Applications of Integration Day 1


 Antony McDonald
 2 years ago
 Views:
Transcription
1 Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under the curve y = 1 x from x = 1 to x = 5.
2 Example 3 Find the area bounded by y = sec 2 x, x = 0, x = π 4, y = 0 Example 4 Find the area bounded by y = ex, x = 0, x = ln 4, y = 0
3 Example 5 Find the finite area bounded by the curve y = x3 2x 2 3x and the xaxis. Example 6 Find the finite area bounded by the two graphs y = 2 x and x + y = 3.
4 Example 7 Find the area lying between y = 6x x2 and y = x2 2x. Example 8 Example 9 Determine the area of the region Determine the area of the enclosed by y = e x, y = e x, x = ln 4 region bounded by y = e x, y = 1, y = 2, x = 3
5
6 Applications of Integration Day 2 More on Area Between Curves and using the Calculator Example 1 Example 2 Find the area between Find the area enclosed by y = sec 2 x and y = sin x y = 2 x 2 and y = x from x = 0 to x = π 4 Example 3 Find the area enclosed by y = 2cos x and y = x2 1
7 Example 4 Find the area bounded by y = x and the xaxis and the line y = x 2 There exists another way to determine the area above take a horizontal slice.
8 Example 5 Find the area enclosed by y = x3 and x = y 2 2 Example 6 Find the area bounded by x = y 1 ( )( y 3) and the yaxis
9 Summary Concept I Area Region A Region B Region C Region D General Comments Notice that a and b are on the xaxis c and d are on the yaxis When using formulas involving you must solve for for example: When using formulas involving you must solve for for example: Look at the region and decide which way to slice vertical slices horizontal slices Remember if using horizontal slices, write relations in the form If finding total area, sometimes it is necessary to break up your integrals, for example
10 Applications of Integration Day 3 Volumes of Solids by Plane Slices Consider the volume of a cylinder Consider a bounded region and rotating that region around the xaxis. Take a slice and consider its volume. Take the sum of the volumes of each slice. What do you get? Volume formula:
11 Example 1 The region bounded by the curve and the xaxis over the interval [1,4] is rotated about the xaxis to generate a solid. Find the volume of the solid. Find the volume of the solid generated by revolving the region bounded by the lines and curves about the xaxis Example 3 Example 4
12 Find the volume of the solid generated by revolving the region bounded by the lines and curves about the yaxis Example 5 Example 6 Example 7 The region bounded by the curve xaxis to generate a solid. Find the volume of the solid. is rotated about the
13 Applications of Integration Day 4 Volume formulas for rotation around the axes (based on crosssectional areas): Disk Washer f(x) around the xaxis region between f(x) and g(x) around x axis f(y) around the yaxis region between f(y) and g(y) around yaxis More Volumes using the Washer Method Example 1 (1985 exam) The region bounded by the curves y = ex and y = e x is rotated about the x axis to generate a solid. Find the volume of the solid.
14 Example 2 Find the volume of the solid generated by revolving the region bounded y = x 2 and y = 2x about the yaxis Rotating Regions about lines parallel to the axes. First idea think of distances in terms of x and y values below and pick a point (x, y). Consider distances from the point to the lines y=3 and y= 1. Then consider distances from point to the lines x = 2 and x = 2 Graph y = x2
15 Example 3 The region bounded by the curve y = 2 x2 and y = 1 is rotated about the line y = 1 to generate a solid. Find the volume of the solid. Example 4 5 Example The region bounded by y = x2, x = 0, x = 2 Take the same region in Eg 4 is rotated about the line y = 1 to generate rotate it about the line y = 4 a solid. Find the volume of the solid. to generate a solid. Find the volume
16 Example 5 The region bounded by y = x 2 + 2, y = 1 2 x +1, x = 0, x =1 is rotated about the line y = 3 to generate a solid. Find the volume of the solid. Example 6!!!!!!!! Example 7!!!! The region bounded by y = x2, x = 0, x = 2!! Same region in Eg. 6 is rotated is rotated about the line x = 3 to generate generate a solid. Find the volume of the solid. about the line x = 3 to solid find the volume.
17 Example 8 The region bounded by y = following lines. Find each volume. x, y = 3, x = 0 is rotated about each of the a) xaxis b) yaxis c) the line y = 3 d) the line x = 9 Additional Questions: *use your graphing calculator to help you find the region and intersection points when necessary. 1. Find the area of the region bounded by a) the graph of x = ( y 1) ( y 4) and the yaxis b) the graphs of y = e0.2x and y = cos x between x = 0 and x = 5 c) the graphs of y = x3 4x and y = 3x2 4x 4 (finite region) 2. Let and a) Find coordinates of intersection and sketch the graph b) Find the area bounded by the graphs 3. Find the area of the region bounded by the graphs of and (draw the region)
18 4 A = ( y 1) ( y 4)dy = 4.5 Solutions: 1. a) 1 5 A = ( cos x e 0.2x )dx = 9.55 b) 0 2 A = (( x 3 4x) ( 3x 2 4x 4) )dx = 6.75 c) 1 2. a) = 1.177, b) A = (cosx x 2 +1) dx = A = (x 3 x +1 e x 2 ) dx + (e x 2 x 3 + x 1) dx =
19 Applications of Integration Day 5 First idea: Draw on the axes below. Pick a point on the curve and label it: P (x, y). a) Draw the line y = 1 and find the distance d from the point P to the line b) Draw the line y = 4 and find the distance d from the point P to the line c) Draw the line x = 1 and find the distance d from the point P to the line d) Draw the line x = 3 and find the distance d from the point P to the line Second idea: Review area formulas Circle Semicircle Square Isosceles Right triangle Equilateral triangle
20 Third idea: Consider the solids we have been working with up to this point. Suppose we sliced the solid and pulled out a cross section what would it look like? Look at the volume formula how could this formula be viewed in terms of the crosssectional area Consider another type of solid. This solid is not formed by rotation of a region about a line. Considering a region as its base, then describing its crosssectional shape forms a solid with a known cross section. How could we write the general formula for the volume of such a shape? Can you imagine a solid having a circular base and each crosssection perpendicular to the xaxis is a square. Make a solid that looks like this. Draw what it must look like. Find its volume.
21 Example 1 Find the volume of the solid where the base is the circle and a) the cross section perpendicular to the xaxis is an isosceles right triangle with one leg in the plane of the base. b) the cross section perpendicular to the xaxis is an equilateral triangle with one leg in the plane of the base.
22 Example 2 Find the volume of the solid where the base is the region bounded by and the xaxis between x = 0 and x = π a) the cross section perpendicular to the xaxis is a square b) the cross section perpendicular to the xaxis is a semicircle
23 Example 3 Find the volume of the solid where the base is the region bounded by, and the line x = ln 2. a) the cross section perpendicular to the xaxis is a square b) the cross section perpendicular to the xaxis is a semicircle There is another way to describe a solid instead of taking a base describe it solely in terms of its cross sectional area.
24 Example 5 Find the volume of an object that lies within [2, 5] and its cross section at each point is a square of length x. Example 6 Find the volume of an object that lies between x = 0 and x = π/2 and its cross section is a square of length cos x.
25 Concept I!Area Calculus 12 AP Unit V Review of Big Ideas Region A Region B Region C Region D General Comments! Notice that a and b are on the xaxis c and d are on the yaxis! When using formulas involving you must solve for for example:! When using formulas involving you must solve for for example:! Look at the region and decide which way to slice!! vertical slices!! horizontal slices! Remember if using horizontal slices, write relations in the form! If finding total volume, sometimes it is necessary to break up your integrals, for example (See Assignments V1 and V2 for good questions on area to practice)
26 Concept II! Volume A! Volumes of Revolution General Comments! These are based on circular slices therefore all of the formulas involve! Try to visualize the slice, then rotate it around the line specified!!  then decide if it is a disk!!  or if it is a washer! When revolving around the xaxis take vertical slices! When revolving around the yaxis take horizontal slices Region A Region B Region C Region D When revolving Region A When revolving Region B When revolving Region C When revolving Region D around xaxis visualize the around yaxis visualize the around xaxis visualize the around yaxis visualize the disk disk washer washer Revolving Regions Around lines Parallel to the Axes When revolving around lines parallel to the xaxis and yaxis draw a diagram to determine the value of R and r in the formula for the washer. It is often helpful to think of a point on the curve in the form and to use the distances x and y in your formulas for R and r, then translate them using the function.!!!!!! For example, in the above diagram if the region being rotated is under the curve between x = 0 and x = 2!!  rotate region about the line y = 3! R = 3 + y r = 3
27 !!  rotate region about the line y = 5! R = 5 r = 5 y! In both cases, substitute these values into the formula: B! Volumes of Known Cross Section General Comments! These are based on slices of different shapes, usually squares, right triangles, equilateral triangles or! semicircles therefore you must know the following area formulas square! right triangle!! equilateral triangle semicircle! Try to draw the slice, remove it from the diagram, then decide on the length of s or r.! For example, consider a solid with a base given by the region bounded by the curve, the xaxis! and the lines x = 0 and x = 1.5 If the known cross sections are squares then: Take a vertical slice Since Therefore The important thing in any volume question is to be able to visualize the crosssection by taking a slice. (See Assignments V3, V4, and Additional questions for good questions on volume to practice ) Also do Calculus AP Volume Worksheet Assignment V5 on the next page.
28 Calculus AP Volume Worksheet Assignment V5 A. In questions 1 to 5, consider the region bounded by, the xaxis, and the lines x = 0 and x = 1. Find the volume of the following solids. (Draw a diagram for each question). 1. The solid obtained by rotating the region about the xaxis. 2. The solid obtained by rotating the region about the line y = The solid obtained by rotating the region about the line y = The solid whose base is the given region and whose crosssections perpendicular to the xaxis are squares. 5. The solid whose base is the given region and whose crosssections perpendicular to the xaxis are semicircles. B. For questions 6 to 7, write out the integral that you would use to calculate the volume. Then, use your calculator to find the numerical integral. 6. The region bounded by, rotated about the line x = The region bounded by, rotated about the line x = 2. C. Find the volume of the solid described in questions 8 to 13. Write out a formula for the crosssectional area, the write the integral you would use to calculate the volume, then use your calculator to find the numerical integral. 8. The cross section is a square with side length x; the object lies between x = 0 and x = The cross section is a square with side length, the object lies between x = 0 and x = π. 10. The base is the circle, the cross section perpendicular to the xaxis is a square. 11. The base is the circle, the cross section perpendicular to the xaxis is an equilateral triangle. 12. The base is the circle, the cross section perpendicular to the xaxis is an isosceles right triangle with one leg in the plane of the base. 13. The base is the circle, the cross section perpendicular to the xaxis is a semicircle whose diameter is in the plane of the base.
29
7.3 Volumes Calculus
7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are
More informationEngineering Math II Spring 2015 Solutions for Class Activity #2
Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the xaxis. Then find
More informationPROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving
More informationThis function is symmetric with respect to the yaxis, so I will let  /2 /2 and multiply the area by 2.
INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,
More informationAP CALCULUS AB 2008 SCORING GUIDELINES
AP CALCULUS AB 2008 SCORING GUIDELINES Question 1 Let R be the region bounded by the graphs of y = sin( π x) and y = x 4 x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line
More informationGRAPHING IN POLAR COORDINATES SYMMETRY
GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry  yaxis,
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More informationWeek #15  Word Problems & Differential Equations Section 8.1
Week #15  Word Problems & Differential Equations Section 8.1 From Calculus, Single Variable by HughesHallett, Gleason, McCallum et. al. Copyright 25 by John Wiley & Sons, Inc. This material is used by
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More information4 More Applications of Definite Integrals: Volumes, arclength and other matters
4 More Applications of Definite Integrals: Volumes, arclength and other matters Volumes of surfaces of revolution 4. Find the volume of a cone whose height h is equal to its base radius r, by using the
More information(b)using the left hand end points of the subintervals ( lower sums ) we get the aprroximation
(1) Consider the function y = f(x) =e x on the interval [, 1]. (a) Find the area under the graph of this function over this interval using the Fundamental Theorem of Calculus. (b) Subdivide the interval
More informationAP Calculus AB 2004 FreeResponse Questions
AP Calculus AB 2004 FreeResponse Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
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 informationVolumes of Revolution
Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize dimensional solids and a specific procedure to sketch a solid of revolution. Students
More informationMath 1B, lecture 5: area and volume
Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in
More informationSAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions
SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions All questions in the Math Level 1 and Math Level Tests are multiplechoice questions in which you are asked to choose the
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
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 informationContents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationAnalyzing Piecewise Functions
Connecting Geometry to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 04/9/09 Analyzing Piecewise Functions Objective: Students will analyze attributes of a piecewise function including
More informationDouble integrals. Notice: this material must not be used as a substitute for attending the lectures
ouble integrals Notice: this material must not be used as a substitute for attending the lectures . What is a double integral? Recall that a single integral is something of the form b a f(x) A double integral
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More informationAP Calculus AB 2008 FreeResponse Questions
AP Calculus AB 2008 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationSection 1.8 Coordinate Geometry
Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of
More information*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Noncalculator) MONDAY, 21 MAY 1.00 PM 2.30 PM NATIONAL QUALIFICATIONS 2012
X00//0 NTIONL QULIFITIONS 0 MONY, MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Noncalculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (40 marks) Instructions for completion
More informationExample 1. Example 1 Plot the points whose polar coordinates are given by
Polar Coordinates A polar coordinate system, gives the coordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points
More informationDefinition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =
Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a
More informationAP Calculus AB 2009 FreeResponse Questions
AP Calculus AB 2009 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More informationSection 6.4. Lecture 23. Section 6.4 The Centroid of a Region; Pappus Theorem on Volumes. Jiwen He. Department of Mathematics, University of Houston
Section 6.4 Lecture 23 Section 6.4 The Centroid of a Region; Pappus Theorem on Volumes Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math1431 Jiwen
More informationAP Calculus AB 2006 FreeResponse Questions
AP Calculus AB 2006 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationSection summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2
Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationNational Quali cations 2015
H National Quali cations 05 X77/76/ WEDNESDAY, 0 MAY 9:00 AM 0:0 AM Mathematics Paper (NonCalculator) Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given only
More informationAP Calculus BC 2003 FreeResponse Questions
AP Calculus BC 2003 FreeResponse Questions The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Thursday, January 29, 2004 9:15 a.m. to 12:15 p.m.
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and
More informationAP Calculus BC 2004 FreeResponse Questions
AP Calculus BC 004 FreeResponse Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
More informationAP Calculus BC 2007 FreeResponse Questions
AP Calculus BC 7 FreeResponse Questions 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 informationAP Calculus AB 2013 FreeResponse Questions
AP Calculus AB 2013 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More information3. Double Integrals 3A. Double Integrals in Rectangular Coordinates
3. Double Integrals 3A. Double Integrals in ectangular Coordinates 3A1 Evaluate each of the following iterated integrals: c) 2 1 1 1 x 2 (6x 2 +2y)dydx b) x 2x 2 ydydx d) π/2 π 1 u (usint+tcosu)dtdu u2
More informationAP Calculus AB 2007 FreeResponse Questions
AP Calculus AB 2007 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationMULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then
MULTIPLE INTEGALS 1. ouble Integrals Let be a simple region defined by a x b and g 1 (x) y g 2 (x), where g 1 (x) and g 2 (x) are continuous functions on [a, b] and let f(x, y) be a function defined on.
More informationUnit 10: Quadratic Relations
Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
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 informationAP Calculus AB 2012 FreeResponse Questions
AP Calculus AB 1 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationDate: Period: Symmetry
Name: Date: Period: Symmetry 1) Line Symmetry: A line of symmetry not only cuts a figure in, it creates a mirror image. In order to determine if a figure has line symmetry, a figure can be divided into
More informationConjectures. Chapter 2. Chapter 3
Conjectures Chapter 2 C1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C2 Vertical Angles Conjecture If two angles are vertical
More information12 Surface Area and Volume
12 Surface Area and Volume 12.1 ThreeDimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids
More informationGRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?
GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.
More information6 Applications of the
CHAPTER 6 Applications of the Integral 6. Area Between Two Curves Preliminary Questions. Suppose that f (x) andg(x). True or False: the integral b ( f (x) g(x)) dx is still a equal to the area between
More informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More informationDetermine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s
Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,
More informationAP Calculus AB 2006 Scoring Guidelines
AP Calculus AB 006 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 informationSurface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry
Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel
More informationMEI STRUCTURED MATHEMATICS INTRODUCTION TO ADVANCED MATHEMATICS, C1. Practice Paper C1C
MEI Mathematics in Education and Industry MEI STRUCTURED MATHEMATICS INTRODUCTION TO ADVANCED MATHEMATICS, C Practice Paper CC Additional materials: Answer booklet/paper Graph paper MEI Examination formulae
More informationApplications of the definite integral to calculating volume and length
Chapter 5 Applications of the definite integral to calculating volume and length In this chapter we consider applications of the definite integral to calculating geometric quantities such as volumes. The
More informationLines That Pass Through Regions
: Student Outcomes Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, and if it does, they describe
More informationAP Calculus AB 2012 Scoring Guidelines
AP Calculus AB Scoring Guidelines The College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in 9, the College
More informationSURFACE AREA AND VOLUME
SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationAP Calculus AB. Practice Exam. Advanced Placement Program
Advanced Placement Program AP Calculus AB Practice Exam The questions contained in this AP Calculus AB Practice Exam are written to the content specifications of AP Exams for this subject. Taking this
More informationMaximum / Minimum Problems
171 CHAPTER 6 Maximum / Minimum Problems Methods for solving practical maximum or minimum problems will be examined by examples. Example Question: The material for the square base of a rectangular box
More information4.4 Transforming Circles
Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationDraft copy. Circles, cylinders and prisms. Circles
12 Circles, cylinders and prisms You are familiar with formulae for area and volume of some plane shapes and solids. In this chapter you will build on what you learnt in Mathematics for Common Entrance
More informationThe Inscribed Angle Alternate A Tangent Angle
Student Outcomes Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays is
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationa b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12.
MA123 Elem. Calculus Fall 2015 Exam 3 20151119 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACTapproved calculator during
More informationTopics Covered on Geometry Placement Exam
Topics Covered on Geometry Placement Exam  Use segments and congruence  Use midpoint and distance formulas  Measure and classify angles  Describe angle pair relationships  Use parallel lines and transversals
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More information100 Math Facts 6 th Grade
100 Math Facts 6 th Grade Name 1. SUM: What is the answer to an addition problem called? (N. 2.1) 2. DIFFERENCE: What is the answer to a subtraction problem called? (N. 2.1) 3. PRODUCT: What is the answer
More information10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres. 10.4 Day 1 Warmup
10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres 10.4 Day 1 Warmup 1. Which identifies the figure? A rectangular pyramid B rectangular prism C cube D square pyramid 3. A polyhedron
More informationAP Calculus AB 2003 Scoring Guidelines
AP Calculus AB Scoring Guidelines The materials included in these files are intended for use y AP teachers for course and exam preparation; permission for any other use must e sought from the Advanced
More informationAP Calculus AB 2003 Scoring Guidelines Form B
AP Calculus AB Scoring Guidelines Form B The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More informationPractice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
More informationa b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12.
MA123 Elem. Calculus Fall 2015 Exam 2 20151022 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACTapproved calculator during
More information( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those
1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationa. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F
FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all
More informationAP Calculus AB 2009 Scoring Guidelines
AP Calculus AB 9 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in 19,
More informationWarmUp y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.
CST/CAHSEE: WarmUp Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and
More informationAP Calculus BC 2009 FreeResponse Questions Form B
AP Calculus BC 009 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 informationCONNECT: Volume, Surface Area
CONNECT: Volume, Surface Area 1. VOLUMES OF SOLIDS A solid is a threedimensional (3D) object, that is, it has length, width and height. One of these dimensions is sometimes called thickness or depth.
More information1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.
1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationAP Calculus BC 2006 FreeResponse Questions
AP Calculus BC 2006 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationObjectives. Materials
Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI84 Plus / TI83 Plus Graph paper Introduction One of the ways
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More information