University of Alabama Department of Physics and Astronomy PH 101 LeClair Summer Exam 1 Solutions

Size: px
Start display at page:

Download "University of Alabama Department of Physics and Astronomy PH 101 LeClair Summer Exam 1 Solutions"

Transcription

1 University of Alabama Department of Pysics and Astronomy PH 101 LeClair Summer 2011 Exam 1 Solutions 1. A motorcycle is following a car tat is traveling at constant speed on a straigt igway. Initially, te car and te motorcycle are bot traveling at te same speed of 18 m/s, and te distance between tem is 92.0 m. After 2.50 s, te motorcycle starts to accelerate at a rate of 4.00 m/s 2. How long does it take from te moment te motorcycle starts to accelerate until it catces up to te car? Solution: Tis was also problem 5b from omework 2 (HW average: 85.9%, exam average 83.7%). Since te car and motorcycle are moving at te same velocity until te moment te motorcycle accelerates, tey will maintain a separation of x o =92 m until te moment te motorcycle accelerates. In tis case, te time of 2.50 s is not relevant for solving te problem - it is only telling you wen someone started measuring te motorcycle s and car s positions, and as suc represents an arbitrary coice of origin. It is not a ead start, like in previous problems similar to tis one, since bot veicles are in motion at te start of te problem. Let te motorcycle s position at te moment it accelerates be x = 0, and let time t = 0 be te moment wen te motorcycle begins accelerating. We will call te initial velocity of bot veicles v o, and te motorcycle s acceleration a m. Te motorcycle s position as a function of time is ten x m (t) = v o t at2 (1) wic correctly captures te initial condition x m (0)=0. Te car starts out a distance x o aead of te motorcycle, and continues at constant speed v o, so its position is x c (t) = x o + v o t (2) wic correctly captures te starting point x c (0)=x o. Te motorcycle catces te car wen teir positions are equal: x c = x m x o + v o t = v o t at2 x o = 1 2 at2 (3) (4) (5) 2xo t = 6.78 s (6) a

2 Note tat te initial velocity v o cancels out since bot objects start out wit te same initial velocity - only te relative velocity matters, wic is entirely determined by te motorcycle s acceleration. An equivalent question would be ow long does it take for te motorcycle to accelerate across te gap of distance x o? 2. A pilot flies orizontally at 1300 km/, at eigt =35 m above initially level ground. However, at time t=0, te pilot begins to fly over ground sloping upward at angle θ=4.3. If te pilot does not cange te airplane s eading, at wat time t does te plane strike te ground? Solution: (exam average 83.3%) Tis is a problem from a different textbook. Given: Te initial velocity and eigt of a plane flying toward an upward slope of angle θ. Find: How long before te plane its te slope? At time t=0, te plane is at te beginning of te slope, a eigt above level ground. Assuming te plane continues at te same orizontal speed, we wis to find te time at wic te plane its te slope. Given te plane s velocity and eigt and te slope s angle, we can relate te orizontal distance to intercept te ramp to te plane s eigt. Sketc: Assume a sperical plane (it doesn t matter). If te plane is at altitude, it will it te ramp after covering a orizontal distance d, were tan θ=/d. v θ d Relevant equations: We can relate te orizontal distance to intersect te ramp to te plane s altitude using te known slope of ground: tan θ = d (7) We can determine ow long te orizontal distance d will be covered given te plane s constant orizontal speed v: d = vt (8) Symbolic solution: Combining our equations above, te time t it takes for te plane to it te

3 slope is t = d v = v tan θ (9) Numeric solution: Using te numbers given, and converting units, t = v tan θ = 35 m 1300 km/ (1000 m/km) (1 /3600 s) (tan s (10) ) 3. A block of mass m = 5.00 kg is pulled along a orizontal frictionless floor by a cord tat exerts a force of magnitude F = 12.0 N at an angle of 65 wit respect to orizontal. (a) Wat is te magnitude of te block s acceleration? (b) Te force magnitude F is slowly increased. Wat is its value just before te block is lifted off te floor? Solution: (exam average 51.9%) Tis is a problem from a different textbook. Given: A block pulled along a frictionless floor by a force making an angle θ wit te orizontal. Find: Te block s acceleration, te maximum force before te block leaves te floor, and te block s acceleration at tat point. Sketc: Let te x and y axes be orizontal and vertical, respectively. We ave only te block s weigt, te normal force, and te applied force. n θ F y mg x Relevant equations: We need only Newton s second law and geometry. Symbolic solution: Along te vertical direction, a force balance must give zero for te block to remain on te floor. Tis immediately yields te normal force. Fy = F n mg + F sin θ = 0 = F n = mg F sin θ (11)

4 A orizontal force balance gives us te acceleration: X Fx = F cos θ = max = ax = F cos θ m (12) If te magnitude of te force is increased, te block will leave te floor as te normal force becomes zero: Fn = mg F sin θ = 0 = F= mg sin θ (13) At tat point, its acceleration will be ax = F g cos θ = m tan θ (14) Numeric solution: Using te numbers given, te initial acceleration is ax = F 12.0 N cos θ = cos m/s2 m 5.00 kg (15) At te point te block is about to leave te floor, te required force is (5.00 kg) 9.81 m/s2 mg 54 N F= = sin θ sin 65 (16) 4. Consider te figure below. Let = 1 m, θ = 15, and m = 0.5 kg. Tere is a coefficient of kinetic friction µk = 0.2 between te mass and te inclined plane, and te mass m starts out at te very top of te incline wit a velocity of vi = 0.1 m/s. Wat is te speed of te mass at te bottom of te ramp? Solution: (exam average 84.7%) Tis is a problem from te PH105 textbook. First focus on te block and incline, and draw te free body diagram for tose two. Let te x axis point down te incline, and te y axis up, normal (perpendicular) to te incline. In te y direction,

5 we ave te normal force, and part of te weigt of te block: ΣF y = F n mg cos θ = ma y = 0 F n = mg cos θ (17) In te x direction, we ave te oter component of te weigt, and opposing tat we ave friction: ΣF x = mg sin θ f k = mg sin θ µ k F n = mg sin θ µ k mg cos θ = ma x (18) a x = g (sin θ µ k cos θ) (19) Given te acceleration a x, te initial velocity v i = 0.1 m/s, and te lengt of te ramp being / sin θ, we can readily calculate te speed at te bottom of te ramp, wic we ll call v f : ( v 2 f = v2 i + 2a x x = v 2 i + 2g (sin θ µ k cos θ) v f = ) ( = v 2i + 2g 1 µ ) k tan θ (20) sin θ ( v 2i + 2g 1 µ ) k 2.23 m/s (21) tan θ 5. A baseball leaves te bat at 30.0 above te orizontal and is caugt by an outfielder 375 ft from ome plate at te same eigt from wic it left. Wat is te initial speed of te ball? Solution: (exam average 89.2%) We know te launc angle and range of te projectile over level ground. For tis situation, we ve already derived te range equation, so we may as well use it: R = v2 o sin 2θ g (22) were θ = 30.0 is te launc angle, v o te launc velocity, and R = 375 ft 114 m is te range. Solving for v o, v o = gr 36 m/s 118 ft/s 81 mp (23) sin 2θ 6. An advertisement claims tat a particular automobile can stop on a dime. Wat net force would actually be necessary to stop a 850 kg automobile traveling initially at 45.0 km/ in a distance equal to te diameter of a dime, wic is 1.8 cm. Hint: watc te units! Solution: (exam average 76.4%) We know te initial velocity v o = 45 km/ = 12.5 m/s, te final velocity v f = 0, and te distance over wic te car accelerated, x = m. Tis is enoug to

6 get us te net acceleration, wic is ten enoug to get us te net force using Newton s second law. First te acceleration: v 2 f = v2 o + 2a x a = v2 o 2 x (24) (25) Te minus sign reminds us tat te acceleration is in te direction opposite x. If tis is te net acceleration, te net force causing it must follow F=ma, so te net force is in magnitude (i.e., we don t care about te sign) F net = ma = mv2 o 2 x N 415 tons (26) A scary, unsurvivable amount of force, equivalent to pulling about 440 g s (were 50 g s generally means serious injury or deat). One could also solve tis wit te work-energy teorem: te force F stopping te veicle troug a displacement x does work W = F x, and tis must be te same as te car s cange in kinetic energy, wic is 1 2 mv2 o. Equating and solving for F, we arrive at te same result.

Example force problems

Example force problems PH 105 / LeClair Fall 2015 Example force problems 1. An advertisement claims that a particular automobile can stop on a dime. What net force would actually be necessary to stop a 850 kg automobile traveling

More information

Derivative as Instantaneous Rate of Change

Derivative as Instantaneous Rate of Change 43 Derivative as Instantaneous Rate of Cange Consider a function tat describes te position of a racecar moving in a straigt line away from some starting point Let y s t suc tat t represents te time in

More information

INTRODUCTION AND MATHEMATICAL CONCEPTS

INTRODUCTION AND MATHEMATICAL CONCEPTS Capter 1 INTRODUCTION ND MTHEMTICL CONCEPTS PREVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips

More information

Section 15.6 Directional Derivatives and the Gradient Vector

Section 15.6 Directional Derivatives and the Gradient Vector Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,

More information

INTRODUCTION AND MATHEMATICAL CONCEPTS

INTRODUCTION AND MATHEMATICAL CONCEPTS INTODUCTION ND MTHEMTICL CONCEPTS PEVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips of sine,

More information

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).

1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t). . Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use

More information

Exam 1 Review Solutions

Exam 1 Review Solutions Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),

More information

Some Review Problems for First Midterm Mathematics 1300, Calculus 1

Some Review Problems for First Midterm Mathematics 1300, Calculus 1 Some Review Problems for First Midterm Matematics 00, Calculus. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd,

More information

Phy 231 Sp 02 Homework #6 Page 1 of 4

Phy 231 Sp 02 Homework #6 Page 1 of 4 Py 231 Sp 02 Homework #6 Page 1 of 4 6-1A. Te force sown in te force-time diagram at te rigt versus time acts on a 2 kg mass. Wat is te impulse of te force on te mass from 0 to 5 sec? (a) 9 N-s (b) 6 N-s

More information

E p = mgh (if h i=0) E k = ½ mv 2 Ek is measured in Joules (J); m is measured in kg; v is measured in m/s. Energy Continued (E)

E p = mgh (if h i=0) E k = ½ mv 2 Ek is measured in Joules (J); m is measured in kg; v is measured in m/s. Energy Continued (E) nergy Continued () Gravitational Potential nergy: - e energy stored in an object due to its distance above te surface of te art. - e energy stored depends on te mass of te object, te eigt above te surface,

More information

106 PHYS - CH6 - Part2

106 PHYS - CH6 - Part2 106 PHYS - CH6 - Part Conservative Forces (a) A force is conservative if work one by tat force acting on a particle moving between points is inepenent of te pat te particle takes between te two points

More information

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim

More information

Problem Set 7: Potential Energy and Conservation of Energy AP Physics C Supplementary Problems

Problem Set 7: Potential Energy and Conservation of Energy AP Physics C Supplementary Problems Proble Set 7: Potential Energy and Conservation of Energy AP Pysics C Suppleentary Probles 1. Approxiately 5.5 x 10 6 kg of water drops 50 over Niagara Falls every second. (a) Calculate te aount of potential

More information

= h. Geometrically this quantity represents the slope of the secant line connecting the points

= h. Geometrically this quantity represents the slope of the secant line connecting the points Section 3.7: Rates of Cange in te Natural and Social Sciences Recall: Average rate of cange: y y y ) ) ), ere Geometrically tis quantity represents te slope of te secant line connecting te points, f (

More information

Math 34A Practice Final Solutions Fall 2007

Math 34A Practice Final Solutions Fall 2007 Mat 34A Practice Final Solutions Fall 007 Problem Find te derivatives of te following functions:. f(x) = 3x + e 3x. f(x) = x + x 3. f(x) = (x + a) 4. Is te function 3t 4t t 3 increasing or decreasing wen

More information

2.8 The Derivative as a Function

2.8 The Derivative as a Function .8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open

More information

Excerpt from "Calculus" 2013 AoPS Inc.

Excerpt from Calculus 2013 AoPS Inc. Excerpt from "Calculus" 03 AoPS Inc. Te term related rates refers to two quantities tat are dependent on eac oter and tat are canging over time. We can use te dependent relationsip between te quantities

More information

1 Limits and Continuity

1 Limits and Continuity 1 Limits and Continuity 1.0 Tangent Lines, Velocities, Growt In tion 0.2, we estimated te slope of a line tangent to te grap of a function at a point. At te end of tion 0.3, we constructed a new function

More information

Mathematics 105 Calculus I. Exam 1. February 13, Solution Guide

Mathematics 105 Calculus I. Exam 1. February 13, Solution Guide Matematics 05 Calculus I Exam February, 009 Your Name: Solution Guide Tere are 6 total problems in tis exam. On eac problem, you must sow all your work, or oterwise torougly explain your conclusions. Tere

More information

Key Concepts. Important Techniques. 1. Average rate of change slope of a secant line. You will need two points ( a, the formula: to find value

Key Concepts. Important Techniques. 1. Average rate of change slope of a secant line. You will need two points ( a, the formula: to find value AB Calculus Unit Review Key Concepts Average and Instantaneous Speed Definition of Limit Properties of Limits One-sided and Two-sided Limits Sandwic Teorem Limits as x ± End Beaviour Models Continuity

More information

Derivatives. By: OpenStaxCollege

Derivatives. By: OpenStaxCollege By: OpenStaxCollege Te average teen in te United States opens a refrigerator door an estimated 25 times per day. Supposedly, tis average is up from 10 years ago wen te average teenager opened a refrigerator

More information

Work and Energy. Introduction. Work. PHY energy - J. Hedberg

Work and Energy. Introduction. Work. PHY energy - J. Hedberg Work and Energy PHY 207 - energy - J. Hedberg - 2017 1. Introduction 2. Work 3. Kinetic Energy 4. Potential Energy 5. Conservation of Mecanical Energy 6. Ex: Te Loop te Loop 7. Conservative and Non-conservative

More information

Section 3: The Derivative Definition of the Derivative

Section 3: The Derivative Definition of the Derivative Capter 2 Te Derivative Business Calculus 85 Section 3: Te Derivative Definition of te Derivative Returning to te tangent slope problem from te first section, let's look at te problem of finding te slope

More information

Solutions Manual for Precalculus An Investigation of Functions

Solutions Manual for Precalculus An Investigation of Functions Solutions Manual for Precalculus An Investigation of Functions David Lippman, Melonie Rasmussen 1 st Edition Solutions created at Te Evergreen State College and Soreline Community College 1.1 Solutions

More information

Grade: 11 International Physics Olympiad Qualifier Set: 2

Grade: 11 International Physics Olympiad Qualifier Set: 2 Grade: 11 International Pysics Olympiad Qualifier Set: 2 --------------------------------------------------------------------------------------------------------------- Max Marks: 60 Test ID: 12111 Time

More information

2.1 THE DEFINITION OF DERIVATIVE

2.1 THE DEFINITION OF DERIVATIVE 2.1 Te Derivative Contemporary Calculus 2.1 THE DEFINITION OF DERIVATIVE 1 Te grapical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of te derivative

More information

Section 2: The Derivative Definition of the Derivative

Section 2: The Derivative Definition of the Derivative Capter 2 Te Derivative Applied Calculus 80 Section 2: Te Derivative Definition of te Derivative Suppose we drop a tomato from te top of a 00 foot building and time its fall. Time (sec) Heigt (ft) 0.0 00

More information

Physics 121, April 1, Equilibrium. Physics 121. April 1, Physics 121. April 1, Course Information. Discussion of Exam # 2

Physics 121, April 1, Equilibrium. Physics 121. April 1, Physics 121. April 1, Course Information. Discussion of Exam # 2 Pysics 121, April 1, 2008. Pysics 121. April 1, 2008. Course Information Discussion of Exam # 2 Topics to be discussed today: Requirements for Equilibrium Gravitational Equilibrium Sample problems Pysics

More information

REVIEW LAB ANSWER KEY

REVIEW LAB ANSWER KEY REVIEW LAB ANSWER KEY. Witout using SN, find te derivative of eac of te following (you do not need to simplify your answers): a. f x 3x 3 5x x 6 f x 3 3x 5 x 0 b. g x 4 x x x notice te trick ere! x x g

More information

MVT and Rolle s Theorem

MVT and Rolle s Theorem AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state

More information

Tangent Lines-1. Tangent Lines

Tangent Lines-1. Tangent Lines Tangent Lines- Tangent Lines In geometry, te tangent line to a circle wit centre O at a point A on te circle is defined to be te perpendicular line at A to te line OA. Te tangent lines ave te special property

More information

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit

More information

REVISING MECHANICS (LIVE) 30 JUNE 2015 Exam Questions

REVISING MECHANICS (LIVE) 30 JUNE 2015 Exam Questions REVISING MECHANICS (LIVE) 30 JUNE 2015 Exam Questions Question 1 (Adapted from DBE November 2014, Question 2) Two blocks of masses 20 kg and 5 kg respectively are connected by a light inextensible string,

More information

MATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart

MATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart MATH 121 - Fall 08 Review Problems for te Midterm Eamination Covers [1.1, 4.3] in Stewart 1. (a) Use te definition of te derivative to find f (3) wen f() = π 1 2. (b) Find an equation of te tangent line

More information

2.3. Applying Newton s Laws of Motion. Objects in Equilibrium

2.3. Applying Newton s Laws of Motion. Objects in Equilibrium Appling Newton s Laws of Motion As ou read in Section 2.2, Newton s laws of motion describe ow objects move as a result of different forces. In tis section, ou will appl Newton s laws to objects subjected

More information

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a? Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is

More information

1 Power is transferred through a machine as shown. power input P I machine. power output P O. power loss P L. What is the efficiency of the machine?

1 Power is transferred through a machine as shown. power input P I machine. power output P O. power loss P L. What is the efficiency of the machine? 1 1 Power is transferred troug a macine as sown. power input P I macine power output P O power loss P L Wat is te efficiency of te macine? P I P L P P P O + P L I O P L P O P I 2 ir in a bicycle pump is

More information

The Derivative The rate of change

The Derivative The rate of change Calculus Lia Vas Te Derivative Te rate of cange Knowing and understanding te concept of derivative will enable you to answer te following questions. Let us consider a quantity wose size is described by

More information

HOMEWORK HELP 2 FOR MATH 151

HOMEWORK HELP 2 FOR MATH 151 HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,

More information

Chapter 1 Functions and Graphs. Section 1.5 = = = 4. Check Point Exercises The slope of the line y = 3x+ 1 is 3.

Chapter 1 Functions and Graphs. Section 1.5 = = = 4. Check Point Exercises The slope of the line y = 3x+ 1 is 3. Capter Functions and Graps Section. Ceck Point Exercises. Te slope of te line y x+ is. y y m( x x y ( x ( y ( x+ point-slope y x+ 6 y x+ slope-intercept. a. Write te equation in slope-intercept form: x+

More information

Continuity and Differentiability Worksheet

Continuity and Differentiability Worksheet Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;

More information

1 Solutions to the in class part

1 Solutions to the in class part NAME: Solutions to te in class part. Te grap of a function f is given. Calculus wit Analytic Geometry I Exam, Friday, August 30, 0 SOLUTIONS (a) State te value of f(). (b) Estimate te value of f( ). (c)

More information

Lab 6 Derivatives and Mutant Bacteria

Lab 6 Derivatives and Mutant Bacteria Lab 6 Derivatives and Mutant Bacteria Date: September 27, 20 Assignment Due Date: October 4, 20 Goal: In tis lab you will furter explore te concept of a derivative using R. You will use your knowledge

More information

On my honor, I have neither given nor received unauthorized aid on this examination.

On my honor, I have neither given nor received unauthorized aid on this examination. Instructor(s): iel/uric PHYSICS DEPARTENT PHY 2053 Exam 1 October 3, 2012 Name (print, last first): Signature: On my onor, I ave neiter given nor receive unautorize ai on tis examination. YOUR TEST NUBER

More information

Average Rate of Change

Average Rate of Change Te Derivative Tis can be tougt of as an attempt to draw a parallel (pysically and metaporically) between a line and a curve, applying te concept of slope to someting tat isn't actually straigt. Te slope

More information

Derivatives of Exponentials

Derivatives of Exponentials mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function

More information

Random sample problems

Random sample problems UNIVERSITY OF ALABAMA Department of Physics and Astronomy PH 125 / LeClair Spring 2009 Random sample problems 1. The position of a particle in meters can be described by x = 10t 2.5t 2, where t is in seconds.

More information

1. (a) 3. (a) 4 3 (b) (a) t = 5: 9. (a) = 11. (a) The equation of the line through P = (2, 3) and Q = (8, 11) is y 3 = 8 6

1. (a) 3. (a) 4 3 (b) (a) t = 5: 9. (a) = 11. (a) The equation of the line through P = (2, 3) and Q = (8, 11) is y 3 = 8 6 A Answers Important Note about Precision of Answers: In many of te problems in tis book you are required to read information from a grap and to calculate wit tat information. You sould take reasonable

More information

Math 1210 Midterm 1 January 31st, 2014

Math 1210 Midterm 1 January 31st, 2014 Mat 110 Midterm 1 January 1st, 01 Tis exam consists of sections, A and B. Section A is conceptual, wereas section B is more computational. Te value of every question is indicated at te beginning of it.

More information

1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x.

1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x. Problem. Let f x x. Using te definition of te derivative prove tat f x x Solution. Te function f x is only defined wen x 0, so we will assume tat x 0 for te remainder of te solution. By te definition of

More information

1. Questions (a) through (e) refer to the graph of the function f given below. (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist

1. Questions (a) through (e) refer to the graph of the function f given below. (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist Mat 1120 Calculus Test 2. October 18, 2001 Your name Te multiple coice problems count 4 points eac. In te multiple coice section, circle te correct coice (or coices). You must sow your work on te oter

More information

Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0

Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0 3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,

More information

Preface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Preface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser

More information

1watt=1W=1kg m 2 /s 3

1watt=1W=1kg m 2 /s 3 Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory

More information

Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006

Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2

More information

Week 4 Homework/Recitation: 9/21/2017 Chapter4: Problems 3, 5, 11, 16, 24, 38, 52, 77, 78, 98. is shown in the drawing. F 2

Week 4 Homework/Recitation: 9/21/2017 Chapter4: Problems 3, 5, 11, 16, 24, 38, 52, 77, 78, 98. is shown in the drawing. F 2 Week 4 Homework/Recitation: 9/1/017 Chapter4: Problems 3, 5, 11, 16, 4, 38, 5, 77, 78, 98. 3. Two horizontal forces, F 1 and F, are acting on a box, but only F 1 is shown in the drawing. F can point either

More information

Two-Dimensional Motion and Vectors

Two-Dimensional Motion and Vectors CHAPTER 3 VECTOR quantities: Two-imensional Motion and Vectors Vectors ave magnitude and direction. (x, y) Representations: y (x, y) (r, ) x Oter vectors: velocity, acceleration, momentum, force Vector

More information

MA119-A Applied Calculus for Business Fall Homework 4 Solutions Due 9/29/ :30AM

MA119-A Applied Calculus for Business Fall Homework 4 Solutions Due 9/29/ :30AM MA9-A Applied Calculus for Business 006 Fall Homework Solutions Due 9/9/006 0:0AM. #0 Find te it 5 0 + +.. #8 Find te it. #6 Find te it 5 0 + + = (0) 5 0 (0) + (0) + =.!! r + +. r s r + + = () + 0 () +

More information

Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk

Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 1 Te Tangent Line Problem Te idea of a tangent line first arises in geometry in te context of a circle. But before we jump into a discussion of

More information

Polynomial Functions. Linear Functions. Precalculus: Linear and Quadratic Functions

Polynomial Functions. Linear Functions. Precalculus: Linear and Quadratic Functions Concepts: definition of polynomial functions, linear functions tree representations), transformation of y = x to get y = mx + b, quadratic functions axis of symmetry, vertex, x-intercepts), transformations

More information

Pre-Calculus Review Preemptive Strike

Pre-Calculus Review Preemptive Strike Pre-Calculus Review Preemptive Strike Attaced are some notes and one assignment wit tree parts. Tese are due on te day tat we start te pre-calculus review. I strongly suggest reading troug te notes torougly

More information

Figure 5.1a, b IDENTIFY: Apply to the car. EXECUTE: gives.. EVALUATE: The force required is less than the weight of the car by the factor.

Figure 5.1a, b IDENTIFY: Apply to the car. EXECUTE: gives.. EVALUATE: The force required is less than the weight of the car by the factor. 51 IDENTIFY: for each object Apply to each weight and to the pulley SET UP: Take upward The pulley has negligible mass Let be the tension in the rope and let be the tension in the chain EXECUTE: (a) The

More information

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225 THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:

More information

1. A sphere with a radius of 1.7 cm has a volume of: A) m 3 B) m 3 C) m 3 D) 0.11 m 3 E) 21 m 3

1. A sphere with a radius of 1.7 cm has a volume of: A) m 3 B) m 3 C) m 3 D) 0.11 m 3 E) 21 m 3 1. A sphere with a radius of 1.7 cm has a volume of: A) 2.1 10 5 m 3 B) 9.1 10 4 m 3 C) 3.6 10 3 m 3 D) 0.11 m 3 E) 21 m 3 2. A 25-N crate slides down a frictionless incline that is 25 above the horizontal.

More information

Section 2.4: Definition of Function

Section 2.4: Definition of Function Section.4: Definition of Function Objectives Upon completion of tis lesson, you will be able to: Given a function, find and simplify a difference quotient: f ( + ) f ( ), 0 for: o Polynomial functions

More information

Lesson 6: The Derivative

Lesson 6: The Derivative Lesson 6: Te Derivative Def. A difference quotient for a function as te form f(x + ) f(x) (x + ) x f(x + x) f(x) (x + x) x f(a + ) f(a) (a + ) a Notice tat a difference quotient always as te form of cange

More information

. Compute the following limits.

. Compute the following limits. Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim

More information

f a h f a h h lim lim

f a h f a h h lim lim Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point

More information

Mathematics 123.3: Solutions to Lab Assignment #5

Mathematics 123.3: Solutions to Lab Assignment #5 Matematics 3.3: Solutions to Lab Assignment #5 Find te derivative of te given function using te definition of derivative. State te domain of te function and te domain of its derivative..: f(x) 6 x Solution:

More information

UNIVERSITY OF MANITOBA DEPARTMENT OF MATHEMATICS MATH 1510 Applied Calculus I FIRST TERM EXAMINATION - Version A October 12, :30 am

UNIVERSITY OF MANITOBA DEPARTMENT OF MATHEMATICS MATH 1510 Applied Calculus I FIRST TERM EXAMINATION - Version A October 12, :30 am DEPARTMENT OF MATHEMATICS MATH 1510 Applied Calculus I October 12, 2016 8:30 am LAST NAME: FIRST NAME: STUDENT NUMBER: SIGNATURE: (I understand tat ceating is a serious offense DO NOT WRITE IN THIS TABLE

More information

A = h w (1) Error Analysis Physics 141

A = h w (1) Error Analysis Physics 141 Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.

More information

Outline. MS121: IT Mathematics. Limits & Continuity Rates of Change & Tangents. Is there a limit to how fast a man can run?

Outline. MS121: IT Mathematics. Limits & Continuity Rates of Change & Tangents. Is there a limit to how fast a man can run? Outline MS11: IT Matematics Limits & Continuity & 1 Limits: Atletics Perspective Jon Carroll Scool of Matematical Sciences Dublin City University 3 Atletics Atletics Outline Is tere a limit to ow fast

More information

PHYSICS 1050 Mid-term Test 1 University of Wyoming 15 February 2005

PHYSICS 1050 Mid-term Test 1 University of Wyoming 15 February 2005 Name: (4 points) PHYSICS 1050 Mid-term Test 1 University of Wyoming 15 February 2005 Tis test is open-note and open-book. Tis means tat any reference material is permitted during te test. Calculators also

More information

Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.

Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4. December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need

More information

Combining functions: algebraic methods

Combining functions: algebraic methods Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)

More information

Honors Calculus Midterm Review Packet

Honors Calculus Midterm Review Packet Name Date Period Honors Calculus Midterm Review Packet TOPICS THAT WILL APPEAR ON THE EXAM Capter Capter Capter (Sections. to.6) STRUCTURE OF THE EXAM Part No Calculators Miture o multiple-coice, matcing,

More information

Numerical Differentiation

Numerical Differentiation Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function

More information

Part 2: Introduction to Open-Channel Flow SPRING 2005

Part 2: Introduction to Open-Channel Flow SPRING 2005 Part : Introduction to Open-Cannel Flow SPRING 005. Te Froude number. Total ead and specific energy 3. Hydraulic jump. Te Froude Number Te main caracteristics of flows in open cannels are tat: tere is

More information

Physics 8 Wednesday, October 19, Troublesome questions for HW4 (5 or more people got 0 or 1 points on them): 1, 14, 15, 16, 17, 18, 19. Yikes!

Physics 8 Wednesday, October 19, Troublesome questions for HW4 (5 or more people got 0 or 1 points on them): 1, 14, 15, 16, 17, 18, 19. Yikes! Physics 8 Wednesday, October 19, 2011 Troublesome questions for HW4 (5 or more people got 0 or 1 points on them): 1, 14, 15, 16, 17, 18, 19. Yikes! Troublesome HW4 questions 1. Two objects of inertias

More information

Excursions in Computing Science: Week v Milli-micro-nano-..math Part II

Excursions in Computing Science: Week v Milli-micro-nano-..math Part II Excursions in Computing Science: Week v Milli-micro-nano-..mat Part II T. H. Merrett McGill University, Montreal, Canada June, 5 I. Prefatory Notes. Cube root of 8. Almost every calculator as a square-root

More information

JANE PROFESSOR WW Prob Lib1 Summer 2000

JANE PROFESSOR WW Prob Lib1 Summer 2000 JANE PROFESSOR WW Prob Lib Summer 2000 Sample WeBWorK problems. WeBWorK assignment Derivatives due 2//06 at 2:00 AM..( pt) If f x 9, find f 0. 2.( pt) If f x 7x 27, find f 5. 3.( pt) If f x 7 4x 5x 2,

More information

11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR

11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR SECTION 11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 633 wit speed v o along te same line from te opposite direction toward te source, ten te frequenc of te sound eard b te observer is were c is

More information

The Derivative as a Function

The Derivative as a Function Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a + )

More information

June : 2016 (CBCS) Body. Load

June : 2016 (CBCS) Body. Load Engineering Mecanics st Semester : Common to all rances Note : Max. marks : 6 (i) ttempt an five questions (ii) ll questions carr equal marks. (iii) nswer sould be precise and to te point onl (iv) ssume

More information

Click here to see an animation of the derivative

Click here to see an animation of the derivative Differentiation Massoud Malek Derivative Te concept of derivative is at te core of Calculus; It is a very powerful tool for understanding te beavior of matematical functions. It allows us to optimize functions,

More information

A. B. C. D. E. v x. ΣF x

A. B. C. D. E. v x. ΣF x Q4.3 The graph to the right shows the velocity of an object as a function of time. Which of the graphs below best shows the net force versus time for this object? 0 v x t ΣF x ΣF x ΣF x ΣF x ΣF x 0 t 0

More information

This chapter covers all kinds of problems having to do with work in physics terms. Work

This chapter covers all kinds of problems having to do with work in physics terms. Work Chapter 7 Working the Physics Way In This Chapter Understanding work Working with net force Calculating kinetic energy Handling potential energy Relating kinetic energy to work This chapter covers all

More information

0.1 Differentiation Rules

0.1 Differentiation Rules 0.1 Differentiation Rules From our previous work we ve seen tat it can be quite a task to calculate te erivative of an arbitrary function. Just working wit a secon-orer polynomial tings get pretty complicate

More information

Work and Energy. Chapter 7

Work and Energy. Chapter 7 Work and Energy Chapter 7 Scalar Product of Two Vectors Definition of the scalar, or dot, product: A B A Alternatively, we can write: x B x A y B y A z B z Work Work Done by a Constant Force The work done

More information

1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2

1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2 MTH - Spring 04 Exam Review (Solutions) Exam : February 5t 6:00-7:0 Tis exam review contains questions similar to tose you sould expect to see on Exam. Te questions included in tis review, owever, are

More information

Pre-lab Quiz/PHYS 224 Earth s Magnetic Field. Your name Lab section

Pre-lab Quiz/PHYS 224 Earth s Magnetic Field. Your name Lab section Pre-lab Quiz/PHYS 4 Eart s Magnetic Field Your name Lab section 1. Wat do you investigate in tis lab?. For a pair of Helmoltz coils described in tis manual and sown in Figure, r=.15 m, N=13, I =.4 A, wat

More information

Lines, Conics, Tangents, Limits and the Derivative

Lines, Conics, Tangents, Limits and the Derivative Lines, Conics, Tangents, Limits and te Derivative Te Straigt Line An two points on te (,) plane wen joined form a line segment. If te line segment is etended beond te two points ten it is called a straigt

More information

The distance between City C and City A is just the magnitude of the vector, namely,

The distance between City C and City A is just the magnitude of the vector, namely, Pysics 11 Homework III Solutions C. 3 - Problems 2, 15, 18, 23, 24, 30, 39, 58. Problem 2 So, we fly 200km due west from City A to City B, ten 300km 30 nort of west from City B to City C. (a) We want te

More information

Version PREVIEW Semester 1 Review Slade (22222) 1

Version PREVIEW Semester 1 Review Slade (22222) 1 Version PREVIEW Semester 1 Review Slade () 1 This print-out should have 48 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. Holt SF 0Rev 10A

More information

232 Calculus and Structures

232 Calculus and Structures 3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE

More information

Continuity and Differentiability of the Trigonometric Functions

Continuity and Differentiability of the Trigonometric Functions [Te basis for te following work will be te definition of te trigonometric functions as ratios of te sides of a triangle inscribed in a circle; in particular, te sine of an angle will be defined to be te

More information

NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,

NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example, NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing

More information

Force 10/01/2010. (Weight) MIDTERM on 10/06/10 7:15 to 9:15 pm Bentley 236. (Tension)

Force 10/01/2010. (Weight) MIDTERM on 10/06/10 7:15 to 9:15 pm Bentley 236. (Tension) Force 10/01/2010 = = Friction Force (Weight) (Tension), coefficient of static and kinetic friction MIDTERM on 10/06/10 7:15 to 9:15 pm Bentley 236 2008 midterm posted for practice. Help sessions Mo, Tu

More information

Final exam: Tuesday, May 11, 7:30-9:30am, Coates 143

Final exam: Tuesday, May 11, 7:30-9:30am, Coates 143 Final exam: Tuesday, May 11, 7:30-9:30am, Coates 143 Approximately 7 questions/6 problems Approximately 50% material since last test, 50% everyting covered on Exams I-III About 50% of everyting closely

More information

Physics 111: Mechanics Lecture 5

Physics 111: Mechanics Lecture 5 Physics 111: Mechanics Lecture 5 Bin Chen NJIT Physics Department Forces of Friction: f q When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion.

More information