SL Calculus Practice Problems

Size: px
Start display at page:

Download "SL Calculus Practice Problems"

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 end-points. (ii) Write down the approimate coordinates of the positive -intercept, the maimum point and the end-points. (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 radio-active substance at time t hours is given by. m = 4e 0.2t.

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t. REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

More information

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

Mathematics. Total marks 100. Section I Pages marks Attempt Questions 1 10 Allow about 15 minutes for this section 04 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators ma be

More information

Functions and Graphs

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

More information

INTEGRATION FINDING AREAS

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

More information

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

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

AP Calculus AB 2005 Scoring Guidelines Form B

AP Calculus AB 2005 Scoring Guidelines Form B AP Calculus AB 5 coring Guidelines Form B The College Board: Connecting tudents to College uccess The College Board is a not-for-profit membership association whose mission is to connect students to college

More information

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

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

More information

AP Calculus AB 2004 Scoring Guidelines

AP Calculus AB 2004 Scoring Guidelines AP Calculus AB 4 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from

More information

CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often

CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often 7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.

More information

13 Graphs, Equations and Inequalities

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

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

x = y + 2, and the line

x = 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 information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

2008 AP Calculus AB Multiple Choice Exam

2008 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 information

HI-RES STILL TO BE SUPPLIED

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

More information

14.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes. Learning Style

14.3. 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 information

Higher. Functions and Graphs. Functions and Graphs 18

Higher. 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 information

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

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

More information

t hours This is the distance in miles travelled in 2 hours when the speed is 70mph. = 22 yards per second. = 110 yards.

t 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. Velocit-time graphs Constant velocit The sketch shows the velocit-time graph for a car that is travelling along a motorwa at a stead 7 mph. 7 The

More information

AP Calculus BC 2008 Scoring Guidelines

AP Calculus BC 2008 Scoring Guidelines AP Calculus BC 8 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

More information

Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Exam #3 Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

More information

CHAPTER 54 SOME APPLICATIONS OF DIFFERENTIATION

CHAPTER 54 SOME APPLICATIONS OF DIFFERENTIATION CHAPTER 5 SOME APPLICATIONS OF DIFFERENTIATION EXERCISE 0 Page 65. An alternating current, i amperes, is given b i = 0 sin πft, where f is the frequenc in hertz and t the time in seconds. Determine the

More information

Graphing and transforming functions

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

More information

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

D.3. Angles and Degree Measure. Review of Trigonometric Functions APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

More information

Chapter 3. Curve Sketching. By the end of this chapter, you will

Chapter 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 information

Section 10-5 Parametric Equations

Section 10-5 Parametric Equations 88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

More information

4 Non-Linear relationships

4 Non-Linear relationships NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

More information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

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

More information

Section C Non Linear Graphs

Section 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 information

C3: Functions. Learning objectives

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

More information

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

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions: Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

More information

Trigonometric functions

Trigonometric 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 information

CIRCLE COORDINATE GEOMETRY

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

More information

Catholic Schools Trial Examination 2004 Mathematics

Catholic Schools Trial Examination 2004 Mathematics 0 Catholic Trial HSC Examination Mathematics Page Catholic Schools Trial Examination 0 Mathematics a If x 5 = 5000, find x correct to significant figures. b Express 0. + 0.. in the form b a, where a and

More information

Section 14.5 Directional derivatives and gradient vectors

Section 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 information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

DIFFERENTIATION OPTIMIZATION PROBLEMS

DIFFERENTIATION 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 information

Further Calculus Past Papers Unit 3 Outcome 2

Further 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 information

Higher Mathematics Homework A

Higher 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 information

Double Integrals in Polar Coordinates

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

More information

Practice 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. 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 information

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

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

More information

AP Calculus AB 2010 Free-Response Questions Form B

AP Calculus AB 2010 Free-Response Questions Form B AP Calculus AB 2010 Free-Response Questions Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

More information

9.5 CALCULUS AND POLAR COORDINATES

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

More information

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

If (a)(b) 5 0, then a 5 0 or b 5 0. chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic

More information

The Quadratic Function

The 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 information

3 Unit Circle Trigonometry

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

More information

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

*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Non-calculator) NATIONAL QUALIFICATIONS 2014 TUESDAY, 6 MAY 1.00 PM 2.30 PM X00//0 NTIONL QULIFITIONS 0 TUESY, 6 MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Non-calculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (0 marks) Instructions for completion

More information

AP Calculus AB 2007 Scoring Guidelines Form B

AP 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 not-for-profit membership association whose mission is to connect students to

More information

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

Math 21a Old Exam One Fall 2003 Solutions Spring, 2009 1 (a) Find the curvature κ(t) of the curve r(t) = cos t, sin t, t at the point corresponding to t = Hint: You ma use the two formulas for the curvature κ(t) = T (t) r (t) = r (t) r (t) r (t) 3 Solution:

More information

Exponential and Logarithmic Functions

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

More information

Solution for Final Review Problems 1

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

More information

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

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

More information

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

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

More information

Analyzing the Graph of a Function

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

More information

SECTION 2-5 Combining Functions

SECTION 2-5 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 information

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

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

More information

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

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

A Summary of Curve Sketching. Analyzing the Graph of a Function

A 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 information

Coordinate Geometry. Positive gradients: Negative gradients:

Coordinate 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 information

x 2. 4x x 4x 1 x² = 0 x = 0 There is only one x-intercept. (There can never be more than one y-intercept; do you know why?)

x 2. 4x x 4x 1 x² = 0 x = 0 There is only one x-intercept. (There can never be more than one y-intercept; 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 information

Click here for answers.

Click 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 information

Give a formula for the velocity as a function of the displacement given that when s = 1 metre, v = 2 m s 1. (7)

Give 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 information

Quadratic Equations in One Unknown

Quadratic 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 information

Contents. 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... 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 information

Filling in Coordinate Grid Planes

Filling 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 information

Quadratic Functions. MathsStart. Topic 3

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

More information

Integral Calculus - Exercises

Integral 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 information

4.4 Logarithmic Functions

4.4 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 information

When I was 3.1 POLYNOMIAL FUNCTIONS

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

More information

Solving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter

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

More information

NATIONAL QUALIFICATIONS

NATIONAL 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 information

2.4 Inequalities with Absolute Value and Quadratic Functions

2.4 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 information

MATH Area Between Curves

MATH 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 information

Mark Howell Gonzaga High School, Washington, D.C.

Mark 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 information

Implicit Differentiation

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

More information

Scholarship 2014 Calculus

Scholarship 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 information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section 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 information

THE PARABOLA section. Developing the Equation

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

More information

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

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

More information

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

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

More information

HSC Mathematics - Extension 1. Workshop E4

HSC 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 information

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

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

More information

General tests Algebra

General 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 information

Algebra Module A47. The Parabola. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Algebra 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 information

P1. Plot the following points on the real. P2. Determine which of the following are solutions

P1. 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 information

Lesson 9.1 Solving Quadratic Equations

Lesson 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 y-values. 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.

(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 information

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

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

More information

ALGEBRA. Generate points and plot graphs of functions

ALGEBRA. 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 information

Motion 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. 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 information

College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran

College 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 information

Section 5-9 Inverse Trigonometric Functions

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

More information

Calculus AB 2014 Scoring Guidelines

Calculus 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 information

Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review 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 information

Mathematics Paper 1 (Non-Calculator)

Mathematics Paper 1 (Non-Calculator) H National Qualifications CFE Higher Mathematics - Specimen Paper A Duration hour and 0 minutes Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full

More information

Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal

Physics 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 3-D We have defined the velocit and acceleration of a particle as the first and second

More information

Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Calculus: Module 12. Applications of differentiation

Supporting 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 information

Lesson 6.1 Exercises, pages

Lesson 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 information

4.1 Radian and Degree Measure

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

More information