Math 103: Secants, Tangents and Derivatives
|
|
|
- Kimberly Nichols
- 7 years ago
- Views:
Transcription
1 Math 103: Secants, Tangents and Derivatives Ryan Blair University of Pennsylvania Thursday September 27, 2011 Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
2 Outline 1 Review 2 Secant Lines and Tangent Lines 3 Derivative Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
3 Review Limits Involving Infinity 1 Limits as x approaches or. 2 Limits at infinite discontinuities 3 Horizontal asymptotes 4 Vertical asymptotes Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
4 Horizontal asymptotes Review A line y = b is a horizontal asymptote of the graph of y = f (x) if lim x f (x) = b or lim x f (x) = b Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
5 Horizontal asymptotes Review A line y = b is a horizontal asymptote of the graph of y = f (x) if lim x f (x) = b or lim x f (x) = b Example: Find the horozontal asymptotes of y = x+1+sin(x) x Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
6 Secant lines Secant Lines and Tangent Lines A line in the plane is a secant line to a circle if it meets the circle in exactly two points. A line in the plane is a secant line to the graph of y=f(x) if it meets the graph of y = f (x) in at least two points. Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
7 Secant lines Secant Lines and Tangent Lines A line in the plane is a secant line to a circle if it meets the circle in exactly two points. A line in the plane is a secant line to the graph of y=f(x) if it meets the graph of y = f (x) in at least two points. Find the secant line to y = x 3 2x + 1 between x = 0 and x = 1. Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
8 Tangent lines Secant Lines and Tangent Lines A line in the plane is a tangent line to a circle if it meets the circle in exactly one point. Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
9 Tangent lines Secant Lines and Tangent Lines A line in the plane is a tangent line to a circle if it meets the circle in exactly one point. A line in the plane is a tangent line to the graph of y=f(x) if it meets the graph of y = f (x) in exactly one point locally and it kisses the graph at that point Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
10 Tangent lines Secant Lines and Tangent Lines A line in the plane is a tangent line to a circle if it meets the circle in exactly one point. A line in the plane is a tangent line to the graph of y=f(x) if it meets the graph of y = f (x) in exactly one point locally and it kisses the graph at that point kisses = (near the point of intersection the graph is completely contain to one side of the tangent line) Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
11 Secant Lines and Tangent Lines Since we don t have two points we need a new idea to find the slope of a tangent line. Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
12 Secant Lines and Tangent Lines Since we don t have two points we need a new idea to find the slope of a tangent line. Find the slope of secant lines to f (x) = 1 on the 1+x following intervals: 1 [1, 3] 2 [1, 2] 3 [1, 1.5] Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
13 Secant Lines and Tangent Lines Since we don t have two points we need a new idea to find the slope of a tangent line. Find the slope of secant lines to f (x) = 1 on the 1+x following intervals: 1 [1, 3] 2 [1, 2] 3 [1, 1.5] Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
14 Secant Lines and Tangent Lines Since we don t have two points we need a new idea to find the slope of a tangent line. Find the slope of secant lines to f (x) = 1 on the 1+x following intervals: 1 [1, 3] 2 [1, 2] 3 [1, 1.5] Tangent lines are the limits of secant lines Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
15 Tangent Lines Secant Lines and Tangent Lines The tangent line to a curve y = f (x) at a point (a, f (a)) is the line through (a, f (a)) with the slope lim h 0 f (a + h) f (a) h Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
16 Tangent Lines Secant Lines and Tangent Lines The tangent line to a curve y = f (x) at a point (a, f (a)) is the line through (a, f (a)) with the slope lim h 0 f (a + h) f (a) h Find the slope of the line tangent to y = sin(x) at x = 0. Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
17 Application: Instantaneous Velocity If s(t) is a position function defined in terms of time t, then the instantaneous velocity at time t = a is given by v(a) = lim h 0 s(a + h) s(a) h Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
18 Application: Instantaneous Velocity If s(t) is a position function defined in terms of time t, then the instantaneous velocity at time t = a is given by v(a) = lim h 0 s(a + h) s(a) h ExampleSuppose a penny is dropped from the top of DRL which is 19.6 meters high. The position of the penny in terms of hight above the street is given by s(t) = t 2. At what speed is the penny traveling when it hits the ground. Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, / 11
19 Derivative as a function Given any function f (x) we want to find a new function that, for any x-value, outputs the slope of f (x) at that value. Ryan Blair (U Penn) Math 103: Secants, Tangents and DerivativesThursday September 27, / 11
20 Derivative as a function Given any function f (x) we want to find a new function that, for any x-value, outputs the slope of f (x) at that value. f (x) = lim h 0 f (x + h) f (x) h Ryan Blair (U Penn) Math 103: Secants, Tangents and DerivativesThursday September 27, / 11
21 Derivative as a function Given any function f (x) we want to find a new function that, for any x-value, outputs the slope of f (x) at that value. f (x) = lim h 0 f (x + h) f (x) h Notation.Other ways of writing the derivative of y = f (x). f (x) = y = dy dx = df dx = d dx f (x) = Df (x) = D xf (x) Ryan Blair (U Penn) Math 103: Secants, Tangents and DerivativesThursday September 27, / 11
22 The Sandwich Theorem Theorem If f (x) g(x) h(x) when x is near c and lim x c f (x) = lim x c h(x) = L then lim x c g(x) = L Ryan Blair (U Penn) Math 103: Secants, Tangents and DerivativesThursday September 27, / 11
Average rate of change of y = f(x) with respect to x as x changes from a to a + h:
L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,
Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
Lecture 3: Derivatives and extremes of functions
Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16
Average rate of change
Average rate of change 1 1 Average rate of change A fundamental philosophical truth is that everything changes. 1 Average rate of change A fundamental philosophical truth is that everything changes. In
Calculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim
correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
Microeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
= f x 1 + h. 3. Geometrically, the average rate of change is the slope of the secant line connecting the pts (x 1 )).
Math 1205 Calculus/Sec. 3.3 The Derivative as a Rates of Change I. Review A. Average Rate of Change 1. The average rate of change of y=f(x) wrt x over the interval [x 1, x 2 ]is!y!x ( ) - f( x 1 ) = y
MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010
MATH 11 FINAL EXAM FALL 010-011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic
AP CALCULUS AB 2009 SCORING GUIDELINES
AP CALCULUS AB 2009 SCORING GUIDELINES Question 5 x 2 5 8 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 in
Section 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
2.2 Derivative as a Function
2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x
PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
2.1: The Derivative and the Tangent Line Problem
.1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position
18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 8.0 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Unit : Derivatives A. What
Instantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:
AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be
Mark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
Section 1: Instantaneous Rate of Change and Tangent Lines Instantaneous Velocity
Chapter 2 The Derivative Business Calculus 74 Section 1: Instantaneous Rate of Change and Tangent Lines Instantaneous Velocity Suppose we drop a tomato from the top of a 100 foot building and time its
AP Calculus BC 2006 Free-Response Questions
AP Calculus BC 2006 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to
x 2 y 2 +3xy ] = d dx dx [10y] dy dx = 2xy2 +3y
![x 2 y 2 +3xy ] = d dx dx [10y] dy dx = 2xy2 +3y](/thumbs/40/21392919.jpg "x 2 y 2 +3xy ] = d dx dx [10y] dy dx = 2xy2 +3y")
MA7 - Calculus I for thelife Sciences Final Exam Solutions Spring -May-. Consider the function defined implicitly near (,) byx y +xy =y. (a) [7 points] Use implicit differentiation to find the derivative
Calculus. Contents. Paul Sutcliffe. Office: CM212a.
Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical
Slope and Rate of Change
Chapter 1 Slope and Rate of Change Chapter Summary and Goal This chapter will start with a discussion of slopes and the tangent line. This will rapidly lead to heuristic developments of limits and the
x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
AP Calculus AB 2004 Free-Response Questions
AP Calculus AB 2004 Free-Response Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
2.2. Instantaneous Velocity
2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical
AP Calculus BC 2013 Free-Response Questions
AP Calculus BC 013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in
f(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
Tangent Lines and Rates of Change
Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
The Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)
Calculus AB 2014 Scoring Guidelines
P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official
Problems 1-21 could be on the no Derive part. Sections 1.2, 2.2, 2.3, 3.1, 3.3, 3.4, 4.1, 4.2
MTH 120 Practice Test #1 Sections 1.2, 2.2, 2.3, 3.1, 3.3, 3.4, 4.1, 4.2 Use the properties of limits to help decide whether the limit eists. If the limit eists, find its value. 1) lim 5 2) lim 3 2-25
2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
MATH 221 FIRST SEMESTER CALCULUS. fall 2007
MATH 22 FIRST SEMESTER CALCULUS fall 2007 Typeset:December, 2007 2 Math 22 st Semester Calculus Lecture notes version.0 (Fall 2007) This is a self contained set of lecture notes for Math 22. The notes
Math 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
AP Calculus BC 2001 Free-Response Questions
AP Calculus BC 001 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must
Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
Mathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
Section 5.4 More Trigonometric Graphs. Graphs of the Tangent, Cotangent, Secant, and Cosecant Function
Section 5. More Trigonometric Graphs Graphs of the Tangent, Cotangent, Secant, and Cosecant Function 1 REMARK: Many curves have a U shape near zero. For example, notice that the functions secx and x +
MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
APPLYING PRE-CALCULUS/CALCULUS SHARP EL-9600. Graphing Calculator DAVID P. LAWRENCE
APPLYING PRE-CALCULUS/CALCULUS SHARP U S I N G T H E EL-9600 Graphing Calculator DAVID P. LAWRENCE Applying PRE-CALCULUS/CALCULUS using the SHARP EL-9600 GRAPHING CALCULATOR David P. Lawrence Southwestern
MATH 221 FIRST SEMESTER CALCULUS. fall 2009
MATH 22 FIRST SEMESTER CALCULUS fall 2009 Typeset:June 8, 200 MATH 22 st SEMESTER CALCULUS LECTURE NOTES VERSION 2.0 (fall 2009) This is a self contained set of lecture notes for Math 22. The notes were
Exercises and Problems in Calculus. John M. Erdman Portland State University. Version August 1, 2013
Exercises and Problems in Calculus John M. Erdman Portland State University Version August 1, 2013 c 2010 John M. Erdman E-mail address: erdman@pdx.edu Contents Preface ix Part 1. PRELIMINARY MATERIAL
100. In general, we can define this as if b x = a then x = log b
Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,
If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule.
Oriented Surfaces and Flux Integrals Let be a surface that has a tangent plane at each of its nonboundary points. At such a point on the surface two unit normal vectors exist, and they have opposite directions.
CK-12 Geometry: Parts of Circles and Tangent Lines
CK-12 Geometry: Parts of Circles and Tangent Lines Learning Objectives Define circle, center, radius, diameter, chord, tangent, and secant of a circle. Explore the properties of tangent lines and circles.
(b)using the left hand end points of the subintervals ( lower sums ) we get the aprroximation
(1) Consider the function y = f(x) =e x on the interval [, 1]. (a) Find the area under the graph of this function over this interval using the Fundamental Theorem of Calculus. (b) Subdivide the interval
How To Understand The Theory Of Algebraic Functions
Homework 4 3.4,. Show that x x cos x x holds for x 0. Solution: Since cos x, multiply all three parts by x > 0, we get: x x cos x x, and since x 0 x x 0 ( x ) = 0, then by Sandwich theorem, we get: x 0
Graphing calculators Transparencies (optional)
What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity Teacher Version Lesson Objective: Length of Activity: Students will: Recognize piecewise functions and the notation used
Derivatives as Rates of Change
Derivatives as Rates of Change One-Dimensional Motion An object moving in a straight line For an object moving in more complicated ways, consider the motion of the object in just one of the three dimensions
AP Calculus AB 2005 Free-Response Questions
AP Calculus AB 25 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to
MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions
MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial
Calculus 1: Sample Questions, Final Exam, Solutions
Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.
x), etc. In general, we have
BASIC CALCULUS REFRESHER. Introduction. Ismor Fischer, Ph.D. Dept. of Statistics UW-Madison This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in
Definition of derivative
Definition of derivative Contents 1. Slope-The Concept 2. Slope of a curve 3. Derivative-The Concept 4. Illustration of Example 5. Definition of Derivative 6. Example 7. Extension of the idea 8. Example
Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
f(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =
Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))
The Derivative. Philippe B. Laval Kennesaw State University
The Derivative Philippe B. Laval Kennesaw State University Abstract This handout is a summary of the material students should know regarding the definition and computation of the derivative 1 Definition
Readings this week. 1 Parametric Equations Supplement. 2 Section 10.1. 3 Sections 2.1-2.2. Professor Christopher Hoffman Math 124
Readings this week 1 Parametric Equations Supplement 2 Section 10.1 3 Sections 2.1-2.2 Precalculus Review Quiz session Thursday equations of lines and circles worksheet available at http://www.math.washington.edu/
5-3 Polynomial Functions. not in one variable because there are two variables, x. and y
y. 5-3 Polynomial Functions State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. 11x 6 5x 5 + 4x 2 coefficient of the
AP Calculus AB 2006 Scoring Guidelines
AP Calculus AB 006 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college
Good Questions. Answer: (a). Both f and g are given by the same rule, and are defined on the same domain, hence they are the same function.
Good Questions Limits 1. [Q] Let f be the function defined by f(x) = sin x + cos x and let g be the function defined by g(u) = sin u + cos u, for all real numbers x and u. Then, (a) f and g are exactly
Area and Arc Length in Polar Coordinates
Area and Arc Length in Polar Coordinates The Cartesian Coordinate System (rectangular coordinates) is not always the most convenient way to describe points, or relations in the plane. There are certainly
Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1)
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. These are: If x(t) represents the
Objective: Use calculator to comprehend transformations.
math111 (Bradford) Worksheet #1 Due Date: Objective: Use calculator to comprehend transformations. Here is a warm up for exploring manipulations of functions. specific formula for a function, say, Given
Circle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving
MATH 34A REVIEW FOR MIDTERM 2, WINTER 2012. 1. Lines. (1) Find the equation of the line passing through (2,-1) and (-2,9). y = 5
MATH 34A REVIEW FOR MIDTERM 2, WINTER 2012 ANSWERS 1. Lines (1) Find the equation of the line passing through (2,-1) and (-2,9). y = 5 2 x + 4. (2) Find the equation of the line which meets the x-axis
Visualizing Differential Equations Slope Fields. by Lin McMullin
Visualizing Differential Equations Slope Fields by Lin McMullin The topic of slope fields is new to the AP Calculus AB Course Description for the 2004 exam. Where do slope fields come from? How should
Scalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
An Introduction to Calculus. Jackie Nicholas
Mathematics Learning Centre An Introduction to Calculus Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 Some rules of differentiation and how to use them
3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?
Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using
SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions
SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions All questions in the Math Level 1 and Math Level Tests are multiple-choice questions in which you are asked to choose the
Single Variable Calculus. Early Transcendentals
Single Variable Calculus Early Transcendentals This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License To view a copy of this license, visit http://creativecommonsorg/licenses/by-nc-sa/30/
1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
Week 1: Functions and Equations
Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter
Rolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
Birmingham City Schools
Activity 1 Classroom Rules & Regulations Policies & Procedures Course Curriculum / Syllabus LTF Activity: Interval Notation (Precal) 2 Pre-Assessment 3 & 4 1.2 Functions and Their Properties 5 LTF Activity:
Algebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
AP Calculus AB 2011 Scoring Guidelines
AP Calculus AB Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 9, the
Also, compositions of an exponential function with another function are also referred to as exponential. An example would be f(x) = 4 + 100 3-2x.
Exponential Functions Exponential functions are perhaps the most important class of functions in mathematics. We use this type of function to calculate interest on investments, growth and decline rates
AP Calculus AB 2013 Free-Response Questions
AP Calculus AB 2013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded
Higher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
6 Further differentiation and integration techniques
56 6 Further differentiation and integration techniques Here are three more rules for differentiation and two more integration techniques. 6.1 The product rule for differentiation Textbook: Section 2.7
Functions. MATH 160, Precalculus. J. Robert Buchanan. Fall 2011. Department of Mathematics. J. Robert Buchanan Functions
Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: determine whether relations between variables are functions, use function
MULTIVARIATE PROBABILITY DISTRIBUTIONS
MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined
Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:
![Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:](/thumbs/27/11099194.jpg "Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:")
Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes
Trees Saved: 5 Air Emissions Eliminated: 1,104 pounds Water Saved: 1,082 gallons Solid Waste Eliminated: 326 pounds
Planet Friendly Publishing Made in the United States Printed on Recycled Paper Text: 10% Cover: 10% Learn more: www.greenedition.org At REA we re committed to producing books in an earth-friendly manner
AP Calculus BC Exam. The Calculus BC Exam. At a Glance. Section I. SECTION I: Multiple-Choice Questions. Instructions. About Guessing.
The Calculus BC Exam AP Calculus BC Exam SECTION I: Multiple-Choice Questions At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Grade 50% Writing Instrument Pencil required
Solutions to Practice Problems for Test 4
olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,