Falling Cat 2. Falling Cat 3. Falling Cats 5. Falling Cat 4. Acceleration due to Gravity Consider a cat falling from a branch
|
|
- Daisy Neal
- 5 years ago
- Views:
Transcription
1 Calculus for the Life Sciences Lecture Notes Velocit and Tangent Joseph M. Mahaff, Department of Mathematics and Statistics Dnamical Sstems Group Computational Sciences Research Center San Diego State Universit San Diego, CA jmahaff Outline Spring 7 (/47) (/47) Falling Cat Objects falling under the influence of gravit are important in classical differential Calculus Sir Isaac Newton s work on gravit was a ke step to the development of Calculus Controvers as to whether Newton or Gottfried Leibnitz was the first to invent Calculus Cat have evolved to be one of the best mammalian predators Domestic cats have been shown to responsible for up to 6% of the deaths of songbirds in some communities The are adapted to hunting in trees Cats have a ver fleible spine for hunting This fleibilit allows them to rotate rapidl during a fall (3/47) (4/47)
2 Falling Cat Falling Cat 3 Humans are fascinated b this abilit of a cat to right itself Jared Diamond Stud of Cats falling out of New York apartments Paradoicall the cats falling from the highest apartments actuall fared better than ones falling from an intermediate height The cat remains tense earl in the fall With greater heights the falling cat relaes and spreads its legs to form a parachute This slows its velocit a little and results in a more even impact From intermediate heights, the cat basicall achieves terminal velocit, but the tension causes increased likelihood of severe or fatal injuries (5/47) Acceleration due to Gravit Consider a cat falling from a branch The earl stages of the fall result from acceleration due to gravit Newton s law of motion sas that mass times acceleration is equal to the sum of all the forces acting on an object Velocit is the derivative of position Acceleration is the derivative of velocit (6/47) Falling Cat 4 5 Suppose that a cat falls from a branch that is 6 feet high The height of the cat satisfies the equation How long does this cat fall? h(t) = 6 6t What is its velocit when it hits the ground? From the equation, the cat hits the ground when h(t) = 6 6t = This occurs when t = However, the velocit at t = requires more work We will show that the cat has a velocit, v() = 3 ft/sec (about.8 mph) (7/47) (8/47)
3 of the Falling Cat Suppose that the height of an object is given b h(t) The between times t and t satisfies: v ave = h(t ) h(t ) t t. Return to the cat falling from a 6 ft tree limb, where h(t) = 6 6t Consider the average velocit of the falling cat between t = and t = : v ave = h() h(.5) = = 4ft/sec..5.5 Consider the average velocit of the falling cat between t =.9 and t = : v ave = h() h(.9) = 3.4 = 3.4ft/sec..9. Consider the average velocit of the falling cat between t =.99 and t = : v ave = h() h(.99).99 =.384. = 3.84ft/sec. (9/47) (/47) Velocit of the Falling Cat Return to the cat falling from a 6 ft tree limb, where h(t) = 6 6t Recall the cat hits the ground at t = sec We find the general secant line between t = z and t =, which relates to the near t = Since h( z) = 6 6( z) = 3z 6z v ave = h() h( z) ( z) = 3z +6z z = 3+6z Consider a ball thrown verticall under the influence of gravit, ignoring air resistance The ball begins at ground level (h() = cm) It is thrown verticall with an initial velocit, v() = 96 cm/sec The acceleration of gravit is g = 98 cm/sec The height of the ball for an time t ( t 4) is given b h(t) = 96t 49t As z, v ave 3, so the cat hits the ground at a velocit of 3 ft/sec (.8 mph) (/47) (/47)
4 3 Graph of the height of a ball for t 4, showing position ever.5 sec Height (cm) 5 5 Height of Ball 3 4 Time (sec) Compute the average velocit between each point on the graph The average velocit is the difference between the heights at two times divided b the length of the time period Associate the average velocit with the midpoint between each time interval Height (t ) Height (t ) Average Time h(t ) h(t ) t a = (t +t )/ v(t a ) = h(t ) h(t ) (t t ) h() = h(.5) = t a =.5/ =.5 v(.5) = 75 h(.5) = h() = 96 t a =.75 v(.75) = 45 h(3) = 47 h(3.5) = t a = 3.5 v(3.5) = 5 (3/47) (4/47) 4 Flight of Ball 5 Graph of the velocit of a ball for t 4, showing velocit ever.5 sec Velocit (cm/sec) of Ball Time (sec) The graph of the height of the ball as a function of time is a parabola The graph of the velocit of the ball as a function of time is a line The average velocit is zero when the ball reaches its maimum height The verte of the parabola (maimum height of the ball) is where the velocit is zero (t-intercept) (5/47) (6/47)
5 6 7 Graph of the height of a ball for t 4, showing position ever. sec Height (cm) Height of Ball 5 How does this affect the average velocit computation? The distance between successive heights is now closer But then the intervening time interval is also closer together The average velocit between t =. and t =.3 has h(t ) = 37.4 cm and h(t ) = cm, so v(.5) = 75 cm/sec, the same as before Time (sec) (7/47) (8/47) Flight of Ball 8 Flight of Ball 9 Graph of the velocit of a ball for t 4, showing velocit ever. sec Velocit (cm/sec) of Ball Time (sec) The average velocit data lie on the same straight line as before v(t) = 96 98t This straight line function is the derivative of the quadratic height function h(t) The calculation suggests that the derivative function is independent of the length of the time interval chosen This is specific to the quadratic nature of the height function Soon we will learn to take derivatives of more functions (9/47) (/47)
6 Eample Leaping Salmon Eample Leaping Salmon A river is dammed, and a salmon ladder is built to enable the salmon to bpass the dam and continue to travel upstream to spawn The vertical walls on the salmon ladder are 6 feet high The salmon has to leap verticall upwards over the wall The height of the salmon during its leap is given b Skip Eample h(t) = v t 6t Let v = ft/sec. Sketch a graph of the height of the salmon h(t), with time, showing clearl the maimum height and when the salmon can clear the wall Find the average velocit of the salmon between t = and t =.5 and associate this velocit with t =.5 Repeat this process for each half-second of the leaping salmon, then sketch a graph of the average velocit as a function of time, t Determine the minimum speed, v, that the salmon needs on eiting the water to climb the salmon ladder (/47) (/47) Eample Leaping Salmon 3 Solution: The function h(t) is a parabola, h(t) = t 6t = 4t(5 4t) The t-intercepts are t = and t =.5 The verte occurs at (.65, 6.5) The salmon can clear the wall when h(t) = 6, so t 6t = 6 or 8t t+3 = This can be factored to give (t )(4t 3) = The salmon can clear the wall at an time < t < 3 4 sec (3/47) Eample Leaping Salmon 4 Graph of h(t) = t 6t h(t) (ft) Leaping Salmon (.65,6.5) (.5,6) t (sec) (4/47)
7 Eample Leaping Salmon 5 Solution (cont): The average velocit of the salmon between t = and t =.5 is given b, v(.5) = h(.5) h().5 = ((.5) 6(.5) ).5 The average velocit of the salmon between t =.5 and t = is given b v(.75) = h() h(.5).5 = = 4 ft/sec = ft/sec Eample Leaping Salmon 6 Graph of average velocit of the salmon satisfing Velocit (ft/sec) v ave (t) = 3t (.5,) Salmon Velocit (.75, 4) (5/47) t (sec) (6/47) Eample Leaping Salmon 7 Solution (cont): The minimum speed, v, that the salmon needs to climb the fish ladder is the one that produces a maimum height of 6 ft h(t) = v t 6t The t value of the verte occurs at t = v ( 6) = v 3 Since we want the verte to be 6 ft, ( v ) ( v ) ( v ) v h = v 6 = = 6. Secant Lines and Tangent Line The average velocit is the same calculation as the slope between the two data points of the height function The slope of the secant line between two points on a curve Geometricall, as the points on the curve get closer together, then the secant line approaches the tangent line The tangent line represents the best linear approimation to the curve near a given point Its slope is the derivative of the function at that point v = ft/sec (7/47) (8/47)
8 Secant Lines and Tangent Line Secant Lines and Tangent Line Definition: A secant line for a curve is a line that connect two points on the curve. Graph showing 5 Tangent Line 4 Definition: A tangent line for a curve is a line that touches the curve at eactl one point and provides the best approimation to the curve at that point. 3 Secant Line (9/47) (3/47) Tangent Line Eample = A tangent line represents the best linear approimation to the curve near a given point Consider the function = Tangent Line = f() Tangent Line = f() = f() Tangent Line Find the equation of the tangent line at the point (,) on the graph A secant line is found b taking two points on the curve and finding the equation of the line through those points Create a sequence of secant lines that converge to the tangent line b taking the two points closer and closer together (3/47) (3/47)
9 Eample = Consider the secant line through the points (,) and (,4) This line has a slope of m = 3, and its equation is = 3 Consider the pair of points on the curve =, (,) and (.5,.5) This line has a slope of m =.5, and its equation is =.5.5 The secant line through the points (,) and (.,.) has a slope of m =. Its equation is =.. Eample = 3 Graph of = with secant lines = = 3 =.5.5 =.. = (,) (, 4) (.,.) (.5,.5) (33/47) (34/47) Eample = 4 General secant line for = at (,) Consider the value = +h for some small h The corresponding value = (+h) = +h+h The slope of the secant line through this point and the point (,) is m = (+h+h ) (+h) The formula for this secant line is = h+h h = (+h) (+h) = +h Eample = 5 The general secant line for = through (,) is = (+h) (+h) As h gets ver small, the secant line gets ver close to the tangent line Its not hard to see that the tangent line for = at (,) is = The slope of the tangent line is m = The value of the derivative of = at = (35/47) (36/47)
10 The geometric view of the tangent line is ver eas to visualize The graph on the left is f() with tangent lines shown, while the graph on the right is the derivative of f() 6 4 Tangent Lines for Cubic Equation Slope of Tangent Lines for Cubic (37/47) Several points of interest The graph on the left is a cubic function, while the graph of its derivative is a quadratic As ou approach a maimum (or minimum) for the cubic function, the value of the derivative goes to zero and the sign of the derivative function changes This is an important application of the derivative (38/47) Eample Secant Lines Consider the function Skip Eample f() = Let all secant lines have the point, =. Other points of the sequence have =, =.5, =., =., and =. Find the derivative of f() at = b finding the slope of the tangent line at = Graph f(), the tangent line, and the secant lines Eample Secant Lines Solution: This eample eamines secant lines for through the point (,) f() = When =, f() =, so the secant line has slope m = and is given b = For =.5, two points on the secant line are (,) and (.5,.75), which gives the secant line =.5.5 (39/47) (4/47)
11 Eample Secant Lines 3 Solution (cont): Continuing the process: When =., two points on the secant line are (,) and (.,.4), which gives the secant line =.. For =., two points on the secant line are (,) and (.,.), which gives the secant line =.. For =., two points on the secant line are (,) and (.,.), which gives the secant line =.. (4/47) Eample Secant Lines 4 Solution (cont): The pattern in the sequence easil gives the tangent line = 4 3 = = =.5.5 =.. =.. = f() = (4/47) Eample Secant Lines 5 Eample Secant Lines 6 Solution (cont): Let s find the slope of the secant line through the points Solution (cont): Since the tangent line has slope m =, the derivative of f() = at = is Since patterns cannot alwas be recognizable, we need a better wa to compute the derivative (,f()) = (,) and (+h,f(+h)) Since f(+h) = (+h) (+h) = h +h, the slope of the secant line is m(h) = (h +h) (+h) = h +h h = +h (43/47) As h, m(h) It follows that the slope of the tangent line is, which is the derivative of f() at = (44/47)
12 Eample Square Root Function Consider the function f() = + Find the slope of the secant line through the points (,f()) and (+h,f(+h)) Let h get small and determine the slope of the tangent line through (,), which gives the value of the derivative of f() at = Eample Square Root Function Solution: The slope of the secant line is m(h) = f(+h) f() (+h) +h+ + 4+h = = h h ( )( ) 4+h 4+h+ = h 4+h+ = = 4+h 4 h( 4+h+) 4+h+ (45/47) (46/47) Eample Square Root Function 3 Solution (cont): The slope of the secant line is m(h) = 4+h+ In the formula above, as h, the slope of secant line, m, approaches m t = = 4+ 4 Since the derivative is related to the limiting case of the slope of the secant lines (the slope of the tangent line, m t ), we see that the derivative of f() at = must be 4 (47/47)
elementary and intermediate Algebra Warm-up Name atfm0303mk2810yes
MATH000 online PLACEMENT TEST 1 QUESTIONS 11-0-13 Fall 013 elementar and intermediate Algebra Warm-up Name atfm0303mkes www.alvarezmathhelp.com website PROGRAMS ALVAREZLAB (SAVE AND EXTRACT TO YOUR COMPUTER)
More informationAFM Final Exam Review #1
AFM Final Exam Review # Name. A home security company offers a security system that uses the numbers 0 through 6, inclusive, for a -digit security code. How many different security codes are possible if
More informationChapter 7 One-Dimensional Search Methods
Chapter 7 One-Dimensional Search Methods An Introduction to Optimization Spring, 2014 1 Wei-Ta Chu Golden Section Search! Determine the minimizer of a function over a closed interval, say. The only assumption
More informationBARUCH COLLEGE MATH 2003 SPRING 2006 MANUAL FOR THE UNIFORM FINAL EXAMINATION
BARUCH COLLEGE MATH 003 SPRING 006 MANUAL FOR THE UNIFORM FINAL EXAMINATION The final examination for Math 003 will consist of two parts. Part I: Part II: This part will consist of 5 questions similar
More informationLecture Notes 1 Part B: Functions and Graphs of Functions
Lecture Notes 1 Part B: Functions and Graphs of Functions In Part A of Lecture Notes #1 we saw man examples of functions as well as their associated graphs. These functions were the equations that gave
More informationTRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.
MATH 143 - COLLEGE ALGEBRA/BUSN - PRACTICE EXAM #1 - FALL 2008 - DR. DAVID BRIDGE TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. Mark the statement as true or false.
More informationMath 1526 Summer 2000 Session 1
Math 1526 Summer 2 Session 1 Lab #2 Part #1 Rate of Change This lab will investigate the relationship between the average rate of change, the slope of a secant line, the instantaneous rate change and the
More informationTest # 1 Review Math MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Test # 1 Review Math 135 Name (Sections 1.3,.,3.7,..1,.3,11.1,11.,11.3,and 11.) _ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor out the greatest
More informationName: Class: Date: in general form.
Write the equation in general form. Mathematical Applications for the Management Life and Social Sciences 11th Edition Harshbarger TEST BANK Full clear download at: https://testbankreal.com/download/mathematical-applications-management-life-socialsciences-11th-edition-harshbarger-test-bank/
More informationST. DAVID S MARIST INANDA
ST. DAVID S MARIST INANDA MATHEMATICS NOVEMBER EXAMINATION GRADE 11 PAPER 1 8 th NOVEMBER 2016 EXAMINER: MRS S RICHARD MARKS: 125 MODERATOR: MRS C KENNEDY TIME: 2 1 Hours 2 NAME: PLEASE PUT A CROSS NEXT
More informationSemester Exam Review
Semester Exam Review Name Date Block MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. For the given equation, find the values of a, b, and c, determine
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
CHAPTER FORM A Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the given ordered pair is a solution of the given equation.
More informationSolution of Equations
Solution of Equations Outline Bisection Method Secant Method Regula Falsi Method Newton s Method Nonlinear Equations This module focuses on finding roots on nonlinear equations of the form f()=0. Due to
More informationUnit 1 Maths Methods (CAS) Exam 2013 Thursday June 6th pm
Name: Teacher: Unit 1 Maths Methods (CAS) Exam 2013 Thursday June 6th 1.50-3.20 pm Reading time: 10 Minutes Writing time: 80 Minutes Instruction to candidates: Students are permitted to bring into the
More informationExample 11: A country s gross domestic product (in millions of dollars) is modeled by the function
Math 1314 Lesson 7 With this group of word problems, the first step will be to determine what kind of problem we have for each problem. Does it ask for a function value (FV), a rate of change (ROC) or
More information( ) 4 ( )! x f) h(x) = 2cos x + 1
Chapter Prerequisite Skills BLM -.. Identifying Types of Functions. Identify the type of function (polynomial, rational, logarithmic, etc.) represented by each of the following. Justify your response.
More informationMATH 830/GRACEY EXAM 4 PRACTICE/CH. 5. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MATH 30/GRACEY EXAM PRACTICE/CH. 5 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the epression with positive eponents onl. Then simplif,
More informationCS 3331 Numerical Methods Lecture 2: Functions of One Variable. Cherung Lee
CS 3331 Numerical Methods Lecture 2: Functions of One Variable Cherung Lee Outline Introduction Solving nonlinear equations: find x such that f(x ) = 0. Binary search methods: (Bisection, regula falsi)
More informationTCM Final Review Packet Name Per.
TCM Final Review Packet Name Per. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Translate the statement into a formula. 1) The total distance traveled,
More informationb) According to the statistics above the graph, the slope is What are the units and meaning of this value?
! Name: Date: Hr: LINEAR MODELS Writing Motion Equations 1) Answer the following questions using the position vs. time graph of a runner in a race shown below. Be sure to show all work (formula, substitution,
More informationMAT Pre-Calculus Class Worksheet - Word Problems Chapter 1
MAT 111 - Pre-Calculus Name Class Worksheet - Word Problems Chapter 1 1. The cost of a Frigbox refrigerator is $950, and it depreciates $50 each year. The cost of a new Arctic Air refrigerator is $1200,
More informationYou may be given raw data concerning costs and revenues. In that case, you ll need to start by finding functions to represent cost and revenue.
Example 2: Suppose a company can model its costs according to the function 3 2 Cx ( ) 0.000003x 0.04x 200x 70, 000 where Cxis ( ) given in dollars and demand can be modeled by p 0.02x 300. a. Find the
More informationName: Practice B Exam 2. October 8, 2014
Department of Mathematics University of Notre Dame Math 10250 Elem. of Calc. I Name: Instructor: Practice B Exam 2 October 8, 2014 This exam is in 2 parts on 10 pages and contains 14 problems worth a total
More informationChapter 6 Diagnostic Test
Chapter 6 Diagnostic Test STUDENT BOOK PAGES 310 364 1. Consider the quadratic relation y = x 2 6x + 3. a) Use partial factoring to locate two points with the same y-coordinate on the graph. b) Determine
More informationCalculus for Business Economics Life Sciences and Social Sciences 13th Edition Barnett SOLUTIONS MANUAL Full download at:
Calculus for Business Economics Life Sciences and Social Sciences 1th Edition Barnett TEST BANK Full download at: https://testbankreal.com/download/calculus-for-business-economics-life-sciencesand-social-sciences-1th-edition-barnett-test-bank/
More informationIn this section we want to review all that we know about polynomials.
R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract
More information3.1 Solutions to Exercises
.1 Solutions to Exercises 1. (a) f(x) will approach + as x approaches. (b) f(x) will still approach + as x approaches -, because any negative integer x will become positive if it is raised to an even exponent,
More information3.1 Solutions to Exercises
.1 Solutions to Exercises 1. (a) f(x) will approach + as x approaches. (b) f(x) will still approach + as x approaches -, because any negative integer x will become positive if it is raised to an even exponent,
More informationUse Scantron 882E to transfer the answers. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
HW Date: Name Use Scantron 88E to transfer the answers. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph shows sales in thousands of dollars
More informationTEST # 1 REVIEW MATH MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
TEST # REVIEW MATH Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Give the domain and range of the relation. ) {(-8, -), (, ), (9, 8), (-, ),
More informationChapter 6: Quadratic Functions & Their Algebra
Chapter 6: Quadratic Functions & Their Algebra Topics: 1. Quadratic Function Review. Factoring: With Greatest Common Factor & Difference of Two Squares 3. Factoring: Trinomials 4. Complete Factoring 5.
More informationLesson 10: Interpreting Quadratic Functions from Graphs and Tables
: Interpreting Quadratic Functions from Graphs and Tables Student Outcomes Students interpret quadratic functions from graphs and tables: zeros ( intercepts), intercept, the minimum or maximum value (vertex),
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Answer: No Correct Answer Was Provided. Provide an appropriate response. ) If a relation eists
More informationCS227-Scientific Computing. Lecture 6: Nonlinear Equations
CS227-Scientific Computing Lecture 6: Nonlinear Equations A Financial Problem You invest $100 a month in an interest-bearing account. You make 60 deposits, and one month after the last deposit (5 years
More informationChapter 2-4 Review. Find the equation of the following graphs. Then state the domain and range: 1a) 1b) 1c)
Chapter - Review Find the equation of the following graphs. Then state the domain and range: a) b) c) a) b) c) a) b) c) Find the domain of the following functions. Write your answer in interval notation:
More informationMath 1 EOC Review Parallel Problems
Math 1 EOC Review Parallel Problems Unit 1 14. A school purchases boxes of t-shirts for a fundraiser. Each box has 120 t-shirts, and the school pays $1500 per box. How much does the school need to charge
More informationLaurie s Notes. Overview of Section 7.6. (1x + 6)(2x + 1)
Laurie s Notes Overview of Section 7.6 Introduction In this lesson, students factor trinomials of the form ax 2 + bx + c. In factoring trinomials, an common factor should be factored out first, leaving
More informationTopic #1: Evaluating and Simplifying Algebraic Expressions
John Jay College of Criminal Justice The City University of New York Department of Mathematics and Computer Science MAT 105 - College Algebra Departmental Final Examination Review Topic #1: Evaluating
More information4.2 Rolle's Theorem and Mean Value Theorem
4.2 Rolle's Theorem and Mean Value Theorem Rolle's Theorem: Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f (a) = f (b), then there is at least one
More informationPre-Calculus Midterm Exam REVIEW January 2013
Pre-Calculus Midterm Eam REVIEW Januar 0 Name: Date: Teacher: Period: Your midterm eamination will consist of: 0 multiple-choice questions (including true/false & matching) these will be completed on the
More informationName: Date: Page 1 of 7. What is Slope? There are four types of slope you can encounter. A slope can be positive, negative, zero, or undefined.
Name: Date: Page of 7 What is Slope? What is slope? If ou have ever walked up or down a hill, then ou have alread eperienced a real life eample of slope. Keeping this fact in mind, b definition, the slope
More informationSolving Problems Involving Cost, Revenue, Profit. Max and Min Problems
Solving Problems Involving Cost, Revenue, Profit The cost function C(x) is the total cost of making x items. If the cost per item is fixed, it is equal to the cost per item (c) times the number of items
More informationWhen Is Factoring Used?
When Is Factoring Used? Name: DAY 9 Date: 1. Given the function, y = x 2 complete the table and graph. x y 2 1 0 1 2 3 1. A ball is thrown vertically upward from the ground according to the graph below.
More informationName Student ID # Instructor Lab Period Date Due. Lab 6 The Tangent
Name Student ID # Instructor Lab Period Date Due Lab 6 The Tangent Objectives 1. To visualize the concept of the tangent. 2. To define the slope of the tangent line. 3. To develop a definition of the tangent
More informationName: Common Core Algebra L R Final Exam 2015 CLONE 3 Teacher:
1) Which graph represents a linear function? 2) Which relation is a function? A) B) A) {(2, 3), (3, 9), (4, 7), (5, 7)} B) {(0, -2), (3, 10), (-2, -4), (3, 4)} C) {(2, 7), (2, -3), (1, 1), (3, -1)} D)
More informationMATH 1015 Final Exam Review Rev 02/2018
MATH 1 Final Exam Review Rev 0/018 ============================================================================== 1)Find the domain and range for the function. 1) 3 1-7 - - - -3 - -1 1 3 7 - -3 - - - -7
More informationPlease show work for all calculated answers. Show work in a neat and organized manner.
Math 083 Review for Final Exam Name Please show work for all calculated answers. Show work in a neat and organized manner. 1) Using the frequency table for a monthly budget, find all of the relative frequencies
More information0 Review: Lines, Fractions, Exponents Lines Fractions Rules of exponents... 5
Contents 0 Review: Lines, Fractions, Exponents 3 0.1 Lines................................... 3 0.2 Fractions................................ 4 0.3 Rules of exponents........................... 5 1 Functions
More information4.3 The money-making machine.
. The money-making machine. You have access to a magical money making machine. You can put in any amount of money you want, between and $, and pull the big brass handle, and some payoff will come pouring
More informationPRELIMINARY EXAMINATION 2018 MATHEMATICS GRADE 12 PAPER 1. Time: 3 hours Total: 150 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
PRELIMINARY EXAMINATION 2018 MATHEMATICS GRADE 12 PAPER 1 Time: 3 hours Total: 150 Examiner: P R Mhuka Moderators: J Scalla E Zachariou PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question
More informationTN 2 - Basic Calculus with Financial Applications
G.S. Questa, 016 TN Basic Calculus with Finance [016-09-03] Page 1 of 16 TN - Basic Calculus with Financial Applications 1 Functions and Limits Derivatives 3 Taylor Series 4 Maxima and Minima 5 The Logarithmic
More informationIn a moment, we will look at a simple example involving the function f(x) = 100 x
Rates of Change Calculus is the study of the way that functions change. There are two types of rates of change: 1. Average rate of change. Instantaneous rate of change In a moment, we will look at a simple
More informationSymmetric Game. In animal behaviour a typical realization involves two parents balancing their individual investment in the common
Symmetric Game Consider the following -person game. Each player has a strategy which is a number x (0 x 1), thought of as the player s contribution to the common good. The net payoff to a player playing
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
12B Practice for the Final Eam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. If = -4 and = -2, evaluate the epression. 12-6 1) + 2 A) - 9 B) 0 C)
More informationMath 115 Sample Final. 5) 1 5 y y y
Math 11 Sample Final Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor completel. If the polnomial is prime, state this. 1) 3 + 82-20 A)
More informationMath Winter 2014 Exam 1 January 30, PAGE 1 13 PAGE 2 11 PAGE 3 12 PAGE 4 14 Total 50
Name: Math 112 - Winter 2014 Exam 1 January 30, 2014 Section: Student ID Number: PAGE 1 13 PAGE 2 11 PAGE 3 12 PAGE 4 14 Total 50 After this cover page, there are 5 problems spanning 4 pages. Please make
More informationChapter 9. Chapters 5 8 Review, pages Analysing Graphs of Linear Relations, pages
1. a) -7 No. Different sets of integers can have the same mean. Eample: {-, -1, 1, -,, -1} and {-, 9, -, 1,, } both have a sum of - and a mean of -7.. a decrease of 31 people per ear 3. 7 s. $7 Chapters
More informationThe Trout Pond Revisited
The Trout Pond Revisited A. MATERIALS NEEDED Worksheet, calculator, ruler B. OBJECTIVE The student will use the knowledge already gained concerning the calculations of slopes of lines to find average and
More information1/20 2/17 3/14 4/29 5/20 Total/100. Exam II- VERSION I Spring 2011
1/20 2/17 3/14 4/29 5/20 Total/100 Do not write in the spaces above. MATH 150-03 Dr. Morton Exam II- VERSION I Spring 2011 Name: Directions: You have 50 minutes in which to complete this exam. Make sure
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) y = - 39x - 80 D) y = x + 8 5
Assn 3.4-3.7 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the equation of the tangent line to the curve when x has the given value. 1)
More informationNOTES ON CALCULUS AND UTILITY FUNCTIONS
DUSP 11.203 Frank Levy Microeconomics Tutorial 1 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding utility
More informationIn the Herb Business, Part I
63 In the Herb Business, Part I A. You have joined a highl respected St Croi herbalist in a business to market her herbal products. Your personal goal is to assure that the business thrives. Researchers
More informationGRAPHS IN ECONOMICS. Appendix. Key Concepts. A Positive Relationship
Appendi GRAPHS IN ECONOMICS Ke Concepts Graphing Data Graphs represent quantit as a distance on a line. On a graph, the horizontal scale line is the -ais, the vertical scale line is the -ais, and the intersection
More informationAPPENDIX F Business and Economic Applications
APPENDIX F Business and Economic Applications Business and Economics Applications Previously, you learned that one of the most common ways to measure change is with respect to time. In this section, you
More informationpar ( 12). His closest competitor, Ernie Els, finished 3 strokes over par (+3). What was the margin of victory?
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Tiger Woods won the 000 U.S. Open golf tournament with a score of 1 strokes under par
More informationFinal Examination Re - Calculus I 21 December 2015
. (5 points) Given the graph of f below, determine each of the following. Use, or does not exist where appropriate. y (a) (b) x 3 x 2 + (c) x 2 (d) x 2 (e) f(2) = (f) x (g) x (h) f (3) = 3 2 6 5 4 3 2
More informationCommon Core Algebra L clone 4 review R Final Exam
1) Which graph represents an exponential function? A) B) 2) Which relation is a function? A) {(12, 13), (14, 19), (11, 17), (14, 17)} B) {(20, -2), (24, 10), (-21, -5), (22, 4)} C) {(34, 8), (32, -3),
More informationLCHL Paper 1 Q2 (25 marks)
Note: The sample answers provided are illustrative of one possible approach to answering the particular question. Students may adopt different but equally valid approaches and should be encouraged to compare
More informationLecture Quantitative Finance Spring Term 2015
implied Lecture Quantitative Finance Spring Term 2015 : May 7, 2015 1 / 28 implied 1 implied 2 / 28 Motivation and setup implied the goal of this chapter is to treat the implied which requires an algorithm
More informationContinuous Distributions
Quantitative Methods 2013 Continuous Distributions 1 The most important probability distribution in statistics is the normal distribution. Carl Friedrich Gauss (1777 1855) Normal curve A normal distribution
More information1) State whether the following are functions and give an explanation for your reason. Give the domain and range for each relation.
MCF M Calc Questions:,,, 7, 8, 9,,,, QUADRATIC FUNCTIONS ) State whether the following are functions and give an eplanation for your reason. Give the domain and range for each relation. a) - 7 b) y c)
More informationChapter 2 Rocket Launch: AREA BETWEEN CURVES
ANSWERS Mathematics (Mathematical Analysis) page 1 Chapter Rocket Launch: AREA BETWEEN CURVES RL-. a) 1,.,.; $8, $1, $18, $0, $, $6, $ b) x; 6(x ) + 0 RL-. a), 16, 9,, 1, 0; 1,,, 7, 9, 11 c) D = (-, );
More informationMath 1101 Exam 1 Practice Problems
Math 1101 Eam 1 Practice Problems These problems are not intended to cover all possible test topics. Rather, the should serve as an activit in preparing for our test, but other stud is required to full
More informationMath 116 Review A ball is thrown upward from the top of a 200-foot cliff. The initial velocity of the ball is 125 feet per
Math 6 Review You may only use a calculator if the problem is labeled calc.. Find the equation of the tangent line that is tangent to the graph of f and parallel to the given line. Page of 5 f x x, line
More informationMATH COLLEGE ALGEBRA/BUSN - PRACTICE EXAM #2 - SUMMER DR. DAVID BRIDGE
MATH 13 - COLLEGE ALGEBRA/BUSN - PRACTICE EXAM # - SUMMER 007 - DR. DAVID BRIDGE MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the piecewise
More information8-7 Solving ax^2 + bx + c = 0
29. BASKETBALL When Jerald shoots a free throw, the ball is 6 feet from the floor and has an initial upward velocity of 20 feet per second. The hoop is 10 feet from the floor. a. Use the vertical motion
More informationTA Handout 2 What is a Derivative and How Can We Make Use of It?
TA Handout What is a Derivative and How Can We Make Use of It? 1 Definition and Intuition A simple wa to think of about a derivative is as a measure of the rate of change in a function That is, given a
More informationBARUCH COLLEGE MATH 2205 SPRING MANUAL FOR THE UNIFORM FINAL EXAMINATION Joseph Collison, Warren Gordon, Walter Wang, April Allen Materowski
BARUCH COLLEGE MATH 05 SPRING 006 MANUAL FOR THE UNIFORM FINAL EXAMINATION Joseph Collison, Warren Gordon, Walter Wang, April Allen Materowski The final examination for Math 05 will consist of two parts.
More informationACTIVITY: Comparing Types of Growth
6.5 Eponential Growth growth? What are the characteristics of eponential ACTIVITY: Comparing Tpes of Growth Work with a partner. Describe the pattern of growth for each sequence and graph. How man of the
More informationFebruary 2 Math 2335 sec 51 Spring 2016
February 2 Math 2335 sec 51 Spring 2016 Section 3.1: Root Finding, Bisection Method Many problems in the sciences, business, manufacturing, etc. can be framed in the form: Given a function f (x), find
More informationMA Notes, Lesson 19 Textbook (calculus part) Section 2.4 Exponential Functions
MA 590 Notes, Lesson 9 Tetbook (calculus part) Section.4 Eponential Functions In an eponential function, the variable is in the eponent and the base is a positive constant (other than the number ). Eponential
More informationName: Period: Date: FOMP 10 Final Review Part 2 v1. Short Answer. Level 1-2 Questions. 1. What expression does the diagram represent?
Period: Date: FOMP 10 Final Review Part 2 v1 Short Answer Level 1-2 Questions 1. What expression does the diagram represent? 2. What is the factored form of the expression 5x 2 45? 3. What value of k makes
More informationMath 098 Exam 1 Preparation- 1.1 to 1.5, Ch 2.1, 2.2, 2.3, 2.5, 3.1, & 3.5 v01 NO BOOK/ NO NOTES/YES CALCUATOR Dressler Winter 2016.
Math 098 Eam 1 Preparation- 1.1 to 1., Ch.1,.,.3,., 3.1, & 3. v01 NO BOOK/ NO NOTES/YES CALCUATOR Dressler Winter 01 Name State the solution set of the inequalit in interval notation and sketch its graph.
More informationLesson 21: Comparing Linear and Exponential Functions Again
: Comparing Linear and Exponential Functions Again Student Outcomes Students create models and understand the differences between linear and exponential models that are represented in different ways. Lesson
More informationNAME: DATE: Algebra 2: Lesson 12-7 Geometric Series Word Problems. DO NOW: Answer the following question in order to prepare for today s lesson.
NAME: DATE: Algebra 2: Lesson 12-7 Geometric Series Word Problems Learning Goals: 1. How do we use the geometric series formula when working with word problems? DO NOW: Answer the following question in
More information1 Maximizing profits when marginal costs are increasing
BEE12 Basic Mathematical Economics Week 1, Lecture Tuesday 9.12.3 Profit maximization / Elasticity Dieter Balkenborg Department of Economics University of Exeter 1 Maximizing profits when marginal costs
More informationFeb. 4 Math 2335 sec 001 Spring 2014
Feb. 4 Math 2335 sec 001 Spring 2014 Propagated Error in Function Evaluation Let f (x) be some differentiable function. Suppose x A is an approximation to x T, and we wish to determine the function value
More information123 PART 1: Solutions to Odd-Numbered Exercises and Practice Tests
3 PART : Solutions to Odd-Numbered Eercises and Practice Tests Section.7 Graphs of Rational Functions You should be able to graphf() - q()" (a) Find the - and -intercepts. (b) Find an vertical or horizontal
More informationContinuous Probability Distributions
8.1 Continuous Probability Distributions Distributions like the binomial probability distribution and the hypergeometric distribution deal with discrete data. The possible values of the random variable
More informationDATA HANDLING Five-Number Summary
DATA HANDLING Five-Number Summary The five-number summary consists of the minimum and maximum values, the median, and the upper and lower quartiles. The minimum and the maximum are the smallest and greatest
More informationLesson 6: Extensions and applications of consumer theory. 6.1 The approach of revealed preference
Microeconomics I. Antonio Zabalza. Universit of Valencia 1 Lesson 6: Etensions and applications of consumer theor 6.1 The approach of revealed preference The basic result of consumer theor (discussed in
More informationLesson 4.5 Real-World Problems: Linear Equations
Lesson 4.5 Real-World Problems: Linear Equations Explain the meaning of the slope and y-intercept in real-world problems. Example A telecommunication company charges their customers a fee for phone calls.
More informationt g(t) h(t) k(t)
Problem 1. Determine whether g(t), h(t), and k(t) could correspond to a linear function or an exponential function, or neither. If it is linear or exponential find the formula for the function, and then
More informationList the quadrant(s) in which the given point is located. 1) (-10, 0) A) On an axis B) II C) IV D) III
MTH 55 Chapter 2 HW List the quadrant(s) in which the given point is located. 1) (-10, 0) 1) A) On an axis B) II C) IV D) III 2) The first coordinate is positive. 2) A) I, IV B) I, II C) III, IV D) II,
More informationMath Review Chapter 1
Math 60 - Review Chapter Name ) A mortgage on a house is $90,000, the interest rate is 8 %, and the loan period is 5 years. What is the monthly payment? ) Joan wants to start an annuity that will have
More informationAnswers to Exercise 8
Answers to Exercise 8 Logistic Population Models 1. Inspect your graph of N t against time. You should see the following: Population size increases slowly at first, then accelerates (the curve gets steeper),
More informationf ( x) a, where a 0 and a 1. (Variable is in the exponent. Base is a positive number other than 1.)
MA 590 Notes, Lesson 9 Tetbook (calculus part) Section.4 Eponential Functions In an eponential function, the variable is in the eponent and the base is a positive constant (other than the number ). Eponential
More informationMLC at Boise State Polynomials Activity 2 Week #3
Polynomials Activity 2 Week #3 This activity will discuss rate of change from a graphical prespective. We will be building a t-chart from a function first by hand and then by using Excel. Getting Started
More informationNormal Probability Distributions
Normal Probability Distributions Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution. Normal curve A normal distribution is a continuous
More informationCompleting the Square. A trinomial that is the square of a binomial. x Square half the coefficient of x. AA65.pdf.
AA65.pdf 6.5 Completing the Square 1. Converting from vertex form to standard form involves expanding the square of the binomial, distributing a, and then isolating y. What method does converting from
More information