# Chapter (AB/BC, non-calculator) (a) Write an equation of the line tangent to the graph of f at x 2.

Save this PDF as:

Size: px
Start display at page:

Download "Chapter (AB/BC, non-calculator) (a) Write an equation of the line tangent to the graph of f at x 2."

## Transcription

1 Chapter 1. (AB/BC, non-calculator) Let f( x) x 3 4. (a) Write an equation of the line tangent to the graph of f at x. (b) Find the values of x for which the graph of f has a horizontal tangent. (c) Find f ''( x ).

2 . (AB/BC, non-calculator) Let f( x) 4x 3 and f ( x) gx ( ). x (a) What is the slope of the graph of f at x 3? Show the work that leads to your answer. (b) Write an equation of the line tangent to the graph of g at x 3. (c) What is the slope of the line normal to the graph of g at x 3?

3 3. (AB/BC, non-calculator) Evaluate each limit analytically. (Note: Finding the answer should not involve a lengthy algebraic process.) (a) sin( x h) sin x lim h0 h (b) lim h0 x h x h 3 3 (c) lim h 0 16 h 4 h (d) 1 1 lim 5 h 5 h0 h

4 4. (AB/BC, calculator neutral) Given: x f ( x ) f ( x) gx ( ) g ( x) (a) If f ( x) hx ( ), find h '(). gx ( ) (b) If j( x) f( g( x)), find j '(). (c) If kx ( ) f( x), find k '(5).

5 5. (AB/BC, non-calculator) Given: f ( x) x (a) Find the slope of the normal line to the graph of f at x 3. (b) Two lines passing through the point (3,8) will be tangent to the graph of f. Find an equation for each of these lines.

6 6. (AB/BC, calculator neutral) The accompanying diagram shows the graph of the velocity in the line x 4. ft for a particle moving along sec v(t) t (a) During which time interval is the particle: (i) moving upward. (ii) moving downward. (iii) at rest. (b) State the acceleration of the particle at the specified times. Include units. (i) t 0.75 (ii) t 4.

7 7. (AB/BC, non-calculator) Given: gx ( ) f( x) tan x kx, where k is a real number. f is differentiable for all x; f 4 ; f. 4 4 (a) For what values of x, if any, in the interval 0 x will the derivative of g fail to exist? Justify your answer. (b) If g 6, find the value of k. 4

8 8. (AB/BC, calculator neutral) The table provided below shows the position of a particle, S, at several times, t, as the particle moves along a straight line, where t is measured in seconds and S is measured in meters. t St () Which of the following best estimates the velocity of the particle at t 3? (a) m 9. s (b) m 7.8 s (c) m 5.6 s

9 9. (AB/BC, non-calculator) If y xyx, then dy dx x (a) 6y 3 x 3y (b) 3x 6y x 3 (c) 6y x (d) 6y 3x 3y x (e) 6y

10 10. (AB/BC, non-calculator) The volume of a cylinder with radius r and height h is given by V r h. The radius and height of the cylinder are increasing at constant rates. The radius is expanding at 1cm and the height is 3sec increasing at 1cm. At what rate, in cubic cm per second, is the volume of the cylinder sec increasing when its height is 9 cm and the radius is 4 cm? (a) 3 (b) 6 (c) 8 3 (d) 4 3 (e) 18

11 Chapter (Solutions) Question 1 f x x. Let ( ) 3 4 (a) Write an equation of the line tangent to the graph of f at x. (b) Find the values of x for which the graph of f has a horizontal tangent. (c) Find f ''( x ). f x x x (a) '( ) Point:,1 1: derivative 4: 1: point 1: slope 1: equation m 16 y1 16( x ) (b) 8 xx ( 3) 0 3: 1: derivative equal to 0 : answers x0; x 3 3 (c) f ''( x) 48x x 3 8x 3 : answer

12 Question Let f( x) 4x3and f ( x) gx ( ). x (a) What is the slope of the graph of f at x 3? Show the work that leads to your answer. (b) Write an equation of the line tangent to the graph of g at x 3. (c) What is the slope of the line normal to the graph of g at x 3? (a) f '( x) 4x 3 3: : derivative 1: answer f '(3) 3 (b) x g xf x f x x 1 g '(3) 9 :derivative 5: 1: evaluates g(3) 1: (3,1) 1: equation 3, g 3 3,1 y1 1 x3 y 1 x (c) 9 1: answer

13 Question 3 Evaluate each limit analytically. (Note: Finding the answer should not involve a lengthy algebraic process.) (a) sin( x h) sin x lim h 0 h (b) lim h0 x h x h 3 3 (c) lim h0 16 h 4 h (d) 1 1 lim 5 h 5 h0 h (a) f ( x) cos x 1:answer (b) f( x) 1 : answer 3 3x (c) 1 f( x) 3: :derivative x 1: answer 1 f (16) 8

14 Question 3 (cont.) 1 (d) f( x) 3: x : derivative 1: answer 1 f (5) 5

15 Question 4 Given: x f ( x ) f ( x) gx ( ) g ( x) (a) If f ( x) hx ( ), find h '(). gx ( ) (b) If jx ( ) f( gx ( )), find j '(). (c) If kx ( ) f( x), find k '(5). (a) gxf ( ) ( x) f( xg ) ( x) h( x) 3: : derivative 1: answer gx ( ) 1 h() 5 (b) j( x) f( g( x)) g( x) 3: :derivative 1: answer j() 14

16 Question 4 (cont.) (c) 1 k( x) f( x) 3: :derivative f( x) 1: answer k '(5) 1

17 Question 5 Given: f ( x) x (a) Find the slope of the normal line to the graph of f at x 3. (b) Two lines passing through the point (3,8) will be tangent to the graph of f. Find an equation for each of these lines. (a) f '( x) x f '( 3) 6 : 1: derivative 1: slope 1 m 6 (b) 8 x 3 x x x4; x 1: equation 7: : points : slopes : equations 4,16 ;, 4 m8; m 4 y16 8( x4); y4 4( x )

18 Question 6 The accompanying diagram shows the graph of the velocity in the line x 4. ft for a particle moving along sec v(t) t (a) During which time interval is the particle: (i) moving upward. (ii) moving downward. (iii) at rest. (b) State the acceleration of the particle at the specified times. Include units. (i) t 0.75 (ii) t 4.

19 Question 6 (cont.) (a) (i) 0t 1; t 5 5 4: : intervals : find t-intercept on 4,5 (ii) t 1: answer 5 (iii) 1t 1: answer ft (b) (i) : sec 1:answer 1: units ft (ii) 5 1: answer sec

20 Question 7 Given: gx ( ) f( x) tan x kx, where k is a real number. f is differentiable for all x; f 4 ; f. 4 4 (a) For what values of x, if any, in the interval 0 x will the derivative of g fail to exist? Justify your answer. (b) If g 6, find the value of k. 4 (a) The derivative of g will fail to exist at x and 3 x 4: :values : justification because g is not continuous at these values. (b) g'( x) f( x)sec xtan x f '( x) k sec tan ' f f k 5: 3: derivative : solution 4 1 k 6 k 0

21 Questions c S(3.) S(.7) m s 9. b dy dy y x y x dx dx dy x 3y dx 3x 6y (3 3 ) a V r h dv dh dr r rh dt dt dt dv dt 3 dv 3 dt

### AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:

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

### Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson.

Lecture 17 MATH10070 - Introduction to Calculus maths.ucd.ie/modules/math10070 Kevin Hutchinson 28th October 2010 Z Chain Rule (I): If y = f (u) and u = g(x) dy dx = dy du du dx Z Chain rule (II): d dx

### If f is continuous on [a, b], then the function g defined by. f (t) dt. is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function g defined by g(x) = x a f (t) dt a x b is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

### = 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

### Section 1. Movement. So if we have a function x = f(t) that represents distance as a function of time, then dx is

Worksheet 4.4 Applications of Integration Section 1 Movement Recall that the derivative of a function tells us about its slope. What does the slope represent? It is the change in one variable with respect

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

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

### Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa

Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,

### AP Calculus AB 2006 Free-Response Questions

AP Calculus AB 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 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

### 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

### AP Calculus AB 2008 Free-Response Questions

AP Calculus AB 2008 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

### Learning Objectives for Math 165

Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

### Chapter 2 Differentiation

Chapter 2 Differentiation SECTION 2.1 The Derivative and the Tangent Line Problem Calculus: Chapter 2 Section 2.1 Finding the Slope of a Secant Line m y x m sec f ( c x) f ( c) ( c x) c m sec f ( c x)

### Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate

### 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 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

### 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

### 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

### Derivatives and Graphs. Review of basic rules: We have already discussed the Power Rule.

Derivatives and Graphs Review of basic rules: We have already discussed the Power Rule. Product Rule: If y = f (x)g(x) dy dx = Proof by first principles: Quotient Rule: If y = f (x) g(x) dy dx = Proof,

### 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 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

### AP 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

### 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

### Test # 2 Review. function y sin6x such that dx. per second. Find dy. f(x) 3x 2 6x 8 using the limiting process. dt = 2 centimeters. dt when x 7.

Name: Class: Date: ID: A Test # 2 Review Short Answer 1. Find the slope m of the line tangent to the graph of the function g( x) 9 x 2 at the point 4, 7ˆ. 2. A man 6 feet tall walks at a rate of 2 ft per

### 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: 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

### AP Calculus AB 2012 Scoring Guidelines

AP Calculus AB Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 9, the College

### The graph of a function is given. Choose the answer that represents the graph of its derivative. 1) 1)

Assignment 4 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a function is given. Choose the answer that represents the graph of

### 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

### 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

### AP Calculus BC 2004 Free-Response Questions

AP Calculus BC 004 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

### Homework #5 Solutions

Homework # Solutions Problems Bolded problems are worth 2 points. Section 2.3: 10, 16, 26 Section 2.4: 2, 6, 10, 22, 28 Section 3.1: 4, 14, 24, 28, 36, 38, 0, 60 2.3.10. On May 9, 2007, CBS Evening News

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

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

### AP Calculus BC 2003 Free-Response Questions

AP Calculus BC 2003 Free-Response 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

### Math 103: Secants, Tangents and Derivatives

Math 103: Secants, Tangents and Derivatives Ryan Blair University of Pennsylvania Thursday September 27, 2011 Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011

### 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

### Week #15 - Word Problems & Differential Equations Section 8.1

Week #15 - Word Problems & Differential Equations Section 8.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 25 by John Wiley & Sons, Inc. This material is used by

### 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

### Solution for Final Review Problems 1

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

### 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

### 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

### 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 BC 2012 Free-Response Questions

AP Calculus BC 0 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

### Section 3.8. Finding Maximum and Minimum Values. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section 3.8 Finding Maximum and Minimum Values Problems involving finding the maximum or minimum value of a quantity occur frequently in mathematics and in

### VI. Transcendental Functions. x = ln y. In general, two functions f, g are said to be inverse to each other when the

VI Transcendental Functions 6 Inverse Functions The functions e x and ln x are inverses to each other in the sense that the two statements y = e x, x = ln y are equivalent statements In general, two functions

### People s Physics book 3e Ch 25-1

The Big Idea: In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate

### MATH 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

### Derivatives as Rates of Change

Derivatives as Rates of Change One-Dimensional Motion An object moving in a straight line For an object moving in more complicated ways, consider the motion of the object in just one of the three dimensions

### 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

### Homework #1 Solutions

MAT 303 Spring 203 Homework # Solutions Problems Section.:, 4, 6, 34, 40 Section.2:, 4, 8, 30, 42 Section.4:, 2, 3, 4, 8, 22, 24, 46... Verify that y = x 3 + 7 is a solution to y = 3x 2. Solution: From

### 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

### Math 41: Calculus Final Exam December 7, 2009

Math 41: Calculus Final Exam December 7, 2009 Name: SUID#: Select your section: Atoshi Chowdhury Yuncheng Lin Ian Petrow Ha Pham Yu-jong Tzeng 02 (11-11:50am) 08 (10-10:50am) 04 (1:15-2:05pm) 03 (11-11:50am)

### Introduction to Calculus

Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative

### M1120 Class 5. Dan Barbasch. September 4, Dan Barbasch () M1120 Class 5 September 4, / 16

M1120 Class 5 Dan Barbasch September 4, 2011 Dan Barbasch () M1120 Class 5 September 4, 2011 1 / 16 Course Website http://www.math.cornell.edu/ web1120/index.html Dan Barbasch () M1120 Class 5 September

### Draft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then

CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function,

### 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

### 261 Practice problems for chapter 3

61 Practice problems for chapter 3 Disclaimer; the actual eam ma have different questions. Even though answers are given to this set, on an actual eam ou will be epected to show much more reasoning to

### 3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13

Contents 1 Limits 2 2 Derivatives 3 2.1 Difference Quotients......................................... 3 2.2 Average Rate of Change...................................... 4 2.3 Derivative Rules...........................................

### Motion in One-Dimension

This test covers one-dimensional kinematics, including speed, velocity, acceleration, motion graphs, with some problems requiring a knowledge of basic calculus. Part I. Multiple Choice 1. A rock is released

### The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result.

30 5.6 The chain rule The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result." Example. g(x) = x 2 and f(x) = (3x+1).

### AP CALCULUS AB 2009 SCORING GUIDELINES (Form B)

AP CALCULUS AB 09 SCORING GUIDELINES (Form B) Question 6 t (seconds) vt () (meters per second) 0 8 25 32 3 5 10 8 4 7 The velocity of a particle moving along the x-axis is modeled by a differentiable function

### IB Mathematics HL, Year 1 (Paper 1)

Puxi Campus High School Examinations Semester Two June 2010 IB Mathematics HL, Year 1 (Paper 1) Student Name: Thursday, June 3 rd, 2010 12:45-2:00 Time: 1 hour, 15 minutes Mr. Surowski Instructions to

### 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,

### 1. [20 Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, + or, or Does Not Exist.

Answer Key, Math, Final Eamination, December 9, 9. [ Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, + or, or Does Not Eist. (a lim + 6

### AP Calculus BC Exam. The Calculus BC Exam. At a Glance. Section I. SECTION I: Multiple-Choice Questions. Instructions. About Guessing.

The Calculus BC Exam AP Calculus BC Exam SECTION I: Multiple-Choice Questions At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Grade 50% Writing Instrument Pencil required

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of an exponential function is given. Match the graph to one of the following functions.

### x + 5 x 2 + x 2 dx Answer: ln x ln x 1 + c

. Evaluate the given integral (a) 3xe x2 dx 3 2 e x2 + c (b) 3 x ln xdx 2x 3/2 ln x 4 3 x3/2 + c (c) x + 5 x 2 + x 2 dx ln x + 2 + 2 ln x + c (d) x sin (πx) dx x π cos (πx) + sin (πx) + c π2 (e) 3x ( +

### AP Calculus BC 2009 Free-Response Questions

AP Calculus BC 009 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

### AP Calculus AB 2009 Scoring Guidelines

AP Calculus AB 9 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 19,

### PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS

PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,

### The Derivative and the Tangent Line Problem. The Tangent Line Problem

The Derivative and the Tangent Line Problem Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. 1. The tangent line problem 2. The velocity

### 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

### 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

### Test Information Guide: College-Level Examination Program 2013-14

Test Information Guide: College-Level Examination Program 01-14 Calculus X/ 01 The College Board. All righte reserved. Ccilege Board, College-Level Examination Program, CLEF', and the acorn logo are registered

### 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

### 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,

### AP Calculus AB 2007 Free-Response Questions

AP Calculus AB 2007 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

### Worksheet # 1: Review

Worksheet # 1: Review 1. (MA 113 Exam 1, Problem 1, Spring 27). Find the equation of the line that passes through (1, 2) and is parallel to the line 4x + 2y = 11. Put your answer in y = mx + b form. 2.

### AP Calculus BC 2009 Free-Response Questions Form B

AP Calculus BC 009 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.

### 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

### Implicit Differentiation and Related Rates

Implicit Differentiation and Related Rates PART I: Implicit Differentiation Implicit means implied or understood though not directly expressed The equation has an implicit meaning. It implicitly describes

### 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

### 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

### 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

### Student Performance Q&A:

Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

### Chapter 2: Rocket Launch

Chapter 2: Rocket Launch Lesson 2.1.1. 2-1. Domain:!" x " Range: 2! y! " y-intercept! y = 2 no x-intercepts 2-2. Time Hours sitting Amount Earned 8PM 1 4 9PM 2 4*2hrs = 8 10PM 3 4*3hrs = 12 11:30PM 4.5

### APPLICATION OF DERIVATIVES

6. Overview 6.. Rate of change of quantities For the function y f (x), d (f (x)) represents the rate of change of y with respect to x. dx Thus if s represents the distance and t the time, then ds represents

### W i f(x i ) x. i=1. f(x i ) x = i=1

Work Force If an object is moving in a straight line with position function s(t), then the force F on the object at time t is the product of the mass of the object times its acceleration. F = m d2 s dt

### Absolute Maxima and Minima

Absolute Maxima and Minima Definition. A function f is said to have an absolute maximum on an interval I at the point x 0 if it is the largest value of f on that interval; that is if f( x ) f() x for all

### Extra Problems for Midterm 2

Extra Problems for Midterm Sudesh Kalyanswamy Exercise (Surfaces). Find the equation of, and classify, the surface S consisting of all points equidistant from (0,, 0) and (,, ). Solution. Let P (x, y,

### Work. Example. If a block is pushed by a constant force of 200 lb. Through a distance of 20 ft., then the total work done is 4000 ft-lbs. 20 ft.

Work Definition. If a constant force F is exerted on an object, and as a result the object moves a distance d in the direction of the force, then the work done is Fd. Example. If a block is pushed by a

### 2014 Leaving Cert Ordinary Level Official Sample Paper 1

2014 Leaving Cert Ordinary Level Official Sample Paper 1 Section A Concepts and Skills 150 marks Question 1 (25 marks) (a) Write 6 2 and 81 1 2 without using indices. Given the relation x 1 = 1 x, we have

### Calculus I Review Solutions. f(x) = L means that, for every ɛ > 0, there is a δ > 0 such that

Calculus I Review Solutions. Finish the definition: (a) x a f(x) = L means that, for every ɛ > 0, there is a δ > 0 such that if 0 < x a < δ then f(x) L < ɛ (b) A function f is continuous at x = a if x

### 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

### AP Calculus BC 2004 Scoring Guidelines

AP Calculus BC Scoring Guidelines 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 sought from

### 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

### 2.2. Instantaneous Velocity

2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical

### Yimin Math Centre. 2/3 Unit Math Homework for Year Motion Part Simple Harmonic Motion The Differential Equation...

2/3 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 9 Motion Part 3 1 9.1 Simple Harmonic Motion The Differential Equation................... 1 9.2 Projectile Motion