DOCSLIB.ORG

Chapter One 1.1 Linear Functions & the Tangent Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Chapter One Limits and Rates of Change 1.1 Linear Functions & the Tangent Problem Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT A linear function is a function f of the form f (x) = mx + b, m and b are constants This equation has a slope of m & a y-intercept of b. Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT The slope of a non-vertical line that passes through the points P1 (x1, y1 ) and P2 (x2 , y2 ) is defined by Since the slope is the ratio of the change in y (∆y) to the change !y y " y in x (∆x). m = = 2 1 !x x2 " x1 In calculus, we interpret this to be The rate of change of y with respect to x. Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT Find a linear equation whose graph passes through the points (-2,-2) and (3,4) ! !y y2 " y1 4 ! 2 4 + 2 6 m = = = ! = = !x x2 " x1 3! 2 3+ 2 5 In order to find an equation of any line we need the slope of the line and one point. We now have both of these: 6 m = and either point (-2,-2) or (3,4) 5 Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT Find a linear equation whose graph passes through the points (-2,-2) and (3,4) Equation of a line is defined as: y ! y1 = m(x ! x1 ) where m is slope and x1 & y1 are fixed co-ordinates on the line 6 6 m = and point (3,4) m = and point (-2,-2) 5 5 6 6 y ! 2 x ! 2 y ! 4 = (x ! 3) ! = ( ! ) 5 5 6 18 6 12 y ! 4 = x ! y + 2 = x + 5 5 5 5 20 10 +4 + -2 - 5 5 6 2 6 2 y = x + y= x + 5 5 5 5 Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT The concept of rate of change is critical to our understanding of calculus! Rate of change explains several concepts in mathematics and science. The simplest and easiest example is slope. How does x change as y changes… A linear function is given by y = 6 - 5x. If x increases by 2, how does y change? !y We are given that = " 5 !x " !y% ( !x$ ' = 5 !x The rate of change is # !x& ( ) !y " slope = = " 5 !y = 5!x !x Since !x = 2 Therefore we can state: + " !x = 2 !y = 5(2) When x increases by 2 !y = "10 y will decrease by 10. Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT The Tangent Problem The word tangent comes from the latin word tangens, which means touching. For a simple curve, such as a circle, a tangent is a line that intersects the circle once and only once. Things become more complicated when we introduce other curves. The curve C at right has a single point P that has two lines through it. Line t is tangent at P but intersects the curve at another point (still a tangent line at P). Line l intersects at P but it is not considered tangent because of its intersection (not touching) Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT Find the equation of the tangent line to the parabola y=x2 at the point P(1,1). We can find the equation of any line as long as we have slope m and a point (x,y). Our problem is we only have a point. In order to find slope, by definition, we need to find the rise over run between two points. The process we go through is to use a set of second points, Q(x,y) on the same curve, close to P(1,1). By calculating the slopes of such lines, known as secant lines, we can approximate the slope of the tangent line. Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT Find the equation of the tangent line to the parabola y = x2 at the point P(1,1). We choose a point x≠1 so that our slope is defined and P≠Q. But since Q lies on the curve y = x2, we can substitute x2 for y y !1 x2 !1 mmPPQQ == m = x !1 PQ x !1 For instance, for the point Q, where x = 1.1, the slope would be 2 2 x !1 1.1 !1 1.21!1 .21 Slope of m = = = = = 2.1 PQ x !1 1.1!1 1.1!1 .1 secant line Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT Find the equation of the tangent line to the parabola y = x2 at the point P(1,1). The following table uses the previous method to determine several values of x close to 1 for mPQ x > 1 mPQ x < 1 mPQ 2 3 0 1 1.5 2.5 .5 1.5 1.1 2.1 .9 1.9 1.01 2.01 .99 1.99 1.001 2.001 .999 1.999 We can now determine that the slope of secant lines Gavin Denham Tuesday, January 28,close 2014 2:27:17 PMto MT x = 1 seem to have a slope that approaches 2 Find the equation of the tangent line to the parabola y=x2 at the point P(1,1). We can now say that the slope of the y = 2x ! 1 line tangent at x=1 has a slope of 2. m = 2 and point (1,1) y ! y1 = m(x ! x1 ) y !1 = 2(x !1) The equation of the tangent line y !1 = 2x ! 2 at the point (1,1) on the curve y = x2 +1 +1 is y = 2x - 1 y = 2x !1 Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT Page 9 Exercises 1-10 odd letters (a,c,e…) Gavin Denham Tuesday, January 28, 2014 2:27:17 PM MT.
Recommended publications
  • Algebra II Level 1

    Algebra II Level 1

    Algebra II 5 credits – Level I Grades: 10-11, Level: I Prerequisite: Minimum grade of 70 in Geometry Level I (or a minimum grade of 90 in Geometry Topics) Units of study include: equations and inequalities, linear equations and functions, systems of linear equations and inequalities, matrices and determinants, quadratic functions, polynomials and polynomial functions, powers, roots, and radicals, rational equations and functions, sequence and series, and probability and statistics. PROFICIENCIES INEQUALITIES -Solve inequalities -Solve combined inequalities -Use inequality models to solve problems -Solve absolute values in open sentences -Solve absolute value sentences graphically LINEAR EQUATION FUNCTIONS -Solve open sentences in two variables -Graph linear equations in two variables -Find the slope of the line -Write the equation of a line -Solve systems of linear equations in two variables -Apply systems of equations to solve real word problems -Solve linear inequalities in two variables -Find values of functions and graphs -Find equations of linear functions and apply properties of linear functions -Graph relations and determine when relations are functions PRODUCTS AND FACTORS OF POLYNOMIALS -Simplify, add and subtract polynomials -Use laws of exponents to multiply a polynomial by a monomial -Calculate the product of two or more polynomials -Demonstrate factoring -Solve polynomial equations and inequalities -Apply polynomial equations to solve real world problems RATIONAL EQUATIONS -Simplify quotients using laws of exponents -Simplify
  • The Linear Algebra Version of the Chain Rule 1

    The Linear Algebra Version of the Chain Rule 1

    Ralph M Kaufmann The Linear Algebra Version of the Chain Rule 1 Idea The differential of a differentiable function at a point gives a good linear approximation of the function – by definition. This means that locally one can just regard linear functions. The algebra of linear functions is best described in terms of linear algebra, i.e. vectors and matrices. Now, in terms of matrices the concatenation of linear functions is the matrix product. Putting these observations together gives the formulation of the chain rule as the Theorem that the linearization of the concatenations of two functions at a point is given by the concatenation of the respective linearizations. Or in other words that matrix describing the linearization of the concatenation is the product of the two matrices describing the linearizations of the two functions. 1. Linear Maps Let V n be the space of n–dimensional vectors. 1.1. Definition. A linear map F : V n → V m is a rule that associates to each n–dimensional vector ~x = hx1, . xni an m–dimensional vector F (~x) = ~y = hy1, . , yni = hf1(~x),..., (fm(~x))i in such a way that: 1) For c ∈ R : F (c~x) = cF (~x) 2) For any two n–dimensional vectors ~x and ~x0: F (~x + ~x0) = F (~x) + F (~x0) If m = 1 such a map is called a linear function. Note that the component functions f1, . , fm are all linear functions. 1.2. Examples. 1) m=1, n=3: all linear functions are of the form y = ax1 + bx2 + cx3 for some a, b, c ∈ R.
  • The Computational Complexity of Polynomial Factorization

    The Computational Complexity of Polynomial Factorization

    Open Questions From AIM Workshop: The Computational Complexity of Polynomial Factorization 1 Dense Polynomials in Fq[x] 1.1 Open Questions 1. Is there a deterministic algorithm that is polynomial time in n and log(q)? State of the art algorithm should be seen on May 16th as a talk. 2. How quickly can we factor x2−a without hypotheses (Qi Cheng's question) 1 State of the art algorithm Burgess O(p 2e ) 1.2 Open Question What is the exponent of the probabilistic complexity (for q = 2)? 1.3 Open Question Given a set of polynomials with very low degree (2 or 3), decide if all of them factor into linear factors. Is it faster to do this by factoring their product or each one individually? (Tanja Lange's question) 2 Sparse Polynomials in Fq[x] 2.1 Open Question α β Decide whether a trinomial x + ax + b with 0 < β < α ≤ q − 1, a; b 2 Fq has a root in Fq, in polynomial time (in log(q)). (Erich Kaltofen's Question) 2.2 Open Questions 1. Finding better certificates for irreducibility (a) Are there certificates for irreducibility that one can verify faster than the existing irreducibility tests? (Victor Miller's Question) (b) Can you exhibit families of sparse polynomials for which you can find shorter certificates? 2. Find a polynomial-time algorithm in log(q) , where q = pn, that solves n−1 pi+pj n−1 pi Pi;j=0 ai;j x +Pi=0 bix +c in Fq (or shows no solutions) and works 1 for poly of the inputs, and never lies.
  • Section P.2 Solving Equations Important Vocabulary

    Section P.2 Solving Equations Important Vocabulary

    Section P.2 Solving Equations 5 Course Number Section P.2 Solving Equations Instructor Objective: In this lesson you learned how to solve linear and nonlinear equations. Date Important Vocabulary Define each term or concept. Equation A statement, usually involving x, that two algebraic expressions are equal. Extraneous solution A solution that does not satisfy the original equation. Quadratic equation An equation in x that can be written in the general form 2 ax + bx + c = 0 where a, b, and c are real numbers with a ¹ 0. I. Equations and Solutions of Equations (Page 12) What you should learn How to identify different To solve an equation in x means to . find all the values of x types of equations for which the solution is true. The values of x for which the equation is true are called its solutions . An identity is . an equation that is true for every real number in the domain of the variable. A conditional equation is . an equation that is true for just some (or even none) of the real numbers in the domain of the variable. II. Linear Equations in One Variable (Pages 12-14) What you should learn A linear equation in one variable x is an equation that can be How to solve linear equations in one variable written in the standard form ax + b = 0 , where a and b and equations that lead to are real numbers with a ¹ 0 . linear equations A linear equation has exactly one solution(s). To solve a conditional equation in x, . isolate x on one side of the equation by a sequence of equivalent, and usually simpler, equations, each having the same solution(s) as the original equation.
  • Vocabulary Eighth Grade Math Module Four Linear Equations

    Vocabulary Eighth Grade Math Module Four Linear Equations

    VOCABULARY EIGHTH GRADE MATH MODULE FOUR LINEAR EQUATIONS Coefficient: A number used to multiply a variable. Example: 6z means 6 times z, and "z" is a variable, so 6 is a coefficient. Equation: An equation is a statement about equality between two expressions. If the expression on the left side of the equal sign has the same value as the expression on the right side of the equal sign, then you have a true equation. Like Terms: Terms whose variables and their exponents are the same. Example: 7x and 2x are like terms because the variables are both “x”. But 7x and 7ͬͦ are NOT like terms because the x’s don’t have the same exponent. Linear Expression: An expression that contains a variable raised to the first power Solution: A solution of a linear equation in ͬ is a number, such that when all instances of ͬ are replaced with the number, the left side will equal the right side. Term: In Algebra a term is either a single number or variable, or numbers and variables multiplied together. Terms are separated by + or − signs Unit Rate: A unit rate describes how many units of the first type of quantity correspond to one unit of the second type of quantity. Some common unit rates are miles per hour, cost per item & earnings per week Variable: A symbol for a number we don't know yet. It is usually a letter like x or y. Example: in x + 2 = 6, x is the variable. Average Speed: Average speed is found by taking the total distance traveled in a given time interval, divided by the time interval.
  • The Evolution of Equation-Solving: Linear, Quadratic, and Cubic

    The Evolution of Equation-Solving: Linear, Quadratic, and Cubic

    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2006 The evolution of equation-solving: Linear, quadratic, and cubic Annabelle Louise Porter Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Porter, Annabelle Louise, "The evolution of equation-solving: Linear, quadratic, and cubic" (2006). Theses Digitization Project. 3069. https://scholarworks.lib.csusb.edu/etd-project/3069 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EVOLUTION OF EQUATION-SOLVING LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degre Master of Arts in Teaching: Mathematics by Annabelle Louise Porter June 2006 THE EVOLUTION OF EQUATION-SOLVING: LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino by Annabelle Louise Porter June 2006 Approved by: Shawnee McMurran, Committee Chair Date Laura Wallace, Committee Member , (Committee Member Peter Williams, Chair Davida Fischman Department of Mathematics MAT Coordinator Department of Mathematics ABSTRACT Algebra and algebraic thinking have been cornerstones of problem solving in many different cultures over time. Since ancient times, algebra has been used and developed in cultures around the world, and has undergone quite a bit of transformation. This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work.
  • 2.3 Solving Linear Equations a Linear Equation Is an Equation in Which All Variables Are Raised to Only the 1 Power. in This

    2.3 Solving Linear Equations a Linear Equation Is an Equation in Which All Variables Are Raised to Only the 1 Power. in This

    2.3 Solving Linear Equations A linear equation is an equation in which all variables are raised to only the 1st power. In this section, we review the algebra of solving basic linear equations, see how these skills can be used in applied settings, look at linear equations with more than one variable, and then use this to find inverse functions of a linear function. Basic Linear Equations To solve a basic linear equation in one variable we use the basic operations of +, -, ´, ¸ to isolate the variable. Example 1 Solve each of the following: (a) 2x + 4 = 5x -12 (b) 3(2x - 4) = 2(x + 5) Applied Problems Example 2 Company A charges a monthly fee of $5.00 plus 99 cents per download for songs. Company B charges a monthly fee of $7.00 plus 79 cents per download for songs. (a) Give a formula for the monthly cost with each of these companies. (b) For what number of downloads will the monthly cost be the same for both companies? Example 3 A small business is considering hiring a new sales rep. to market its product in a nearby city. Two pay scales are under consideration. Pay Scale 1 Pay the sales rep. a base yearly salary of $10000 plus a commission of 8% of the total yearly sales. Pay Scale 2 Pay the sales rep a base yearly salary of $13000 plus a commission of 6% of the total yearly sales. (a) For each scale above, give a function formula to express the total yearly earnings as a function of the total yearly sales.
  • Characterization of Non-Differentiable Points in a Function by Fractional Derivative of Jumarrie Type

    Characterization of Non-Differentiable Points in a Function by Fractional Derivative of Jumarrie Type

    Characterization of non-differentiable points in a function by Fractional derivative of Jumarrie type Uttam Ghosh (1), Srijan Sengupta(2), Susmita Sarkar (2), Shantanu Das (3) (1): Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India; Email: [email protected] (2):Department of Applied Mathematics, Calcutta University, Kolkata, India Email: [email protected] (3)Scientist H+, RCSDS, BARC Mumbai India Senior Research Professor, Dept. of Physics, Jadavpur University Kolkata Adjunct Professor. DIAT-Pune Ex-UGC Visiting Fellow Dept. of Applied Mathematics, Calcutta University, Kolkata India Email (3): [email protected] The Birth of fractional calculus from the question raised in the year 1695 by Marquis de L'Hopital to Gottfried Wilhelm Leibniz, which sought the meaning of Leibniz's notation for the derivative of order N when N = 1/2. Leibnitz responses it is an apparent paradox from which one day useful consequences will be drown. Abstract There are many functions which are continuous everywhere but not differentiable at some points, like in physical systems of ECG, EEG plots, and cracks pattern and for several other phenomena. Using classical calculus those functions cannot be characterized-especially at the non- differentiable points. To characterize those functions the concept of Fractional Derivative is used. From the analysis it is established that though those functions are unreachable at the non- differentiable points, in classical sense but can be characterized using Fractional derivative. In this paper we demonstrate use of modified Riemann-Liouvelli derivative by Jumarrie to calculate the fractional derivatives of the non-differentiable points of a function, which may be one step to characterize and distinguish and compare several non-differentiable points in a system or across the systems.
  • Analysis of XSL Applied to BES

    Analysis of XSL Applied to BES

    Analysis of XSL Applied to BES By: Lim Chu Wee, Khoo Khoong Ming. History (2002) Courtois and Pieprzyk announced a plausible attack (XSL) on Rijndael AES. Complexity of ≈ 2225 for AES-256. Later Murphy and Robshaw proposed embedding AES into BES, with equations over F256. S-boxes involved fewer monomials, and would provide a speedup for XSL if it worked (287 for AES-128 in best case). Murphy and Robshaw also believed XSL would not work. (Asiacrypt 2005) Cid and Leurent showed that “compact XSL” does not crack AES. Summary of Our Results We analysed the application of XSL on BES. Concluded: the estimate of 287 was too optimistic; we obtained a complexity ≥ 2401, even if XSL works. Hence it does not crack BES-128. Found further linear dependencies in the expanded equations, upon applying XSL to BES. Similar dependencies exist for AES – unaccounted for in computations of Courtois and Pieprzyk. Open question: does XSL work at all, for some P? Quick Description of AES & BES AES Structure Very general description of AES (in F256): Input: key (k0k1…ks-1), message (M0M1…M15). Suppose we have aux variables: v0, v1, …. At each step we can do one of three things: Let vi be an F2-linear map T of some previously defined byte: one of the vj’s, kj’s or Mj’s. Let vi = XOR of two bytes. Let vi = S(some byte). Here S is given by the map: x → x-1 (S(0)=0). Output = 16 consecutive bytes vi-15…vi-1vi. BES Structure BES writes all equations over F256.
  • 1 Think About a Linear Function and Its First and Second Derivative The

    1 Think About a Linear Function and Its First and Second Derivative The

    Think about a linear function and its first and second derivative The second derivative of a linear function always equals 0 Anytime you are trying to find the limit of a piecewise function, find the limit approaching from the left and right ­ if they are equal then the limit exists Remember you can't differentiate at a vertical tangent line Differentiabilty implies continuity ­ the graph isn't continuous at x = 0 1 To find the velocity, take the first derivative of the position function. To find where the velocity equals zero, set the first derivative equal to zero 2 From the graph, we know that f (1) = 0; Since the graph is increasing, we know that f '(1) is positive; since the graph is concave down, we know that f ''(1) is negative Possible points of inflection ­ need to set up a table to see where concavity changes (that verifies points of inflection) (­∞, ­1) (­1, 0) (0, 2) (2, ∞) + ­ + + concave up concave down concave up concave up Since the concavity changes when x = ­1 and when x = 0, they are points of inflection. x = 2 is not a point of inflection 3 We need to find the derivative, find the critical numbers and use them in the first derivative test to find where f (x) is increasing and decreasing (­∞, 0) (0, ∞) ­ + Decreasing Increasing 4 This is the graph of y = cos x, shifted right (A) and (C) are the only graphs of sin x, and (A) is the graph of ­sin x, which reflects across the x­axis 5 Take the derivative of the velocity function to find the acceleration function There are no critical numbers, because v '(t) doesn't ever equal zero (you can verify this by graphing v '(t) ­ there are no x­intercepts).
  • Algebra Vocabulary List (Definitions for Middle School Teachers)

    Algebra Vocabulary List (Definitions for Middle School Teachers)

    Algebra Vocabulary List (Definitions for Middle School Teachers) A Absolute Value Function – The absolute value of a real number x, x is ⎧ xifx≥ 0 x = ⎨ ⎩−<xifx 0 http://www.math.tamu.edu/~stecher/171/F02/absoluteValueFunction.pdf Algebra Lab Gear – a set of manipulatives that are designed to represent polynomial expressions. The set includes representations for positive/negative 1, 5, 25, x, 5x, y, 5y, xy, x2, y2, x3, y3, x2y, xy2. The manipulatives can be used to model addition, subtraction, multiplication, division, and factoring of polynomials. They can also be used to model how to solve linear equations. o For more info: http://www.stlcc.cc.mo.us/mcdocs/dept/math/homl/manip.htm http://www.awl.ca/school/math/mr/alg/ss/series/algsrtxt.html http://www.picciotto.org/math-ed/manipulatives/lab-gear.html Algebra Tiles – a set of manipulatives that are designed for modeling algebraic expressions visually. Each tile is a geometric model of a term. The set includes representations for positive/negative 1, x, and x2. The manipulatives can be used to model addition, subtraction, multiplication, division, and factoring of polynomials. They can also be used to model how to solve linear equations. o For more info: http://math.buffalostate.edu/~it/Materials/MatLinks/tilelinks.html http://plato.acadiau.ca/courses/educ/reid/Virtual-manipulatives/tiles/tiles.html Algebraic Expression – a number, variable, or combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents and/or roots. o For more info: http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut11 _simp.htm http://www.math.com/school/subject2/lessons/S2U1L1DP.html Area Under the Curve – suppose the curve y=f(x) lies above the x axis for all x in [a, b].
  • Generic Continuous Functions and Other Strange Functions in Classical Real Analysis

    Generic Continuous Functions and Other Strange Functions in Classical Real Analysis

    Georgia State University ScholarWorks @ Georgia State University Mathematics Theses Department of Mathematics and Statistics 4-17-2008 Generic Continuous Functions and other Strange Functions in Classical Real Analysis Douglas Albert Woolley Follow this and additional works at: https://scholarworks.gsu.edu/math_theses Part of the Mathematics Commons Recommended Citation Woolley, Douglas Albert, "Generic Continuous Functions and other Strange Functions in Classical Real Analysis." Thesis, Georgia State University, 2008. https://scholarworks.gsu.edu/math_theses/44 This Thesis is brought to you for free and open access by the Department of Mathematics and Statistics at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Mathematics Theses by an authorized administrator of ScholarWorks @ Georgia State University. For more information, please contact [email protected]. GENERIC CONTINUOUS FUNCTIONS AND OTHER STRANGE FUNCTIONS IN CLASSICAL REAL ANALYSIS by Douglas A. Woolley Under the direction of Mihaly Bakonyi ABSTRACT In this paper we examine continuous functions which on the surface seem to defy well-known mathematical principles. Before describing these functions, we introduce the Baire Cate- gory theorem and the Cantor set, which are critical in describing some of the functions and counterexamples. We then describe generic continuous functions, which are nowhere differ- entiable and monotone on no interval, and we include an example of such a function. We then construct a more conceptually challenging function, one which is everywhere differen- tiable but monotone on no interval. We also examine the Cantor function, a nonconstant continuous function with a zero derivative almost everywhere. The final section deals with products of derivatives. INDEX WORDS: Baire, Cantor, Generic Functions, Nowhere Differentiable GENERIC CONTINUOUS FUNCTIONS AND OTHER STRANGE FUNCTIONS IN CLASSICAL REAL ANALYSIS by Douglas A.