Function Terminology and Types of Functions
|
|
- Barnard Butler
- 5 years ago
- Views:
Transcription
1 1.2: Rate of Change by Equation, Graph, or Table [AP Calculus AB] Objective: Given a function y = f(x) specified by a graph, a table of values, or an equation, describe whether the y-value is increasing or decreasing as x increases through a particular value, and estimate the instantaneous rate of change of y at that value of x. Function Terminology and Types of Functions Function Terminology of f(x) f is the name of a function. (Note: other letters can also be used to name functions. For example d can be used to name a function that models distance.) x is the independent variable or specific x-value that will be substituted into the function. f(x) is the dependent variable or specific y-value that corresponds to the substituted x-value. Types of Functions Type General Equation *Variables other than x are constants Particular Equation Linear f(x) = mx + b f(x) = 2x 1 Particular Graph Quadratic f(x) = ax 2 + bx + c f(x) = x Polynomial (odd) f(x) = a n x n ax 2 + bx + c f(x) = x 3 6x 2 9x 6
2 Polynomial (even) f(x) = a n x n ax 2 + bx + c f(x) = x 4 Exponential f(x) = ab x f(x) = e x Rational f(x) contains (polynomial) (polynomial) f(x) = x x 1 Absolute Value f(x) contains polynomial f(x) = x Trigonometric or Circular f(x) contians sin x, cos x, tan x, csc x, sec x, or cot x f(x) = 2 sin x Analyzing a Function Using its Derivative The derivative of a function f(x) at x = a is the instantaneous rate of change of f(x) with respect to x at x = a. It is found: Numerically, by taking the limit of the average rate over the interval from a to c as a approaches c. (Note: a limit is an arbitrary number that f(x) approaches) f(x) f(a) x a f (a) = lim x a Graphically, by finding the slope of the line tangent to the graph at x = a
3 The derivative f (x) can help you describe characteristics of the original function f(x). If f (a) is positive, then f(a) is increasing at x = a. If f (a) is negative, then f(a) is decreasing at x = a. If f (a) is zero, then f(a) is neither increasing nor decreasing at x = a. Typically, if f (a) is greater than 1, then the rate is fast and f(a) is changing quickly at x = a. Typically, if f (a) is less than 1, then the rate is slow and f(a) is changing slowly at x = a. (Note: classifying the rate as fast or slow depends on the context of the equation.) Example 1: Using the Derivative to Analyze a Function (Increasing/Decreasing, Fast/Slow) The figure shows the graph of a function. At x = a, x = b, and x = c, state whether y is increasing, decreasing, or neither as x increases. Then state whether the rate of change is fast or slow. Example 2: Using a Function s Graph to Determine the Derivative The figure shows the graph of a function that could represent the height, h(t), in feet, of a soccer ball above the ground as a function of time, t, in seconds since it was kicked into the air. a. Estimate the instantaneous rate of change of h(t) at time t = 5.
4 b. Give the mathematical name of this instantaneous rate, and state why the rate is negative. Example 3: Using a Function s Equation to Determine the Derivative The figure shows a graph of P(x) = 40(0.6 x ), the probability that it rains a number of inches, x, at a particular place during a particular thunderstorm. a. The probability that it rains 1 inch is P(1) = 24%. By how much, and in which direction, does the probability change from x = 1 to x = 1.1? (What is the average rate of change from 1 inch to 1.1 inches? Make sure to include units in your answer.) Why is the rate negative? b. Write an equation for r(x), the average rate of change of P(x) from 1 to x. Make a table of values of r(x) for each 0.01 unit of x from 0.97 to Explain why r(x) is undefined at x = 1.
5 c. The instantaneous rate at x = 1 is the limit that the average rate approaches as x approaches 1. Estimate the instantaneous rate using information from part b. d. Estimate the instantaneous rate at x = 1 by using a value of your choice for x (just as performed in Section 1-1). Example 4: Using a Function s Table of Values to Determine the Derivative A mass is bouncing up and down on a spring hanging from the ceiling. Its distance, y, in feet, from the ceiling is measured by a calculator distance probe each 1/10 s, giving this table of values, in which t is time in seconds. a. How fast is y changing at each time? i. t = 0.3 ii. t = 0.6 iii. t = 1.0 b. At time t = 0.3, is the mass going up or down? Justify your answer.
A Library of Functions
LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us
More informationCalculus I Sample Exam #01
Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6
More informationOne of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the unit circle.
2.24 Tanz and the Reciprocals Derivatives of Other Trigonometric Functions One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the
More informationAim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)
Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x
More informationUnit 1 PreCalculus Review & Limits
1 Unit 1 PreCalculus Review & Limits Factoring: Remove common factors first Terms - Difference of Squares a b a b a b - Sum of Cubes ( )( ) a b a b a ab b 3 3 - Difference of Cubes a b a b a ab b 3 3 3
More information1.2 A List of Commonly Occurring Functions
Arkansas Tech University MATH 2914: Calculus I Dr. Marcel B. Finan 1.2 A List of Commonly Occurring Functions In this section, we discuss the most common functions occurring in calculus. Linear Functions
More informationPreliminaries Lectures. Dr. Abdulla Eid. Department of Mathematics MATHS 101: Calculus I
Preliminaries 2 1 2 Lectures Department of Mathematics http://www.abdullaeid.net/maths101 MATHS 101: Calculus I (University of Bahrain) Prelim 1 / 35 Pre Calculus MATHS 101: Calculus MATHS 101 is all about
More information3. Use absolute value notation to write an inequality that represents the statement: x is within 3 units of 2 on the real line.
PreCalculus Review Review Questions 1 The following transformations are applied in the given order) to the graph of y = x I Vertical Stretch by a factor of II Horizontal shift to the right by units III
More informationSummer Review Packet (Limits & Derivatives) 1. Answer the following questions using the graph of ƒ(x) given below.
Name AP Calculus BC Summer Review Packet (Limits & Derivatives) Limits 1. Answer the following questions using the graph of ƒ() given below. (a) Find ƒ(0) (b) Find ƒ() (c) Find f( ) 5 (d) Find f( ) 0 (e)
More informationChapter 3: Derivatives
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 3: Derivatives Sections: v 2.4 Rates of Change & Tangent Lines v 3.1 Derivative of a Function v 3.2 Differentiability v 3.3 Rules for Differentiation
More informationAP Calculus Summer Packet
AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept
More informationSummer Review for Students Taking Calculus in No calculators allowed. To earn credit: Be sure to show all work in the area provided.
Summer Review for Students Taking Calculus in 2016-2017 No calculators allowed. To earn credit: Be sure to show all work in the area provided. 1 Graph each equation on the axes provided. Include any relevant
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More informationCalculus I Exam 1 Review Fall 2016
Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function
More informationAdvanced Precalculus Summer Assignment
Advanced Precalculus Summer Assignment The following packet of material contains prerequisite skills and concepts that will be drawn upon from Algebra II as well as a preview of Precalculus. This summer
More informationMath 1431 Final Exam Review
Math 1431 Final Exam Review Comprehensive exam. I recommend you study all past reviews and practice exams as well. Know all rules/formulas. Make a reservation for the final exam. If you miss it, go back
More informationCalculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA
Calculus Weijiu Liu Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA 1 Opening Welcome to your Calculus I class! My name is Weijiu Liu. I will guide you
More informationAP * Calculus Review. Limits, Continuity, and the Definition of the Derivative
AP * Calculus Review Limits, Continuity, and the Definition of the Derivative Teacher Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board
More informationCALCULUS OPTIONAL SUMMER WORK
NAME JUNE 016 CALCULUS OPTIONAL SUMMER WORK PART I - NO CALCULATOR I. COORDINATE GEOMETRY 1) Identify the indicated quantities for -8x + 15y = 0. x-int y-int slope ) A line has a slope of 5/7 and contains
More informationProblems with an # after the number are the only ones that a calculator is required for in the solving process.
Instructions: Make sure all problems are numbered in order. All work is in pencil, and is shown completely. Graphs are drawn out by hand. If you use your calculator for some steps, intermediate work should
More informationName Class. (a) (b) (c) 4 t4 3 C
Chapter 4 Test Bank 77 Test Form A Chapter 4 Name Class Date Section. Evaluate the integral: t dt. t C (a) (b) 4 t4 C t C C t. Evaluate the integral: 5 sec x tan x dx. (a) 5 sec x tan x C (b) 5 sec x C
More informationPre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.2 Solving Quadratic Equations
Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.1.1 Solve Simple Equations Involving Absolute Value 0.2 Solving Quadratic Equations 0.2.1 Use the
More informationFinal Exam Review Problems
Final Exam Review Problems Name: Date: June 23, 2013 P 1.4. 33. Determine whether the line x = 4 represens y as a function of x. P 1.5. 37. Graph f(x) = 3x 1 x 6. Find the x and y-intercepts and asymptotes
More informationChapter 1. Functions 1.1. Functions and Their Graphs
1.1 Functions and Their Graphs 1 Chapter 1. Functions 1.1. Functions and Their Graphs Note. We start by assuming that you are familiar with the idea of a set and the set theoretic symbol ( an element of
More informationAP Calculus AB Chapter 1 Limits
AP Calculus AB Chapter Limits SY: 206 207 Mr. Kunihiro . Limits Numerical & Graphical Show all of your work on ANOTHER SHEET of FOLDER PAPER. In Exercises and 2, a stone is tossed vertically into the air
More informationSANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.
More information12) y = -2 sin 1 2 x - 2
Review -Test 1 - Unit 1 and - Math 41 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find and simplify the difference quotient f(x + h) - f(x),
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. (a) 5
More informationA. Incorrect! For a point to lie on the unit circle, the sum of the squares of its coordinates must be equal to 1.
Algebra - Problem Drill 19: Basic Trigonometry - Right Triangle No. 1 of 10 1. Which of the following points lies on the unit circle? (A) 1, 1 (B) 1, (C) (D) (E), 3, 3, For a point to lie on the unit circle,
More informationThis Week. Professor Christopher Hoffman Math 124
This Week Sections 2.1-2.3,2.5,2.6 First homework due Tuesday night at 11:30 p.m. Average and instantaneous velocity worksheet Tuesday available at http://www.math.washington.edu/ m124/ (under week 2)
More informationReview of Topics in Algebra and Pre-Calculus I. Introduction to Functions function Characteristics of a function from set A to set B
Review of Topics in Algebra and Pre-Calculus I. Introduction to Functions A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in set B.
More informationUnit #3 Rules of Differentiation Homework Packet
Unit #3 Rules of Differentiation Homework Packet In the table below, a function is given. Show the algebraic analysis that leads to the derivative of the function. Find the derivative by the specified
More informationa b c d e GOOD LUCK! 3. a b c d e 12. a b c d e 4. a b c d e 13. a b c d e 5. a b c d e 14. a b c d e 6. a b c d e 15. a b c d e
MA Elem. Calculus Fall 07 Exam 07-09- Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACT-approved calculator during the exam,
More information2.1 Limits, Rates of Change and Slopes of Tangent Lines
2.1 Limits, Rates of Change and Slopes of Tangent Lines (1) Average rate of change of y f x over an interval x 0,x 1 : f x 1 f x 0 x 1 x 0 Instantaneous rate of change of f x at x x 0 : f x lim 1 f x 0
More informationfunction independent dependent domain range graph of the function The Vertical Line Test
Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding
More informationAP CALCULUS AB SECTION I, Part A Time 55 Minutes Number of questions 28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM
AP CALCULUS AB SECTION I, Part A Time 55 Minutes Number of questions 28 Time Began: Time Ended: A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM Directions: Solve each of the following problems, using
More informationTOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12
NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus
More informationSection 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point
Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Line through P and Q approaches to the tangent line at P as Q approaches P. That is as a + h a = h gets smaller. Slope of the
More informationSections Practice AP Calculus AB Name
Sections 4.1-4.5 Practice AP Calculus AB Name Be sure to show work, giving written explanations when requested. Answers should be written exactly or rounded to the nearest thousandth. When the calculator
More information80 Wyner PreCalculus Spring 2017
80 Wyner PreCalculus Spring 2017 CHAPTER NINE: DERIVATIVES Review May 16 Test May 23 Calculus begins with the study of rates of change, called derivatives. For example, the derivative of velocity is acceleration
More information1. OBJECTIVE: Linear Equations
CUNY YORK COLLEGE FINAL EXAM REVIEW MATH 120: Precalculus Use the following questions to review for your final examimation for Math 120. Your ability to answer these questions will reflect what you learned
More informationAP Calculus Summer Assignment - Part 1
2017-2018 AP CALCULUS SUMMER ASSIGNMENT Linear Functions AP Calculus Summer Assignment - Part 1 Determine the equation of the line that passes through the given points 1. (0,-1) and (5,9) 2. (-2,-1) and
More informationAP Calculus BC Class Starter January 22, 2018
January 22, 2018 1. Given the function, find the following. (a) Evaluate f(4). (b) The definition of the derivative can be written two ways, as indicated below. Find both forms and evaluate the derivative
More informationAP Calculus Summer Prep
AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have
More informationRules for Differentiation Finding the Derivative of a Product of Two Functions. What does this equation of f '(
Rules for Differentiation Finding the Derivative of a Product of Two Functions Rewrite the function f( = ( )( + 1) as a cubic function. Then, find f '(. What does this equation of f '( represent, again?
More informationSince x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C)
SAT II - Math Level 2 Test #01 Solution 1. x + = 2, then x² + = Since x + = 2, by squaring both side of the equation, (A) - (B) 0 (C) 2 (D) 4 (E) -2 we get x² + 2x 1 + 1 = 4, or simplifying it, x² + 2
More informationProblems with an # after the number are the only ones that a calculator is required for in the solving process.
Instructions: Make sure all problems are numbered in order. (Level : If the problem had an *please skip that number) All work is in pencil, and is shown completely. Graphs are drawn out by hand. If you
More informationAP Calculus I Summer Packet
AP Calculus I Summer Packet This will be your first grade of AP Calculus and due on the first day of class. Please turn in ALL of your work and the attached completed answer sheet. I. Intercepts The -intercept
More informationYou should be comfortable with everything below (and if you aren t you d better brush up).
Review You should be comfortable with everything below (and if you aren t you d better brush up).. Arithmetic You should know how to add, subtract, multiply, divide, and work with the integers Z = {...,,,
More informationFind the domain and range of each function. Use interval notation (parenthesis or square brackets).
Page of 10 I. Functions & Composition of Functions A function is a set of points (x, y) such that for every x, there is one and only one y. In short, in a function, the x-values cannot repeat while the
More informationPolynomials and Rational Functions. Quadratic Equations and Inequalities. Remainder and Factor Theorems. Rational Root Theorem
Pre-Calculus Pre-AP Scope and Sequence - Year at a Glance Pre-Calculus Pre-AP - First Semester Pre-calculus with Limits; Larson/Hostetler Three Weeks 1 st 3 weeks 2 nd 3 weeks 3 rd 3 weeks 4 th 3 weeks
More informationConcepts of graphs of functions:
Concepts of graphs of functions: 1) Domain where the function has allowable inputs (this is looking to find math no-no s): Division by 0 (causes an asymptote) ex: f(x) = 1 x There is a vertical asymptote
More informationUNIT 3: DERIVATIVES STUDY GUIDE
Calculus I UNIT 3: Derivatives REVIEW Name: Date: UNIT 3: DERIVATIVES STUDY GUIDE Section 1: Section 2: Limit Definition (Derivative as the Slope of the Tangent Line) Calculating Rates of Change (Average
More informationREVIEW: MORE FUNCTIONS AP CALCULUS :: MR. VELAZQUEZ
REVIEW: MORE FUNCTIONS AP CALCULUS :: MR. VELAZQUEZ INVERSE FUNCTIONS Two functions are inverses if they undo each other. In other words, composing one function in the other will result in simply x (the
More informationPre-Calculus Exam 2009 University of Houston Math Contest. Name: School: There is no penalty for guessing.
Pre-Calculus Exam 009 University of Houston Math Contest Name: School: Please read the questions carefully and give a clear indication of your answer on each question. There is no penalty for guessing.
More informationFree Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom
Free Response Questions 1969-010 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS In this section, we will learn about: The purpose of mathematical models. MATHEMATICAL MODELS A mathematical
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3 x 9 D) 27. y 4 D) -8x 3 y 6.
Precalculus Review - Spring 018 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplify the exponential expression. Assume that variables represent
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationPre-Calculus MATH 119 Fall Section 1.1. Section objectives. Section 1.3. Section objectives. Section A.10. Section objectives
Pre-Calculus MATH 119 Fall 2013 Learning Objectives Section 1.1 1. Use the Distance Formula 2. Use the Midpoint Formula 4. Graph Equations Using a Graphing Utility 5. Use a Graphing Utility to Create Tables
More informationTest # 1 Review Math MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Test # 1 Review Math 13 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the slope of the curve at the given point P and an equation of the
More informationHonors Pre-calculus Midterm Review
Honors Pre-calculus Midterm Review Name: Chapter 1: Functions and Their Graphs 1. Evaluate the function f(x) = x 2 + 1 at each specified value of the independent variable and simplify. a. f( 3) b. f(x
More informationREQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS
REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS The Department of Applied Mathematics administers a Math Placement test to assess fundamental skills in mathematics that are necessary to begin the study
More informationAP Calculus BC Summer Review
AP Calculus BC 07-08 Summer Review Due September, 07 Name: All students entering AP Calculus BC are epected to be proficient in Pre-Calculus skills. To enhance your chances for success in this class, it
More informationEvaluating Limits Analytically. By Tuesday J. Johnson
Evaluating Limits Analytically By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Rationalizing numerators and/or denominators Trigonometric skills reviews
More informationReview Guideline for Final
Review Guideline for Final Here is the outline of the required skills for the final exam. Please read it carefully and find some corresponding homework problems in the corresponding sections to practice.
More informationReview for Cumulative Test 2
Review for Cumulative Test We will have our second course-wide cumulative test on Tuesday February 9 th or Wednesday February 10 th, covering from the beginning of the course up to section 4.3 in our textbook.
More informationFind the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3)
Final Exam Review AP Calculus AB Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Use the graph to evaluate the limit. 2) lim x
More information2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2
29 April PreCalculus Final Review 1. Find the slope and y-intercept (if possible) of the equation of the line. Sketch the line: y = 3x + 13 2. Determine the domain of the function. Verify your result with
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More information1.2 Mathematical Models: A Catalog of Essential Functions
1.2 Mathematical Models: A Catalog of Essential Functions A is a mathematical description of a real-world phenomenon. e.g., the size of a population, the demand for a product, the speed of a falling object,
More informationSummer Packet Greetings Future AP Calculus Scholar,
Summer Packet 2017 Greetings Future AP Calculus Scholar, I am excited about the work that we will do together during the 2016-17 school year. I do not yet know what your math capability is, but I can assure
More informationAP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm.
AP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm. Name: Date: Period: I. Limits and Continuity Definition of Average
More informationAP Calculus Summer Homework MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
AP Calculus Summer Homework 2015-2016 Part 2 Name Score MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the distance d(p1, P2) between the points
More informationCALCULUS ASSESSMENT REVIEW
CALCULUS ASSESSMENT REVIEW DEPARTMENT OF MATHEMATICS CHRISTOPHER NEWPORT UNIVERSITY 1. Introduction and Topics The purpose of these notes is to give an idea of what to expect on the Calculus Readiness
More information1 x. II. CHAPTER 2: (A) Graphing Rational Functions: Show Asymptotes using dotted lines, Intercepts, Holes(Coordinates, if any.)
FINAL REVIEW-014: Before using this review guide be sure to study your test and quizzes from this year. The final will contain big ideas from the first half of the year (chapters 1-) but it will be focused
More informationSolve the problem. Determine the center and radius of the circle. Use the given information about a circle to find its equation.
Math1314-TestReview2-Spring2016 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Is the point (-5, -3) on the circle defined
More informationMath 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS
Math 473: Practice Problems for Test 1, Fall 011, SOLUTIONS Show your work: 1. (a) Compute the Taylor polynomials P n (x) for f(x) = sin x and x 0 = 0. Solution: Compute f(x) = sin x, f (x) = cos x, f
More informationCurriculum Scope & Sequence
Book: Sullivan Pre-Calculus Enhanced with Graphing Utilities Subject/Grade Level: MATHEMATICS/HIGH SCHOOL Curriculum Scope & Sequence Course: PRE-CALCULUS CP/HONORS ***The goals and standards addressed
More informationCHAIN RULE: DAY 2 WITH TRIG FUNCTIONS. Section 2.4A Calculus AP/Dual, Revised /30/2018 1:44 AM 2.4A: Chain Rule Day 2 1
CHAIN RULE: DAY WITH TRIG FUNCTIONS Section.4A Calculus AP/Dual, Revised 018 viet.dang@humbleisd.net 7/30/018 1:44 AM.4A: Chain Rule Day 1 THE CHAIN RULE A. d dx f g x = f g x g x B. If f(x) is a differentiable
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationCalculus I. 1. Limits and Continuity
2301107 Calculus I 1. Limits and Continuity Outline 1.1. Limits 1.1.1 Motivation:Tangent 1.1.2 Limit of a function 1.1.3 Limit laws 1.1.4 Mathematical definition of a it 1.1.5 Infinite it 1.1. Continuity
More informationSummer Assignment MAT 414: Calculus
Summer Assignment MAT 414: Calculus Calculus - Math 414 Summer Assignment Due first day of school in September Name: 1. If f ( x) = x + 1, g( x) = 3x 5 and h( x) A. f ( a+ ) x+ 1, x 1 = then find: x+ 7,
More informationDirections: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.
AP Calculus AB SUMMER ASSIGNMENT Multiple Choice Section Directions: Please read questions carefully It is recommended that you do the Short Answer Section prior to doing the Multiple Choice Show all work
More information1 + x 2 d dx (sec 1 x) =
Page This exam has: 8 multiple choice questions worth 4 points each. hand graded questions worth 4 points each. Important: No graphing calculators! Any non-graphing, non-differentiating, non-integrating
More information1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.
Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope
More informationx x implies that f x f x.
Section 3.3 Intervals of Increase and Decrease and Extreme Values Let f be a function whose domain includes an interval I. We say that f is increasing on I if for every two numbers x 1, x 2 in I, x x implies
More informationPólya Enrichment Stage Table of Contents
Pólya Enrichment Stage Table of Contents George Pólya (1887-1985) v Preface ix Chapter 1. Functions 1 Chapter 2. Symmetric Polynomials 15 Chapter 3. Geometry 22 Chapter 4. Inequalities 34 Chapter 5. Functional
More informationAP Calculus BC. Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines
AP Calculus BC Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines Essential Questions & Why: Essential Questions: What is the difference between average and instantaneous rates of
More informationAP Calculus Review Assignment Answer Sheet 1. Name: Date: Per. Harton Spring Break Packet 2015
AP Calculus Review Assignment Answer Sheet 1 Name: Date: Per. Harton Spring Break Packet 015 This is an AP Calc Review packet. As we get closer to the eam, it is time to start reviewing old concepts. Use
More informationThe function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and
Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous
More informationDirections: This is a final exam review which covers all of the topics of the course. Please use this as a guide to assist you in your studies.
MATH 1113 Precalculus FINAL EXAM REVIEW irections: This is a final exam review which covers all of the topics of the course. Please use this as a guide to assist you in your studies. Question: 1 QI: 758
More informationChapter 3 Derivatives
Chapter Derivatives Section 1 Derivative of a Function What you ll learn about The meaning of differentiable Different ways of denoting the derivative of a function Graphing y = f (x) given the graph of
More informationPrecalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear.
Precalculus Review Functions to KNOW! 1. Polynomial Functions Types: General form Generic Graph and unique properties Constants Linear Quadratic Cubic Generalizations for Polynomial Functions - The domain
More informationChapter 2 THE DERIVATIVE
Chapter 2 THE DERIVATIVE 2.1 Two Problems with One Theme Tangent Line (Euclid) A tangent is a line touching a curve at just one point. - Euclid (323 285 BC) Tangent Line (Archimedes) A tangent to a curve
More informationSection 1.1: A Preview of Calculus When you finish your homework, you should be able to
Section 1.1: A Preview of Calculus When you finish your homework, you should be able to π Understand what calculus is and how it compares with precalculus π Understand that the tangent line problem is
More informationSection 6.2 Notes Page Trigonometric Functions; Unit Circle Approach
Section Notes Page Trigonometric Functions; Unit Circle Approach A unit circle is a circle centered at the origin with a radius of Its equation is x y = as shown in the drawing below Here the letter t
More informationTrigonometric Identities Exam Questions
Trigonometric Identities Exam Questions Name: ANSWERS January 01 January 017 Multiple Choice 1. Simplify the following expression: cos x 1 cot x a. sin x b. cos x c. cot x d. sec x. Identify a non-permissible
More informationMATH 2053 Calculus I Review for the Final Exam
MATH 05 Calculus I Review for the Final Exam (x+ x) 9 x 9 1. Find the limit: lim x 0. x. Find the limit: lim x + x x (x ).. Find lim x (x 5) = L, find such that f(x) L < 0.01 whenever 0 < x
More informationContinuity. The Continuity Equation The equation that defines continuity at a point is called the Continuity Equation.
Continuity A function is continuous at a particular x location when you can draw it through that location without picking up your pencil. To describe this mathematically, we have to use limits. Recall
More information