MATH CALCULUS I 4.1: Area and Distance
|
|
- Valerie Douglas
- 5 years ago
- Views:
Transcription
1 MATH CALCULUS I 4.1: Area and Distance Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 8
2 The Area and Distance Problems Integral calculus is motivated in part by two problems: The Area Problem: How do we compute the area of an arbitrary region in the plane, or more specifically, how do we compute the area of a region bounded by the graphs of functions? The Distance Problem: Given a velocity function v(t), how do we determine the displacement or distance travelled on a time interval? These problems may not seem related, but they are nearly identical. D.L. White (Kent State University) 2 / 8
3 Constant Velocity Recall that if velocity is constant, then distance is rate times time. Thus if the velocity function is v(t) = r, where r a constant, then on the time interval from t = a to t = b, distance is r(b a). Graphing velocity v as a function of time t, we have: v r b a r a b t The height of the rectangle is r and the width of the rectangle is b a. Hence the area of the rectangle is r(b a). Therefore, if velocity is constant on the interval from t = a to t = b, then distance travelled is equal to the area of the region bounded by the graph of v(t), the t-axis, and the lines t = a and t = b; or, distance travelled equals the area under the graph of v(t). D.L. White (Kent State University) 3 / 8
4 Computing areas of rectangles and computing distance when velocity is constant are really the same computation. What about areas of regions whose boundaries are not straight lines, or distance when the velocity is not constant? Recall the Velocity Problem and Tangent Problem we started with: Given a position function s(t), we could compute average velocity over a time interval or the slope of a secant line to the graph of s(t). To compute instantaneous velocity or the slope of a tangent line, we had to take a limit as the length of the time interval approached zero. D.L. White (Kent State University) 4 / 8
5 In order to compute areas of more general regions or displacements when the velocity is not constant, we need a similar procedure. The general idea is as follows: AREA: Approximate the region under the graph of a function by rectangles, then compute the limit of the sum of the areas of the rectangles as the width of the rectangles approaches 0. DISTANCE: Approximate the velocity function by constant velocity on subintervals of time, then compute the limit of the sum of the displacements on the subintervals as the length of the time intervals approaches 0. D.L. White (Kent State University) 5 / 8
6 More precisely, suppose v(t) has the graph below. Break the interval [a, b] into subintervals. Approximate v(t) by the velocity at the start of each subinterval. Draw a rectangle on each subinterval with height the approximate velocity. v v(t) t a = t 0 t 1 t 2 t 3 t 4 t 5 = b D.L. White (Kent State University) 6 / 8
7 v v(t) t On a given subinterval, we have the following: Exact distance distance at the constant velocity. Area under the graph area of rectangle. Distance at the constant velocity = area of rectangle. The approximations become more and more accurate as the time interval becomes shorter and shorter or as the rectangle becomes more and more narrow. D.L. White (Kent State University) 7 / 8
8 v v(t) t On the whole interval, we have: Exact distance sum of distances at constant velocities. Area under the graph sum of areas of rectangles. Sum of distances at constant velocities = sum of areas of rectangles. The limit of the sum of the distances at the constant velocities as the time intervals become shorter and shorter = exact distance. The limit of the sum of the areas of the rectangles as the rectangles become more and more narrow = exact area. Finally, since the sum of distances at constant velocities equals the sum of areas of rectangles, we have that DISTANCE TRAVELLED = AREA UNDER THE GRAPH. D.L. White (Kent State University) 8 / 8
MATH CALCULUS I 2.2: Differentiability, Graphs, and Higher Derivatives
MATH 12002 - CALCULUS I 2.2: Differentiability, Graphs, and Higher Derivatives Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 /
More informationSlopes and Rates of Change
Slopes and Rates of Change If a particle is moving in a straight line at a constant velocity, then the graph of the function of distance versus time is as follows s s = f(t) t s s t t = average velocity
More informationCalculus I Homework: The Tangent and Velocity Problems Page 1
Calculus I Homework: The Tangent and Velocity Problems Page 1 Questions Example The point P (1, 1/2) lies on the curve y = x/(1 + x). a) If Q is the point (x, x/(1 + x)), use Mathematica to find the slope
More informationLimits and the derivative function. Limits and the derivative function
The Velocity Problem A particle is moving in a straight line. t is the time that has passed from the start of motion (which corresponds to t = 0) s(t) is the distance from the particle to the initial position
More informationMATH CALCULUS I 2.1: Derivatives and Rates of Change
MATH 12002 - CALCULUS I 2.1: Derivatives and Rates of Cange Professor Donald L. Wite Department of Matematical Sciences Kent State University D.L. Wite (Kent State University) 1 / 1 Introduction Our main
More informationSpeed how fast an object is moving (also, the magnitude of the velocity) scalar
Mechanics Recall Mechanics Kinematics Dynamics Kinematics The description of motion without reference to forces. Terminology Distance total length of a journey scalar Time instant when an event occurs
More informationMATH CALCULUS I 1.5: Continuity
MATH 12002 - CALCULUS I 1.5: Continuity Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 12 Definition of Continuity Intuitively,
More information4.1 Areas and Distances. The Area Problem: GOAL: Find the area of the region S that lies under the curve y = f(x) from a to b.
4.1 Areas and Distances The Area Problem: GOAL: Find the area of the region S that lies under the curve y = f(x) from a to b. Easier Problems: Find the area of a rectangle with length l and width w. Find
More informationGraphical Analysis Part III. Motion Graphs. Basic Equations. Velocity is Constant. acceleration is zero. and. becomes
Graphical Analysis Part III Motion Graphs Basic Equations d = vt+ 0 1 at v = v 0 + at Velocity is Constant acceleration is zero and becomes 1 d = v 0 t+ at d = vt 1 Velocity is Constant the slope of d
More informationRemember... Average rate of change slope of a secant (between two points)
3.7 Rates of Change in the Natural and Social Sciences Remember... Average rate of change slope of a secant (between two points) Instantaneous rate of change slope of a tangent derivative We will assume
More informationChapter 3: Introduction to Kinematics
Chapter 3: Introduction to Kinematics Kari Eloranta 2018 Jyväskylän Lyseon lukio Pre Diploma Program Year October 11, 2017 1 / 17 3.1 Displacement Definition of Displacement Displacement is the change
More informationIB Math SL Year 2 Name Date Lesson 10-4: Displacement, Velocity, Acceleration Revisited
Name Date Lesson 10-4: Displacement, Velocity, Acceleration Revisited Learning Goals: How do you apply integrals to real-world scenarios? Recall: Linear Motion When an object is moving, a ball in the air
More informationINTEGRALS. In Chapter 2, we used the tangent and velocity problems to introduce the derivative the central idea in differential calculus.
INTEGRALS 5 INTEGRALS In Chapter 2, we used the tangent and velocity problems to introduce the derivative the central idea in differential calculus. INTEGRALS In much the same way, this chapter starts
More informationTopic 2.1 Motion. Topic 2.1 Motion. Kari Eloranta Jyväskylän Lyseon lukio. August 18, Kari Eloranta 2017 Topic 2.
Topic 2.1 Motion Kari Eloranta 2017 Jyväskylän Lyseon lukio August 18, 2017 Velocity and Speed 2.1: Kinematic Quanties: Displacement Definition of Displacement Displacement is the change in position. The
More informationIntroduction. Math Calculus 1 section 2.1 and 2.2. Julian Chan. Department of Mathematics Weber State University
Math 1210 Calculus 1 section 2.1 and 2.2 Julian Chan Department of Mathematics Weber State University 2013 Objectives Objectives: to tangent lines to limits What is velocity and how to obtain it from the
More informationRemember... Average rate of change slope of a secant (between two points)
3.7 Rates of Change in the Natural and Social Sciences Remember... Average rate of change slope of a secant (between two points) Instantaneous rate of change slope of a tangent derivative We will assume
More informationWhen does the function assume this value?
Calculus Write your questions and thoughts here! 9.3 Average Value Name: Recall: Average Rate of Change: Mean Value Theorem (MVT) for Derivatives: Notes Average Value of a Function: 1 1. Find the average
More informationScience One Integral Calculus
Science One Integral Calculus January 018 Happy New Year! Differential Calculus central idea: The Derivative What is the derivative f (x) of a function f(x)? Differential Calculus central idea: The Derivative
More informationParametric Functions and Vector Functions (BC Only)
Parametric Functions and Vector Functions (BC Only) Parametric Functions Parametric functions are another way of viewing functions. This time, the values of x and y are both dependent on another independent
More information( ) 4 and 20, find the value. v c is equal to this average CALCULUS WORKSHEET 1 ON PARTICLE MOTION
CALCULUS WORKSHEET 1 ON PARTICLE MOTION Work these on notebook paper. Use your calculator only on part (f) of problems 1. Do not use your calculator on the other problems. Write your justifications in
More informationThe Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus Objectives Evaluate a definite integral using the Fundamental Theorem of Calculus. Understand and use the Mean Value Theorem for Integrals. Find the average value of
More informationWorksheet 1. What You Need to Know About Motion Along the x-axis (Part 1)
Curriculum Module: Calculus: Motion Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1) In discussing motion, there are three closely related concepts that you need to keep straight.
More informationLimits, Rates of Change, and Tangent Lines
Limits, Rates of Change, and Tangent Lines jensenrj July 2, 2018 Contents 1 What is Calculus? 1 2 Velocity 2 2.1 Average Velocity......................... 3 2.2 Instantaneous Velocity......................
More informationLesson 3 Velocity Graphical Analysis
Physics 2 Lesson 3 Velocity Graphical Analysis I. Pearson Textbook Reference Refer to pages 11 to 2. II. Position-time Graphs Position-time graphs indicate the position of an object relative to a reference
More information2.1 How Do We Measure Speed? Student Notes HH6ed
2.1 How Do We Measure Speed? Student Notes HH6ed Part I: Using a table of values for a position function The table below represents the position of an object as a function of time. Use the table to answer
More informationF (x) is an antiderivative of f(x) if F (x) = f(x). Lets find an antiderivative of f(x) = x. We know that d. Any ideas?
Math 24 - Calculus for Management and Social Science Antiderivatives and the Indefinite Integral: Notes So far we have studied the slope of a curve at a point and its applications. This is one of the fundamental
More informationPosition-Time Graphs
Position-Time Graphs Suppose that a man is jogging at a constant velocity of 5.0 m / s. A data table representing the man s motion is shown below: If we plot this data on a graph, we get: 0 0 1.0 5.0 2.0
More informationAVERAGE VALUE AND MEAN VALUE THEOREM
AVERAGE VALUE AND MEAN VALUE THEOREM Section 4.4A Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 1 MATERIALS NEEDED A. Grid Paper B. Compass
More information2.1 Tangent Lines and Rates of Change
.1 Tangent Lines and Rates of Change Learning Objectives A student will be able to: Demonstrate an understanding of the slope of the tangent line to the graph. Demonstrate an understanding of the instantaneous
More informationThe area under a curve. Today we (begin to) ask questions of the type: How much area sits under the graph of f(x) = x 2 over the interval [ 1, 2]?
The area under a curve. Today we (begin to) ask questions of the type: How much area sits under the graph of f(x) = x 2 over the interval [ 1, 2]? Before we work on How we will figure out Why velocity
More informationCALCULUS 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
More informationAP Calculus Worksheet: Chapter 2 Review Part I
AP Calculus Worksheet: Chapter 2 Review Part I 1. Given y = f(x), what is the average rate of change of f on the interval [a, b]? What is the graphical interpretation of your answer? 2. The derivative
More informationThe Mean Value Theorem Rolle s Theorem
The Mean Value Theorem In this section, we will look at two more theorems that tell us about the way that derivatives affect the shapes of graphs: Rolle s Theorem and the Mean Value Theorem. Rolle s Theorem
More informationSections 2.1, 2.2 and 2.4: Limit of a function Motivation:
Sections 2.1, 2.2 and 2.4: Limit of a function Motivation: There are expressions which can be computed only using Algebra, meaning only using the operations +,, and. Examples which can be computed using
More informationKINEMATICS IN ONE DIMENSION p. 1
KINEMATICS IN ONE DIMENSION p. 1 Motion involves a change in position. Position can be indicated by an x-coordinate on a number line. ex/ A bumblebee flies along a number line... x = 2 when t = 1 sec 2
More informationMath 131. Rolle s and Mean Value Theorems Larson Section 3.2
Math 3. Rolle s and Mean Value Theorems Larson Section 3. Many mathematicians refer to the Mean Value theorem as one of the if not the most important theorems in mathematics. Rolle s Theorem. Suppose f
More informationAP 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
More informationToday s Agenda. Upcoming Homework Section 2.1: Derivatives and Rates of Change
Today s Agenda Upcoming Homework Section 2.1: Derivatives and Rates of Change Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Wed., 9 September 2015 1 / 9 Upcoming Homework Written HW B:
More informationWEEK 7 NOTES AND EXERCISES
WEEK 7 NOTES AND EXERCISES RATES OF CHANGE (STRAIGHT LINES) Rates of change are very important in mathematics. Take for example the speed of a car. It is a measure of how far the car travels over a certain
More informationCalculus Review. v = x t
Calculus Review Instructor : Kim 1. Average Rate of Change and Instantaneous Velocity To find the average velocity(v ) of a particle, we need to find the particle s displacement (=change in position) divided
More information1 y = Recitation Worksheet 1A. 1. Simplify the following: b. ( ) a. ( x ) Solve for y : 3. Plot these points in the xy plane:
Math 13 Recitation Worksheet 1A 1 Simplify the following: a ( ) 7 b ( ) 3 4 9 3 5 3 c 15 3 d 3 15 Solve for y : 8 y y 5= 6 3 3 Plot these points in the y plane: A ( 0,0 ) B ( 5,0 ) C ( 0, 4) D ( 3,5) 4
More informationDay 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value
AP Calculus Unit 6 Basic Integration & Applications Day 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value b (1) v( t) dt p( b) p( a), where v(t) represents the velocity and
More informationFor those of you who are taking Calculus AB concurrently with AP Physics, I have developed a
AP Physics C: Mechanics Greetings, For those of you who are taking Calculus AB concurrently with AP Physics, I have developed a brief introduction to Calculus that gives you an operational knowledge of
More informationParticle Motion. Typically, if a particle is moving along the x-axis at any time, t, x()
Typically, if a particle is moving along the x-axis at any time, t, x() t represents the position of the particle; along the y-axis, yt () is often used; along another straight line, st () is often used.
More informationMath 131. The Derivative and the Tangent Line Problem Larson Section 2.1
Math 131. The Derivative and the Tangent Line Problem Larson Section.1 From precalculus, the secant line through the two points (c, f(c)) and (c +, f(c + )) is given by m sec = rise f(c + ) f(c) f(c +
More informationMath 131 Exam II "Sample Questions"
Math 11 Exam II "Sample Questions" This is a compilation of exam II questions from old exams (written by various instructors) They cover chapters and The solutions can be found at the end of the document
More informationDisplacement, Velocity and Acceleration in one dimension
Displacement, Velocity and Acceleration in one dimension In this document we consider the general relationship between displacement, velocity and acceleration. Displacement, velocity and acceleration are
More informationAP CALCULUS BC 2009 SCORING GUIDELINES
AP CALCULUS BC 2009 SCORING GUIDELINES Question 5 x 2 5 8 1 f ( x ) 1 4 2 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points
More informationINTEGRALS5 INTEGRALS
INTEGRALS5 INTEGRALS INTEGRALS In Chapter 2, we used the tangent and velocity problems to introduce the derivative the central idea in differential calculus. INTEGRALS In much the same way, this chapter
More informationMath 1241, Spring 2014 Section 3.3. Rates of Change Average vs. Instantaneous Rates
Math 1241, Spring 2014 Section 3.3 Rates of Change Average vs. Instantaneous Rates Average Speed The concept of speed (distance traveled divided by time traveled) is a familiar instance of a rate of change.
More informationThe Basics of Physics with Calculus Part II. AP Physics C
The Basics of Physics with Calculus Part II AP Physics C The AREA We have learned that the rate of change of displacement is defined as the VELOCITY of an object. Consider the graph below v v t lim 0 dx
More informationThe University of Sydney Math1003 Integral Calculus and Modelling. Semester 2 Exercises and Solutions for Week
The University of Sydney Math3 Integral Calculus and Modelling Semester 2 Exercises and Solutions for Week 2 2 Assumed Knowledge Sigma notation for sums. The ideas of a sequence of numbers and of the limit
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.9 Antiderivatives In this section, we will learn about: Antiderivatives and how they are useful in solving certain scientific problems.
More informationDerivatives and Shapes of Curves
MATH 1170 Section 43 Worksheet NAME Derivatives and Shapes of Curves In Section 42 we discussed how to find the extreme values of a function using the derivative These results say, In Chapter 2, we discussed
More informationCHAPTER 9 MOTION ALONG A STRAIGHT LINE FORM 5 PAPER 2
PPER. particle moves in a straight line and passes through a fixed point O, with a velocity of m s. Its acceleration, a m s, t seconds after passing through O is given by a 8 4t. The particle stops after
More informationParticle Motion. Typically, if a particle is moving along the x-axis at any time, t, x()
Typically, if a particle is moving along the x-axis at any time, t, x() t represents the position of the particle; along the y-axis, yt () is often used; along another straight line, st () is often used.
More informationWhich car/s is/are undergoing an acceleration?
Which car/s is/are undergoing an acceleration? Which car experiences the greatest acceleration? Match a Graph Consider the position-time graphs below. Each one of the 3 lines on the position-time graph
More informationSection 2.1: The Derivative and the Tangent Line Problem Goals for this Section:
Section 2.1: The Derivative and the Tangent Line Problem Goals for this Section: Find the slope of the tangent line to a curve at a point. Day 1 Use the limit definition to find the derivative of a function.
More informationPhysic 231 Lecture 3. Main points of today s lecture. for constant acceleration: a = a; assuming also t0. v = lim
Physic 231 Lecture 3 Main points of today s lecture Δx v = ; Δ t = t t0 for constant acceleration: a = a; assuming also t0 = 0 Δ x = v v= v0 + at Δx 1 v = lim Δ x = Δ t 0 ( v+ vo ) t 2 Δv 1 2 a = ; Δ v=
More informationINTEGRATION: AREAS AND RIEMANN SUMS MR. VELAZQUEZ AP CALCULUS
INTEGRATION: AREAS AND RIEMANN SUMS MR. VELAZQUEZ AP CALCULUS APPROXIMATING AREA For today s lesson, we will be using different approaches to the area problem. The area problem is to definite integrals
More informationChapter 2. Motion along a straight line
Chapter 2 Motion along a straight line 2.2 Motion We find moving objects all around us. The study of motion is called kinematics. Examples: The Earth orbits around the Sun A roadway moves with Earth s
More informationPre-Test Developed by Sean Moroney and James Petersen UNDERSTANDING THE VELOCITY CURVE. The Velocity Curve in Calculus
in Calculus UNDERSTANDING THE VELOCITY CURVE Pre-Test Developed by Sean Moroney and James Petersen Introductory Calculus - in Calculus the Pre-Test Learning about the Velocity Curve During the course of
More informationLimits and Continuity. 2 lim. x x x 3. lim x. lim. sinq. 5. Find the horizontal asymptote (s) of. Summer Packet AP Calculus BC Page 4
Limits and Continuity t+ 1. lim t - t + 4. lim x x x x + - 9-18 x-. lim x 0 4-x- x 4. sinq lim - q q 5. Find the horizontal asymptote (s) of 7x-18 f ( x) = x+ 8 Summer Packet AP Calculus BC Page 4 6. x
More informationTangent Lines and Derivatives
The Derivative and the Slope of a Graph Tangent Lines and Derivatives Recall that the slope of a line is sometimes referred to as a rate of change. In particular, we are referencing the rate at which the
More informationaverage speed instantaneous origin resultant average velocity position particle model scalar
REPRESENTING MOTION Vocabulary Review Write the term that correctly completes the statement. Use each term once. average speed instantaneous origin resultant average velocity position particle model scalar
More information2.1 How Do We Measure Speed? Student Notes HH6ed. Time (sec) Position (m)
2.1 How Do We Measure Speed? Student Notes HH6ed Part I: Using a table of values for a position function The table below represents the position of an object as a function of time. Use the table to answer
More informationAcceleration. 3. Changing Direction occurs when the velocity and acceleration are neither parallel nor anti-parallel
Acceleration When the velocity of an object changes, we say that the object is accelerating. This acceleration can take one of three forms: 1. Speeding Up occurs when the object s velocity and acceleration
More informationWhat is a Vector? A vector is a mathematical object which describes magnitude and direction
What is a Vector? A vector is a mathematical object which describes magnitude and direction We frequently use vectors when solving problems in Physics Example: Change in position (displacement) Velocity
More information12 Rates of Change Average Rates of Change. Concepts: Average Rates of Change
12 Rates of Change Concepts: Average Rates of Change Calculating the Average Rate of Change of a Function on an Interval Secant Lines Difference Quotients Approximating Instantaneous Rates of Change (Section
More informationFor a function f(x) and a number a in its domain, the derivative of f at a, denoted f (a), is: D(h) = lim
Name: Section: Names of collaborators: Main Points: 1. Definition of derivative as limit of difference quotients 2. Interpretation of derivative as slope of graph 3. Interpretation of derivative as instantaneous
More information2.4 Rates of Change and Tangent Lines Pages 87-93
2.4 Rates of Change and Tangent Lines Pages 87-93 Average rate of change the amount of change divided by the time it takes. EXAMPLE 1 Finding Average Rate of Change Page 87 Find the average rate of change
More information2.2 THE DERIVATIVE 2.3 COMPUTATION OF DERIVATIVES: THE POWER RULE 2.4 THE PRODUCT AND QUOTIENT RULES 2.6 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
Differentiation CHAPTER 2 2.1 TANGENT LINES AND VELOCITY 2.2 THE DERIVATIVE 2.3 COMPUTATION OF DERIVATIVES: THE POWER RULE 2.4 THE PRODUCT AND QUOTIENT RULES 25 2.5 THE CHAIN RULE 2.6 DERIVATIVES OF TRIGONOMETRIC
More informationWorksheet 1.8: Geometry of Vector Derivatives
Boise State Math 275 (Ultman) Worksheet 1.8: Geometry of Vector Derivatives From the Toolbox (what you need from previous classes): Calc I: Computing derivatives of single-variable functions y = f (t).
More informationChapter 3. Accelerated Motion
Chapter 3 Accelerated Motion Chapter 3 Accelerated Motion In this chapter you will: Develop descriptions of accelerated motions. Use graphs and equations to solve problems involving moving objects. Describe
More informationAnna D Aloise May 2, 2017 INTD 302: Final Project. Demonstrate an Understanding of the Fundamental Concepts of Calculus
Anna D Aloise May 2, 2017 INTD 302: Final Project Demonstrate an Understanding of the Fundamental Concepts of Calculus Analyzing the concept of limit numerically, algebraically, graphically, and in writing.
More information2/18/2019. Position-versus-Time Graphs. Below is a motion diagram, made at 1 frame per minute, of a student walking to school.
Position-versus-Time Graphs Below is a motion diagram, made at 1 frame per minute, of a student walking to school. A motion diagram is one way to represent the student s motion. Another way is to make
More informationAB Calculus: Rates of Change and Tangent Lines
AB Calculus: Rates of Change and Tangent Lines Name: The World Record Basketball Shot A group called How Ridiculous became YouTube famous when they successfully made a basket from the top of Tasmania s
More informationMean Value Theorem. is continuous at every point of the closed interval,
Mean Value Theorem The Mean Value Theorem connects the average rate of change (slope of the secant between two points [a and b]) with the instantaneous rate of change (slope of tangent at some point c).
More information1.1 Areas under curves - introduction and examples
Chapter 1 Integral Calculus 1.1 Areas under curves - introduction and examples Problem 1.1.1. A car travels in a straight line for one minute, at a constant speed of 10m/s. How far has the car travelled
More information=.55 = = 5.05
MAT1193 4c Definition of derivative With a better understanding of limits we return to idea of the instantaneous velocity or instantaneous rate of change. Remember that in the example of calculating the
More informationChapter 1: Integral Calculus. Chapter 1: Integral Calculus. Chapter 1: Integral Calculus. Chapter 1: Integral Calculus
Chapter 1: Integral Calculus Chapter 1: Integral Calculus Chapter 1: Integral Calculus B 600m, obviously. Chapter 1: Integral Calculus B 600m, obviously. C 600m, obviously. Is this a trick question? Chapter
More informationSections 5.1: Areas and Distances
Sections.: Areas and Distances In this section we shall consider problems closely related to the problems we considered at the beginning of the semester (the tangent and velocity problems). Specifically,
More informationChapter 1 Problem 28: Agenda. Quantities in Motion. Displacement Isn t Distance. Velocity. Speed 1/23/14
Agenda We need a note-taker! If you re interested, see me after class. Today: HW Quiz #1, 1D Motion Lecture for this week: Chapter 2 (finish reading Chapter 2 by Thursday) Homework #2: continue to check
More informationAP Calculus Exam Format and Calculator Tips:
AP Calculus Exam Format and Calculator Tips: Exam Format: The exam is 3 hours and 15 minutes long and has two sections multiple choice and free response. A graphing calculator is required for parts of
More informationLearning 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
More informationGoal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) RECTANGULAR APPROXIMATION METHODS
AP Calculus 5. Areas and Distances Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) Exercise : Calculate the area between the x-axis and the graph of y = 3 2x.
More informationMath 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 ) +
More informationObjective SWBAT find distance traveled, use rectangular approximation method (RAM), volume of a sphere, and cardiac output.
5.1 Estimating with Finite Sums Objective SWBAT find distance traveled, use rectangular approximation method (RAM), volume of a sphere, and cardiac output. Distance Traveled We know that pondering motion
More informationParticle Motion Problems
Particle Motion Problems Particle motion problems deal with particles that are moving along the x or y axis. Thus, we are speaking of horizontal or vertical movement. The position, velocity, or acceleration
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall, 2016, WEEK 4 JoungDong Kim Week4 Section 2.6, 2.7, 3.1 Limits at infinity, Velocity, Differentiation Section 2.6 Limits at Infinity; Horizontal Asymptotes Definition.
More information1.1 Radical Expressions: Rationalizing Denominators
1.1 Radical Expressions: Rationalizing Denominators Recall: 1. A rational number is one that can be expressed in the form a, where b 0. b 2. An equivalent fraction is determined by multiplying or dividing
More informationFormative Assessment: Uniform Acceleration
Formative Assessment: Uniform Acceleration Name 1) A truck on a straight road starts from rest and accelerates at 3.0 m/s 2 until it reaches a speed of 24 m/s. Then the truck travels for 20 s at constant
More informationPosition-versus-Time Graphs
Position-versus-Time Graphs Below is a motion diagram, made at 1 frame per minute, of a student walking to school. A motion diagram is one way to represent the student s motion. Another way is to make
More informationMath 1314 Test 2 Review Lessons 2 8
Math 1314 Test Review Lessons 8 CASA reservation required. GGB will be provided on the CASA computers. 50 minute exam. 15 multiple choice questions. Do Practice Test for extra practice and extra credit.
More informationThe Definition of Differentiation
CALCULO DIFERENCIAL UNIVERSIDAD LIBRE FACULTAD DE INGENIEIRA DEPARTAMENTO DE CIENCIAS BASICAS LECTURAS EN INGLES The Definition of Differentiation The essence of calculus is the derivative. The derivative
More informationLesson 31 - Average and Instantaneous Rates of Change
Lesson 31 - Average and Instantaneous Rates of Change IBHL Math & Calculus - Santowski 1 Lesson Objectives! 1. Calculate an average rate of change! 2. Estimate instantaneous rates of change using a variety
More informationMEAN VALUE THEOREM. Section 3.2 Calculus AP/Dual, Revised /30/2018 1:16 AM 3.2: Mean Value Theorem 1
MEAN VALUE THEOREM Section 3. Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:16 AM 3.: Mean Value Theorem 1 ACTIVITY A. Draw a curve (x) on a separate sheet o paper within a deined closed
More informationJim Lambers MAT 460 Fall Semester Lecture 2 Notes
Jim Lambers MAT 460 Fall Semester 2009-10 Lecture 2 Notes These notes correspond to Section 1.1 in the text. Review of Calculus Among the mathematical problems that can be solved using techniques from
More informationDetermining Average and Instantaneous Rates of Change
MHF 4UI Unit 9 Day 1 Determining Average and Instantaneous Rates of Change From Data: During the 1997 World Championships in Athens, Greece, Maurice Greene and Donovan Bailey ran a 100 m race. The graph
More informationMath Worksheet 1. f(x) = (x a) 2 + b. = x 2 6x = (x 2 6x + 9) = (x 3) 2 1
Names Date Math 00 Worksheet. Consider the function f(x) = x 6x + 8 (a) Complete the square and write the function in the form f(x) = (x a) + b. f(x) = x 6x + 8 ( ) ( ) 6 6 = x 6x + + 8 = (x 6x + 9) 9
More information