4 Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationGeneral tests Algebra
General tests Algebra Question () : Choose the correct answer :  If = then = a)0 b) 6 c)5 d)4  The shape which represents Y is a function of is :   V A B C D o   y   V o    V o    V o  if
More informationAlgebra Module A47. The Parabola. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Algebra Module A7 The Parabola Copright This publication The Northern Alberta Institute of Technolog. All Rights Reserved. LAST REVISED December, The Parabola Statement of Prerequisite Skills Complete
More informationP1. Plot the following points on the real. P2. Determine which of the following are solutions
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More information(12) and explain your answer in practical terms (say something about apartments, income, and rent!). Solution.
Math 131 Fall 01 CHAPTER 1 EXAM (PRACTICE PROBLEMS  SOLUTIONS) 1 Problem 1. Many apartment complees check your income and credit history before letting you rent an apartment. Let I = f ( r) be the minimum
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationALGEBRA. Generate points and plot graphs of functions
ALGEBRA Pupils should be taught to: Generate points and plot graphs of functions As outcomes, Year 7 pupils should, for eample: Use, read and write, spelling correctl: coordinates, coordinate pair/point,
More informationMotion Graphs. It is said that a picture is worth a thousand words. The same can be said for a graph.
Motion Graphs It is said that a picture is worth a thousand words. The same can be said for a graph. Once you learn to read the graphs of the motion of objects, you can tell at a glance if the object in
More informationCollege Algebra  MAT 161 Page: 1 Copyright 2009 Killoran
College Algera  MAT 161 Page: 1 Copright 009 Killoran Quadratic Functions The graph of f./ D a C C c (where a,,c are real and a 6D 0) is called a paraola. Paraola s are Smmetric over the line that passes
More informationSection 59 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 59 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationCalculus AB 2014 Scoring Guidelines
P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official
More informationReview Sheet for Third Midterm Mathematics 1300, Calculus 1
Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,
More informationMathematics Paper 1 (NonCalculator)
H National Qualifications CFE Higher Mathematics  Specimen Paper A Duration hour and 0 minutes Mathematics Paper (NonCalculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3D We have defined the velocit and acceleration of a particle as the first and second
More informationSupporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Calculus: Module 12. Applications of differentiation
1 Supporting Australian Mathematics Project 2 3 4 5 6 7 8 9 1 11 12 A guide for teachers Years 11 and 12 Calculus: Module 12 Applications of differentiation Applications of differentiation A guide for
More informationLesson 6.1 Exercises, pages
Lesson 6. Eercises, pages 7 80 A. Use technolog to determine the value of each trigonometric ratio to the nearest thousandth. a) sin b) cos ( 6 ) c) cot 7 d) csc 8 0.89 0. tan 7 sin 8 0..0. Sketch each
More information4.1 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
More information