AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:
|
|
|
- Blanche Marsh
- 7 years ago
- Views:
Transcription
1 AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be on the final eam. The eam will be administered on Thursday December 1 during the two hour block period. The eam will consist of multiple choice questions. This is a closed notes and non-calculator eam. 1. Determine the following limit. (Hint: Use the graph of the function.) lim Page 1
2 . Let, 1 f( ). 1, 1 Determine the following limit. (Hint: Use the graph of the function.) lim f( ) 1. Let f ( ) 1 and g( ). Find the limits: (a) lim f( ) (b) lim g ( ) 1 (c) lim g( f ( )) 4 4. Let f and g( ). Find the limits: ( ) + 4 (a) lim f( ) 1 (b) lim g ( ) 1 (c) lim g( f ( )) Page
3 . Find the limit: lim sin 6. Find the limit: lim cos Find the limit: lim tan 4 8. Find the following limit (if it eists). Write a simpler function that agrees with the given function at all but one point. 1 lim 9. Find the following limit (if it eists). Write a simpler function that agrees with the given function at all but one point. lim Determine the limit (if it eists): sin 1 cos lim Determine the limit (if it eists): 4 1cos lim 0 Page
4 1. Find the -values (if any) at which the function f ( ) is not continuous. Which of the discontinuitites are removable? 1. Find the -values (if any) at which the function f( ) is not continuous. Which + 4 of the discontinuitites are removable? Find the -values (if any) at which the function f( ) + 4 is not continuous. Which of the discontinuitites are removable? 1. Find the limit: + 1 lim 16. Find the limit: lim Find the derivative of the following function using the limiting process. f ( ) Find the derivative of the following function using the limiting process. f ( ) Find the derivative of the following function using the limiting process. f ( ) +8 1 Page 4
5 0. Find the derivative of the following function using the limiting process. f( ) Find the derivative of the following function using the limiting process. f( ) 1. Find the derivative of the following function using the limiting process. f ( ) 7 7. Find the slope of the graph of the function at the given value. f ( ) 4 when 4. Find the slope of the graph of the function at the given value. when 4 f ( ) 4 8. Find the slope of the graph of the function at the given value. f ( ) ( 4) when 6. Find the slope of the graph of the function at the given value h(z) z z when z 7. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y ( ) 6 7 Page
6 8. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y 4 ( ) 7 9. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y ( ) Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y ( ) Use the product rule to differentiate. f ( t) t 7 t 6. Use the product rule to differentiate. R( t) t cos t. Use the product rule to differentiate. g( v) v sin v 4. Use the product rule to differentiate. 4 g( s) s cos s Page 6
7 . Use the quotient rule to differentiate. R ( ) Use the quotient rule to differentiate. 7 t Pt () t 7. Use the quotient rule to differentiate. g ( ) sin 8. Find the second derivative of the function. f ( ) 7 9. Find the second derivative of the function. Hs () s 4s s 40. Find the second derivative of the function. Q( t) t sect 41. Find dy/d by implicit differentiation. y Find dy/d by implicit differentiation. 1y y 16 Page 7
8 4. Find dy/d by implicit differentiation y Find dy/d by implicit differentiation. 9 9y y 6 4. Find dy/d by implicit differentiation. 7 7 y y Find dy/d by implicit differentiation. sin 9cos 7y 47. Find dy/d by implicit differentiation and evaluate it at the given point. 4y 6,, Find an equation of the tangent line to the graph of the function given below at the given point. ( y 10) ( 6), 4.80,.00. (The coefficients below are given to two decimal places.) 49. Find an equation of the tangent line to the graph of the function given below at the given point.,, 9 10y y 4 0 (The coefficients below are given to two decimal places.) Page 8
9 0. A point is moving along the graph of the function y 6 6 such that d/dt = centimeters per second. Find dy/dt for the given values of. (a) (b) 4 1. A point is moving along the graph of the function y such that d/dt = 4 centimeters per second. Find dy/dt when 4.. A point is moving along the graph of the function y sin such that d/dt = 8 centimeters per second. Find dy/dt when. 11. Area The radius, r, of a circle is decreasing at a rate of 4 centimeters per minute. Find the rate of change of area, A, when the radius is 6 4. Determine whether Rolle's Theorem can be applied to the function f ( ) ( + )( + )( + 1), 1. If Rolle's Theorem can be on the closed interval applied, find all numbers c in the open interval (, 1) such that f( c) 0. Page 9
10 . Determine whether Rolle's Theorem can be applied to the function f ( ) ( )( ),. If Rolle's Theorem can be applied, on the closed interval find all numbers c in the open interval (,) such that f( c) Determine whether the Mean Value Theorem can be applied to the function f ( ) on the closed interval [4,10]. If the Mean Value Theorem can be applied, find all f(10) f(4) numbers c in the open interval (4,10) such that f() c Determine whether the Mean Value Theorem can be applied to the function f ( ) on the closed interval [0,8]. If the Mean Value Theorem can be applied, find all numbers f(8) f(0) c in the open interval (0,8) such that f() c The function line. s( t) t 1t 6 t + 6 describes the motion of a particle moving along a (a) Find the velocity function of the particle at any time t; (b) Identify the time intervals when the particle is moving in a positive direction; (c) Identify the time intervals when the particle is moving in a negative direction; and (d) Identify the times when the particle changes its direction. 9. The function s( t) 4t t describes the motion of a particle moving along a line. (a) Find the velocity function of the particle at any time t; (b) Identify the time intervals when the particle is moving in a positive direction; (c) Identify the time intervals when the particle is moving in a negative direction; and (d) Identify the times when the particle changes its direction. 60. The function s( t) t 1t describes the motion of a particle moving along a line. (a) Find the velocity function of the particle at any time t; (b) Identify the time intervals when the particle is moving in a positive direction; (c) Identify the time intervals when the particle is moving in a negative direction; and (d) Identify the times when the particle changes its direction. Page 10
![Determine whether the Mean Value Theorem can be applied to the function f ( ) on the closed interval [4,10].](/docs-images/46/21730593/images/page_10.jpg "If the Mean Value Theorem can be applied, find all f(10) f(4) numbers c in the open interval (4,10) such that f() c. 10 4 7.")
11 61. Find the points of inflection and discuss the concavity of the function f ( ) sin + cos on the interval (0, ). 6. Find the points of inflection and discuss the concavity of the function f ( ) 7 cos on the interval [0, ]. 6. Find the limit. 4 8 lim Find the limit lim 7 6. Find two positive numbers with product of 4 and sum that is minimum. 66. Find the length and width of a rectangle that has perimeter 40 meters and a maimum area. 67. Determine the dimensions of a rectangular solid (with a square base) with maimum volume if its surface area is 100 meters. Page 11
12 68. Find the dimensions of the rectangle of maimum area bounded by the -ais and y-ais and the graph of y. 69. A rectangular page is to contain 169 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used. Page 1
13 Answer Key , 1, ,, 1. 1/ 6. 1/ , , Does not eist Continuous everywhere 1. Continuous everywhere (Removable), 4 (Not removable) f ( ) Both A and D 19. f ( ) f( ) f( ) 4. 7 f( ) f () f (4). f () h'() and There are no points at which the graph has a horizontal tangent Page 1
14 t f '( t) 6t t. 4 R'( t) t sin t t cos t. 4 g '( v) v cos v v sin v 4. 4 g '( s) s sin s 4s cos s. 10 R'( ) tt P'( t) t 7. ( )cos sin g'( ) 8. 8 f ( ) H''( s) s 40. Q"( t) t sec t(0 t sec t 10t tan t t tan t) 41. dy d y 4. dy 1y d y dy 1 d 8y dy 9 9y d y dy 7 y 6 7 d 7y 46. dy cos d 6sin 7y 47. dy 9 d y y dy 7 dt dy dt 144 Page 14
15 1. dy 88 dt 801. dy 4cos dt 11. da 48 dt 4. Both A and B. 7 Rolle's Theorem applies; 6. MVT applies; MVT applies; 8. (a) v( t) t 4t 6; (b) 0, 6, ; (c),6 ; (d) t = and t = (a) v( t) 4 t ; (b) 0,1 ; (c) 1, ; (d) t = (a) v( t) t 1; (b) 6, ; (c) 0,6 ; (d) t = None of the above 6. Concave downward on, ; concave upward on 0,,,. Inflection points at and , , Square base side ; height Length 1; width , 1 Page 1
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
Calculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim
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,
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
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
MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
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
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
Section 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
AP Calculus AB 2013 Free-Response Questions
AP Calculus AB 2013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded
correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
Mark 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,
Student Performance Q&A:
Student Performance Q&A: 2008 AP Calculus AB and Calculus BC Free-Response Questions The following comments on the 2008 free-response questions for AP Calculus AB and Calculus BC were written by the Chief
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.
AP Calculus AB 2005 Free-Response Questions
AP Calculus AB 25 Free-Response Questions 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
MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010
MATH 11 FINAL EXAM FALL 010-011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic
= f x 1 + h. 3. Geometrically, the average rate of change is the slope of the secant line connecting the pts (x 1 )).
Math 1205 Calculus/Sec. 3.3 The Derivative as a Rates of Change I. Review A. Average Rate of Change 1. The average rate of change of y=f(x) wrt x over the interval [x 1, x 2 ]is!y!x ( ) - f( x 1 ) = y
AP Calculus AB 2001 Scoring Guidelines
P Calculus Scing Guidelines The materials included in these files are intended f non-commercial use by P teachers f course and eam preparation; permission f any other use must be sought from the dvanced
AP Calculus AB 2011 Scoring Guidelines
AP Calculus AB Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 9, the
PRACTICE 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
3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH. In Isaac Newton's day, one of the biggest problems was poor navigation at sea.
BA01 ENGINEERING MATHEMATICS 01 CHAPTER 3 APPLICATION OF DIFFERENTIATION 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH Introduction to Applications of Differentiation In Isaac Newton's
Readings this week. 1 Parametric Equations Supplement. 2 Section 10.1. 3 Sections 2.1-2.2. Professor Christopher Hoffman Math 124
Readings this week 1 Parametric Equations Supplement 2 Section 10.1 3 Sections 2.1-2.2 Precalculus Review Quiz session Thursday equations of lines and circles worksheet available at http://www.math.washington.edu/
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
AP Calculus BC 2013 Free-Response Questions
AP Calculus BC 013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in
AP Calculus AB 2011 Free-Response Questions
AP Calculus AB 11 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in
Mathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. Dr Tay Seng Chuan
Ground Rules PC11 Fundamentals of Physics I Lectures 3 and 4 Motion in One Dimension Dr Tay Seng Chuan 1 Switch off your handphone and pager Switch off your laptop computer and keep it No talking while
AP 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
MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
AP Calculus AB 2012 Free-Response Questions
AP Calculus AB 1 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in
AP Calculus AB 2006 Scoring Guidelines
AP Calculus AB 006 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
Area of a triangle: The area of a triangle can be found with the following formula: 1. 2. 3. 12in
Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 Solve: Find the area of each triangle. 1. 2. 3. 5in4in 11in 12in 9in 21in 14in 19in 13in
AP Calculus BC 2001 Free-Response Questions
AP Calculus BC 001 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must
AP Calculus BC 2010 Free-Response Questions
AP Calculus BC 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded
A Resource for Free-standing Mathematics Qualifications
To find a maximum or minimum: Find an expression for the quantity you are trying to maximise/minimise (y say) in terms of one other variable (x). dy Find an expression for and put it equal to 0. Solve
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Thursday, August 16, 2012 8:30 to 11:30 a.m.
INTEGRATED ALGEBRA The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name
Math 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1)
Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1) In discussing motion, there are three closely related concepts that you need to keep straight. These are: If x(t) represents the
AP Calculus AB 2004 Free-Response Questions
AP Calculus AB 2004 Free-Response 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
Average rate of change of y = f(x) with respect to x as x changes from a to a + h:
L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,
AP Calculus BC 2006 Free-Response Questions
AP Calculus BC 2006 Free-Response Questions 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
AP 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
MATH 100 PRACTICE FINAL EXAM
MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number
Mathematical Modeling and Optimization Problems Answers
MATH& 141 Mathematical Modeling and Optimization Problems Answers 1. You are designing a rectangular poster which is to have 150 square inches of tet with -inch margins at the top and bottom of the poster
AP CALCULUS AB 2009 SCORING GUIDELINES
AP CALCULUS AB 2009 SCORING GUIDELINES Question 5 x 2 5 8 f ( x ) 1 4 2 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in
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
Trigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
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
AP CALCULUS AB 2007 SCORING GUIDELINES (Form B)
AP CALCULUS AB 2007 SCORING GUIDELINES (Form B) Question 4 Let f be a function defined on the closed interval 5 x 5 with f ( 1) = 3. The graph of f, the derivative of f, consists of two semicircles and
AP 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
FACTORING OUT COMMON FACTORS
278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the
18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 8.0 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Unit : Derivatives A. What
AP Calculus AB Syllabus
Course Overview and Philosophy AP Calculus AB Syllabus The biggest idea in AP Calculus is the connections among the representations of the major concepts graphically, numerically, analytically, and verbally.
AP Calculus AB 2010 Free-Response Questions
AP Calculus AB 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded
Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE
1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object
Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
Surface Area Quick Review: CH 5
I hope you had an exceptional Christmas Break.. Now it's time to learn some more math!! :) Surface Area Quick Review: CH 5 Find the surface area of each of these shapes: 8 cm 12 cm 4cm 11 cm 7 cm Find
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
Visualizing Differential Equations Slope Fields. by Lin McMullin
Visualizing Differential Equations Slope Fields by Lin McMullin The topic of slope fields is new to the AP Calculus AB Course Description for the 2004 exam. Where do slope fields come from? How should
The Derivative. Philippe B. Laval Kennesaw State University
The Derivative Philippe B. Laval Kennesaw State University Abstract This handout is a summary of the material students should know regarding the definition and computation of the derivative 1 Definition
Sandia High School Geometry Second Semester FINAL EXAM. Mark the letter to the single, correct (or most accurate) answer to each problem.
Sandia High School Geometry Second Semester FINL EXM Name: Mark the letter to the single, correct (or most accurate) answer to each problem.. What is the value of in the triangle on the right?.. 6. D.
Solutions to Practice Problems for Test 4
olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,
SAT 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 multiple-choice questions in which you are asked to choose the
How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
Characteristics of the Four Main Geometrical Figures
Math 40 9.7 & 9.8: The Big Four Square, Rectangle, Triangle, Circle Pre Algebra We will be focusing our attention on the formulas for the area and perimeter of a square, rectangle, triangle, and a circle.
5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
Calculus with Parametric Curves
Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function
1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
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...
x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
Version 005 Exam Review Practice Problems NOT FOR A GRADE alexander (55715) 1. Hence
Version 005 Eam Review Practice Problems NOT FOR A GRADE aleander 5575 This print-out should have 47 questions Multiple-choice questions may continue on the net column or page find all choices before answering
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
Math 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
Math 2443, Section 16.3
Math 44, Section 6. Review These notes will supplement not replace) the lectures based on Section 6. Section 6. i) ouble integrals over general regions: We defined double integrals over rectangles in the
Slope and Rate of Change
Chapter 1 Slope and Rate of Change Chapter Summary and Goal This chapter will start with a discussion of slopes and the tangent line. This will rapidly lead to heuristic developments of limits and the
Algebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
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
Good Questions. Answer: (a). Both f and g are given by the same rule, and are defined on the same domain, hence they are the same function.
Good Questions Limits 1. [Q] Let f be the function defined by f(x) = sin x + cos x and let g be the function defined by g(u) = sin u + cos u, for all real numbers x and u. Then, (a) f and g are exactly
Nonlinear Systems of Ordinary Differential Equations
Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System. A dynamical system has a state determined by a collection of real numbers, or more generally
Area Under the Curve. Riemann Sums And the Trapezoidal Rule
Area Under the Curve Riemann Sums And the Trapezoidal Rule Who knew that D=R x T would connect to velocity, and now integration, and the area under a curve? Take a look at the attached applications. Let
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
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
Math 115 Extra Problems for 5.5
Math 115 Extra Problems for 5.5 1. The sum of two positive numbers is 48. What is the smallest possible value of the sum of their squares? Solution. Let x and y denote the two numbers, so that x + y 48.
Solutions to Homework 5
Solutions to Homework 5 1. Let z = f(x, y) be a twice continously differentiable function of x and y. Let x = r cos θ and y = r sin θ be the equations which transform polar coordinates into rectangular
Chapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
Solutions 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
Pizza! Pizza! Assessment
Pizza! Pizza! Assessment 1. A local pizza restaurant sends pizzas to the high school twelve to a carton. If the pizzas are one inch thick, what is the volume of the cylindrical shipping carton for the
CHAPTER 8, GEOMETRY. 4. A circular cylinder has a circumference of 33 in. Use 22 as the approximate value of π and find the radius of this cylinder.
TEST A CHAPTER 8, GEOMETRY 1. A rectangular plot of ground is to be enclosed with 180 yd of fencing. If the plot is twice as long as it is wide, what are its dimensions? 2. A 4 cm by 6 cm rectangle has
12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2
DMA 080 WORKSHEET # (8.-8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) -00 ) 8 27 2/ Use a calculator to approximate the square root to decimal