Slope Fields and Differential Equations

Slope Fields and Differential Equations

Student Stud Session Slope Fields and Differential Equations Students should be able to: Draw a slope field at a specified number of points b hand. Sketch a solution that passes through a given point on

AP Calculus. Slope Fields and Differential Equations. Student Handout

AP Calculus Slope Fields and Differential Equations Student Handout 016-017 EDITION Use the following link or scan the QR code to complete the evaluation for the Stud Session https://www.survemonke.com/r/s_sss

Slope Fields and Differential Equations

Slope Fields and Differential Equations Students should be able to: Draw a slope field at a specified number of points b hand. Sketch a solution that passes through a given point on a slope field. Match

Analyzing f, f, and f Solutions

Analyzing f, f, and f Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate.

Multiple Choice Solutions 1. E (2003 AB25) () xt t t t 2. A (2008 AB21/BC21) 3. B (2008 AB7) Using Fundamental Theorem of Calculus: 1

Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment,

Limits, Continuity, and Differentiability Solutions

Limits, Continuity, and Differentiability Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions

Extra Practice Recovering C

Etra Practice Recovering C 1 Given the second derivative of a function, integrate to get the first derivative, then again to find the equation of the original function. Use the given initial conditions

Differential Equations

Differential Equations Big Ideas Slope fields draw a slope field, sketch a particular solution Separation of variables separable differential equations General solution Particular solution Growth decay

AP CALCULUS AB 2017 SCORING GUIDELINES

AP CALCULUS AB 07 SCORING GUIDELINES Consider the differential equation AP CALCULUS AB 06 SCORING GUIDELINES y =. Question 4 On the aes provided, sketch a slope field for the given differential equation

Fundamental Theorem of Calculus

Students should be able to: Use the fundamental theorem to evaluate definite integrals b f ( d ) Fb ( ) Fa ( ) a Use various forms of the fundamental theorem in application situations. b f ( d ) f ( b

Theorems Solutions. Multiple Choice Solutions

Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment,

du u C sec( u) tan u du secu C e du e C a u a a Trigonometric Functions: Basic Integration du ln u u Helpful to Know:

Integration Techniques for AB Eam Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at

AP Calculus. Particle Motion. Student Handout

AP Calculus Particle Motion Student Handout 016-017 EDITION Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss Copyright 016

AP Calculus. Analyzing a Function Based on its Derivatives

AP Calculus Analyzing a Function Based on its Derivatives Student Handout 016 017 EDITION Click on the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss

Pre-Calculus Module 4

Pre-Calculus Module 4 4 th Nine Weeks Table of Contents Precalculus Module 4 Unit 9 Rational Functions Rational Functions with Removable Discontinuities (1 5) End Behavior of Rational Functions (6) Rational

du u C sec( u) tan u du secu C e du e C a u a a Basic Integration Trigonometric Functions: du ln u u Helpful to Know: Inverse Trigonometric

Student Study Session Eam Solutions We have intentionally included more material than can e covered in most Student Study Sessions to account for groups that are ale to answer the questions at a faster

Graphical Relationships Among f, f,

Graphical Relationships Among f, f, and f The relationship between the graph of a function and its first and second derivatives frequently appears on the AP exams. It will appear on both multiple choice

( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx

Chapter 6 AP Eam Problems Antiderivatives. ( ) + d = ( + ) + 5 + + 5 ( + ) 6 ( + ). If the second derivative of f is given by f ( ) = cos, which of the following could be f( )? + cos + cos + + cos + sin

Student Session Topic: Average and Instantaneous Rates of Change

Student Session Topic: Average and Instantaneous Rates of Change The concepts of average rates of change and instantaneous rates of change are the building blocks of differential calculus. The AP exams

Computing Derivatives Solutions

Stuent Stuy Session Solutions We have intentionally inclue more material than can be covere in most Stuent Stuy Sessions to account for groups that are able to answer the questions at a faster rate. Use

AP Calculus. Limits, Continuity, and Differentiability

AP Calculus Limits, Continuity, and Differentiability Student Handout 016 017 EDITION Click on the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss

AP CALCULUS BC 2008 SCORING GUIDELINES

AP CALCULUS BC 2008 SCORING GUIDELINES Question 4 A particle moves along the x-axis so that its velocity at time t, for 0 t 6, is given by a differentiable function v whose graph is shown above. The velocity

Integration Techniques for the BC exam

Integration Techniques for the B eam For the B eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation

Integration Techniques for the BC exam

Integration Techniques for the B eam For the B eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation

Name Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y

10 Differential Equations Test Form A 1. Find the general solution to the first order differential equation: y 1 yy 0. 1 (a) (b) ln y 1 y ln y 1 C y y C y 1 C y 1 y C. Find the general solution to the

(a) During what time intervals on [0, 4] is the particle traveling to the left?

Chapter 5. (AB/BC, calculator) A particle travels along the -ais for times 0 t 4. The velocity of the particle is given by 5 () sin. At time t = 0, the particle is units to the right of the origin. t /

Math 113 Final Exam Practice Problem Solutions. f(x) = ln x x. lim. lim. x x = lim. = lim 2

Math 3 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice

Section Differential Equations: Modeling, Slope Fields, and Euler s Method

Section.. Differential Equations: Modeling, Slope Fields, and Euler s Method Preliminar Eample. Phsical Situation Modeling Differential Equation An object is taken out of an oven and placed in a room where

(2) Let f(x) = a 2 x if x<2, 4 2x 2 ifx 2. (b) Find the lim f(x). (c) Find all values of a that make f continuous at 2. Justify your answer.

(1) Let f(x) = x x 2 9. (a) Find the domain of f. (b) Write an equation for each vertical asymptote of the graph of f. (c) Write an equation for each horizontal asymptote of the graph of f. (d) Is f odd,

Separable Differential Equations

-1- F D2f0B1I3F sknuwtqa RSAo9fjt3wdaircei TLgL4CK.w j EAMlilW mroidghttso nr3efsgefrpv0eado.q T 9MdaFdQe5 UwDibtuh8 UI0nMf6i3nZ itez scjaaljcduclguesb.k Worksheet b Kuta Software LLC Kuta Software - Infinite

Chapter (AB/BC, non-calculator) (a) Find the critical numbers of g. (b) For what values of x is g increasing? Justify your answer.

Chapter 3 1. (AB/BC, non-calculator) Given g ( ) 2 4 3 6 : (a) Find the critical numbers of g. (b) For what values of is g increasing? Justify your answer. (c) Identify the -coordinate of the critical

AP Calculus. Area Accumulation and Approximation

AP Calculus Area Accumulation and Approximation Student Handout 26 27 EDITION Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss

Differential Equations and Slope Fields

Differential Equations and Slope Fields. Population grows according to the equation : = k where k is a constant and t is measured in ears. f the population doubles ever 10 ears, then the value of k is

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)

. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula

x f(x)

CALCULATOR SECTION. For y y 8 find d point (, ) on the curve. A. D. dy at the 7 E. 6. Suppose silver is being etracted from a.t mine at a rate given by A'( t) e, A(t) is measured in tons of silver and

NO CALCULATORS: 1. Find A) 1 B) 0 C) D) 2. Find the points of discontinuity of the function y of discontinuity.

AP CALCULUS BC NO CALCULATORS: MIDTERM REVIEW. Find lim 7 7 9. B) C) D). Find the points of discontinuit of the function of discontinuit. 9. For each discontinuit identif the tpe A. Removable discontinuit

2004 Free Responses Solutions. Form B

Free Responses Solutions Form B All questions are available from www.collegeboard.com James Rahn www.jamesrahn.com Form B AB Area d 8 B. ( ) π ( ) Volume π d π.7 or.8 or ( ) Volume π 9 y 7. or 68 π Form

AP CALCULUS BC 2016 SCORING GUIDELINES

Consider the differential equation (a) Find in terms of x an. AP CALCULUS BC 06 SCORING GUIDELINES x y. = Question 4 (b) Let y = f ( x) be the particular solution to the given differential equation whose

AP CALCULUS AB/CALCULUS BC 2017 SCORING GUIDELINES

AP CALCULUS AB/CALCULUS BC 07 SCORING GUIDELINES Question 4 H ( 0) = ( 9 7) = 6 4 H ( 0) = 9 An equation for the tangent line is y = 9 6 t. : slope : : tangent line : approimation The internal temperature

1. Find A and B so that f x Axe Bx. has a local minimum of 6 when. x 2.

. Find A and B so that f Ae B has a local minimum of 6 when.. The graph below is the graph of f, the derivative of f; The domain of the derivative is 5 6. Note there is a cusp when =, a horizontal tangent

MATH 150/GRACEY PRACTICE FINAL. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MATH 0/GRACEY PRACTICE FINAL Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Choose the graph that represents the given function without using

Math 370 Exam 2 Review Name

Math 70 Exam 2 Review Name Be sure to complete these problems before the review session. 10 of these questions will count as a quiz in Learning Catalytics. Round 1 will be individual. Round 2 will be in

4.3 Worksheet - Derivatives of Inverse Functions

AP Calculus 3.8 Worksheet 4.3 Worksheet - Derivatives of Inverse Functions All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. What are the following derivatives

AP Calculus (AB/BC) Prerequisite Packet Paint Branch High School Math Department

Updated 6/015 The problems in this packet are designed to help ou review topics from previous math courses that are important to our success in AP Calculus AB / BC. It is important that ou take time during

Math 103 Final Exam Review Problems Rockville Campus Fall 2006

Math Final Eam Review Problems Rockville Campus Fall. Define a. relation b. function. For each graph below, eplain why it is or is not a function. a. b. c. d.. Given + y = a. Find the -intercept. b. Find

121D Practice Test #

D Practice Test # College Algebra / Math D GARAGE (Prof. Vasan) Student Name/ID:. Write the epression as a single logarithm. 5log 8 w + 5 log 8 3log 8 z. Solve for. log + 3 = log + 6 ALEKS D Practice Test

1993 AP Calculus AB: Section I

99 AP Calculus AB: Section I 90 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among

Lesson 6.2 Exercises, pages

Lesson 6.2 Eercises, pages 448 48 A. Sketch each angle in standard position. a) 7 b) 40 Since the angle is between Since the angle is between 0 and 90, the terminal 90 and 80, the terminal arm is in Quadrant.

AP CALCULUS AB 2004 SCORING GUIDELINES (Form B)

004 SORING GUIDELINES (Form B) dy 4 onsider the differential equation ( y. ) d = On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated. (Note: Use

6.2 Indicate whether the function is one-to-one. 16) {(-13, -20), (-10, -20), (13, -8)}

Math 0 Eam Review. Evaluate the epression using the values given in the table. ) (f g)() 7 f() - - - g() - 7 Evaluate the epression using the graphs of = f() and = g(). ) Evaluate (fg)(). 9) H() = - 7

Chapters 8 & 9 Review for Final

Math 203 - Intermediate Algebra Professor Valdez Chapters 8 & 9 Review for Final SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the formula for

x f(x)

CALCULATOR SECTION. For y + y = 8 find d point (, ) on the curve. A. B. C. D. dy at the 7 E. 6. Suppose silver is being etracted from a.t mine at a rate given by A'( t) = e, A(t) is measured in tons of

AP Calculus AB Free-Response Scoring Guidelines

Question pt The rate at which raw sewage enters a treatment tank is given by Et 85 75cos 9 gallons per hour for t 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per

Test # 1 Review Math MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Test # 1 Review Math 13 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the slope of the curve at the given point P and an equation of the

Mathematics Extension 2

0 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators

A) 13 B) 9 C) 22 D) log 9

Math 70 Exam 2 Review Name Be sure to complete these problems before the review session. Participation in our review session will count as a quiz grade. Please bring any questions you have ready to ask!

LSU AP Calculus Practice Test Day

LSU AP Calculus Practice Test Day AP Calculus AB 2018 Practice Exam Section I Part A AP CALCULUS AB: PRACTICE EXAM SECTION I: PART A NO CALCULATORS ALLOWED. YOU HAVE 60 MINUTES. 1. If y = ( 1 + x 5) 3

CALCULUS AB SECTION II, Part A

CALCULUS AB SECTION II, Part A Time 45 minutes Number of problems 3 A graphing calculator is required for some problems or parts of problems. pt 1. The rate at which raw sewage enters a treatment tank

MA 113 Calculus I Fall 2009 Exam 4 December 15, 2009

MA 3 Calculus I Fall 009 Eam December 5, 009 Answer all of the questions - 7 and two of the questions 8-0. Please indicate which problem is not to be graded b crossing through its number in the table below.

1985 AP Calculus AB: Section I

985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of

Student Study Session Topic: Interpreting Graphs

Student Study Session Topic: Interpreting Graphs Starting with the graph of a function or its derivative, you may be asked all kinds of questions without having (or needing) and equation to work with.

MATH 2300 review problems for Exam 3 ANSWERS

MATH 300 review problems for Eam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justif our answer b either computing the sum or b b showing which convergence test ou are

Methods of Integration

U96-b)! Use the substitution u = - to evaluate U95-b)! 4 Methods of Integration d. Evaluate 9 d using the substitution u = + 9. UNIT MATHEMATICS (HSC) METHODS OF INTEGRATION CSSA «8» U94-b)! Use the substitution

Math Worksheet 1 SHOW ALL OF YOUR WORK! f(x) = (x a) 2 + b. = x 2 + 6x + ( 6 2 )2 ( 6 2 )2 + 7 = (x 2 + 6x + 9) = (x + 3) 2 2

Names Date. Consider the function Math 0550 Worksheet SHOW ALL OF YOUR WORK! f() = + 6 + 7 (a) Complete the square and write the function in the form f() = ( a) + b. f() = + 6 + 7 = + 6 + ( 6 ) ( 6 ) +

Math 125 Practice Problems for Test #3

Math Practice Problems for Test # Also stud the assigned homework problems from the book. Donʹt forget to look over Test # and Test #! Find the derivative of the function. ) Know the derivatives of all

All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

AP Calculus 6. Worksheet Da All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. Indefinite Integrals: Remember the first step to evaluating an integral is to

AP Calculus AB/IB Math SL2 Unit 1: Limits and Continuity. Name:

AP Calculus AB/IB Math SL Unit : Limits and Continuity Name: Block: Date:. A bungee jumper dives from a tower at time t = 0. Her height h (in feet) at time t (in seconds) is given by the graph below. In

1. (a) B, D A1A1 N2 2. A1A1 N2 Note: Award A1 for. 2xe. e and A1 for 2x.

1. (a) B, D N (b) (i) f () = e N Note: Award for e and for. (ii) finding the derivative of, i.e. () evidence of choosing the product rule e.g. e e e 4 e f () = (4 ) e AG N0 5 (c) valid reasoning R1 e.g.

Law of Sines, Law of Cosines, Heron s Formula:

PreAP Math Analsis nd Semester Review Law of Sines, Law of Cosines, Heron s Formula:. Determine how man solutions the triangle has and eplain our reasoning. (FIND YOUR FLOW CHART) a. A = 4, a = 4 ards,

Review Exercises for Chapter 2

Review Eercises for Chapter 367 Review Eercises for Chapter. f 1 1 f f f lim lim 1 1 1 1 lim 1 1 1 1 lim 1 1 lim lim 1 1 1 1 1 1 1 1 1 4. 8. f f f f lim lim lim lim lim f 4, 1 4, if < if (a) Nonremovable

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 Prep Session Handout. Integral Defined Functions

AP Calculus Prep Session Handout A continuous, differentiable function can be epressed as a definite integral if it is difficult or impossible to determine the antiderivative of a function using known

Math 231 Final Exam Review

Math Final Eam Review Find the equation of the line tangent to the curve 4y y at the point (, ) Find the slope of the normal line to y ) ( e at the point (,) dy Find d if cos( y) y 4 y 4 Find the eact

The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. n times per year: 1

ALGEBRA B Semester Eam Review The semester B eamination for Algebra will consist of two parts. Part 1 will be selected response. Part will be short answer. Students ma use a calculator. If a calculator

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.

90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y

AP Calculus AB. Scoring Guidelines

17 AP Calculus AB Scoring Guidelines 17 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the

Answer Key. ( 1) n (2x+3) n. n n=1. (2x+3) n. = lim. < 1 or 2x+3 < 4. ( 1) ( 1) 2n n

Math Midterm Eam #3 December, 3 Answer Key. [5 Points] Find the Interval and Radius of Convergence for the following power series. Analyze carefully and with full justification. Use Ratio Test. L lim a

Part 1: Integration problems from exams

. Find each of the following. ( (a) 4t 4 t + t + (a ) (b ) Part : Integration problems from 4-5 eams ) ( sec tan sin + + e e ). (a) Let f() = e. On the graph of f pictured below, draw the approimating

dy x a. Sketch the slope field for the points: (1,±1), (2,±1), ( 1, ±1), and (0,±1).

Chapter 6. d x Given the differential equation: dx a. Sketch the slope field for the points: (,±), (,±), (, ±), and (0,±). b. Find the general solution for the given differential equation. c. Find the

Math 106 Answers to Exam 3a Fall 2015

Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical

CHAPTER 3 Applications of Differentiation

CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. Section. Rolle s Theorem and the Mean Value Theorem. 7 Section. Increasing and Decreasing Functions and the First Derivative

AP Calculus BC 2015 Free-Response Questions

AP Calculus BC 05 Free-Response Questions 05 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central

(i) find the points where f(x) is discontinuous, and classify each point of discontinuity.

Math Final Eam - Practice Problems. A function f is graphed below. f() 5 4 8 7 5 4 4 5 7 8 4 5 (a) Find f(0), f( ), f(), and f(4) Find the domain and range of f (c) Find the intervals where f () is positive

Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 119 Mark Sparks 2012

Unit # Understanding the Derivative Homework Packet f ( h) f ( Find lim for each of the functions below. Then, find the equation of the tangent line to h 0 h the graph of f( at the given value of. 1. f

1A (13) 1. Find an equation for the tangent line to the graph of y = 3 3y +3at the point ( ; 1). The first thing to do is to check that the values =, y =1satisfy the given equation. They do. Differentiating

AP Calculus BC 2011 Free-Response Questions

AP Calculus BC 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

7.4. The Primary Trigonometric Ratios. LEARN ABOUT the Math. Connecting an angle to the ratios of the sides in a right triangle. Tip.

The Primary Trigonometric Ratios GOL Determine the values of the sine, cosine, and tangent ratios for a specific acute angle in a right triangle. LERN OUT the Math Nadia wants to know the slope of a ski

Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.

Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.

UBC-SFU-UVic-UNBC Calculus Exam Solutions 7 June 2007

This eamination has 15 pages including this cover. UBC-SFU-UVic-UNBC Calculus Eam Solutions 7 June 007 Name: School: Signature: Candidate Number: Rules and Instructions 1. Show all your work! Full marks

Math 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2

Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice

Accumulation. area. the function is. f(x)

Left and right Riemann sums Right Riemann Sum Left Riemann Sum Correct justification for over and under approximations: f(x) Left Riemann Sum Right Riemann Sum Increasing (f '(x) > ) Under approximates

Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number.

997 AP Calculus BC: Section I, Part A 5 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number..

Fair Game Review. Chapter = How many calculators are sold when the profit is $425? Solve the equation. Check your solution.

Name Date Chapter 4 Fair Game Review Solve the equation. Check our solution.. 8 3 = 3 2. 4a + a = 2 3. 9 = 4( 3k 4) 7k 4. ( m) 2 5 6 2 = 8 5. 5 t + 8t = 3 6. 3 5h 2 h + 4 = 0 2 7. The profit P (in dollars)

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental

Arkansas Council of Teachers of Mathematics 2013 State Contest Calculus Exam

0 State Contest Calculus Eam In each of the following choose the BEST answer and shade the corresponding letter on the Scantron Sheet. Answer all multiple choice questions before attempting the tie-breaker

ANOTHER FIVE QUESTIONS:

No peaking!!!!! See if you can do the following: f 5 tan 6 sin 7 cos 8 sin 9 cos 5 e e ln ln @ @ Epress sin Power Series Epansion: d as a Power Series: Estimate sin Estimate MACLAURIN SERIES ANOTHER FIVE

CHAPTER 3 Applications of Differentiation

CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First

AP Calculus (BC) Summer Assignment (169 points)

AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion