...2009 ! Module 1: Functions 1 1.1 Objectives: In this module: We introduce with examples, the concept of a function and study some classes of functions. We discuss methods of nding the domain, range, zeros and singularities of some real-valued functions. We look at injective (one to one) and surjective (onto) functions and end the module by exploring the inverse of a function and the composition of functions. We stimulate students through classroom and other interactive activities to understand the concept of a function and then discover the di erences and similarities between the several types of functions introduced. 2 1.2 Learning Outcomes: At the end of this module students should be able to De ne a function; identify di erent types of functions and Identify relations which are not functions; Identify di erent classes of functions, nd the domain and range that makes a rule f (say), a function; Investigate the injectiveness, surjectiveness and the inverse of di erent functions; Establish the composition of two or more functions. 3 1.3 Learning Activities: Students should: 1 Explore notes and exercises, individually and in groups; Explore and use related materials on the Intranet, especially the e-granary and MIT open course ware; Explore and use related materials on the OLI Calculus course on the Internet(http://www.cmu.edu/OLI/courses/); Solve relevant questions from past MTH103 Examinations. 4 1.4 Introduction A function is the means by...
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...real numbers, sequences and series of real numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration. They don’t include multi-variable calculus or contain any problem sets. Optional sections are starred. c John K. Hunter, 2014 Contents Chapter 1. Sets and Functions 1 1.1. Sets 1 1.2. Functions 5 1.3. Composition and inverses of functions 7 1.4. Indexed sets 8 1.5. Relations 11 1.6. Countable and uncountable sets 14 Chapter 2. Numbers 21 2.1. Integers 22 2.2. Rational numbers 23 2.3. Real numbers: algebraic properties 25 2.4. Real numbers: ordering properties 26 2.5. The supremum and infimum 27 2.6. Real numbers: completeness 29 2.7. Properties of the supremum and infimum 31 Chapter 3. Sequences 35 3.1. The absolute value 35 3.2. Sequences 36 3.3. Convergence and limits 39 3.4. Properties of limits 43 3.5. Monotone sequences 45 3.6. The lim sup and lim inf 48 3.7. Cauchy sequences 54 3.8. Subsequences 55 iii iv Contents 3.9. The Bolzano-Weierstrass theorem Chapter 4. Series 4.1. Convergence of series 4.2. The Cauchy condition 4.3. Absolutely convergent series 4.4. The comparison test 4.5. * The Riemann ζ-function 4.6. The ratio and root tests 4.7. Alternating series 4.8. Rearrangements ...
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.... . . . . . . . . . . . . . . . . . . Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summing Consecutive Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Product Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two element subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Important Concepts, Formulas, and Theorems . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Counting Lists, Permutations, and Subsets. . . . . . . . . . . . . . . . . . . . . . Using the Sum and Product Principles . . . . . . . . . . . . . . . . . . . . . . . . Lists and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Bijection Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . k-element permutations of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting subsets of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Important Concepts, Formulas, and Theorems . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . ....
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...from graph of y = f(x). A. 2 left, then 1 up. Bộ môn Toán, ĐH FPT Hà Nội B. 2 right, then 1up. A. x = -1 C. y = -3 D. y = 3 10. Find c so that the function is continuous on R x 2 cx c 1 if g ( x) x 1 2 if x 1 C. 2 right, then 1 down. D. 2 left, then 1 down. 3. Let B. x = 2 f ( g ( x)) 2x 3. Then: A. f ( x) x , g ( x) 2x 3. A. 0 B. f ( x) 2x 3, g ( x) x. 11. If C. f ( x) 2x , g ( x) x 3. f ’(1)=27, find h’(1). D. f ( x) x 3, g ( x) 2x. x 1 A. 3 4. Find limit A. -1/4 x 3 B. 4 C. 1/4 D. -4 s = 3t2 -2t+7. Find average velocity over [2, 4]. B. 24 C. -16 A. (i) (ii) x4-x3+1 (iii) cos x (iv) x4 – x2 B. (iii), (iv) 8. Find limit lim x 3 A. -1 B. ∞ C. (iii) x( x 2) x3 C. -∞ C. 9/4 D. 0 D. 8/27 B. y = 2x-3 C. y = (1/2)x D. y = (-1/2)x + 2 13. The base of a triangle is increasing at a rate of 1cm/s while the area is increasing at 2cm2/s. At what rate is the altitude changing when the base is 5cm and the area is 100cm2? A. -7.2 B. 7.2 C. 40 D. -40 14. Find linear approximation for tan x at a=0. D. (i), (iv) A. L(x) = -x+1 B. L(x) = -x C. L(x) = x+1 C. (1/2)f ’(x) D. f ’(x/2) 7. Which functions are even: (i) sinx B. 27/8 D. -24 f ( x 2h) f ( x ) 6. What is lim ? h 0 h A. f ’(x) B. 2f ’(x) D. 2 h( x) 3 f ( x) , where f(1)=8 and ...
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...space with zero atoms consist of stones. 5. All empty space does not consist of stones. 6. Therefore, stones do not exist. To go even further, I can add another step in order to conclude that ordinary objects do not exist. The additional step would be that if ordinary objects exist, then stones must also exist. Therefore, ordinary objects do not exist. First we must see what objects exists and what objects occupy a position in space and time. Objects in the universe include planets, people, atoms, chairs, and the sun. They are all made out of parts and they are all composite objects. In order to explain the idea of composition we can say that the x’s compose a composite object if and only if the x’s, taken together have a function that none of them have separately, the x’s are physically bonded, and the x’s are inseparable. Nihilism supports Unger’s idea that stones do not exist because nihilism supports the idea that composition never occurs and therefore,...
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...product of the set of real numbers and itself? Ans : The Cartesian product is every ordered pair in all of the quadrants of the coordinate plane. 2. What is the domain and range of the relation {(2, 4),(4, 8),(8, 16)} ? Ans: domain {2,4,8} and range {4,8,16} and the function can be given as f(x)=2x. 3. Decide whether the graph below is a function. Ans: The above shown graph is not a function since for a given x value it has multiple y values at certain points and hence cannot be a function. 4. What is the domain and range of the function f (x) = ? Ans: domain 0 ≤ x and range 0 ≤ f(x) 5. Is the following a function: y = ± x ? Ans : No it is not a function for a given x value it has two Y values . 6. Is the function f (x) = 4x even, odd, or neither? Ans : function f (x) = 4x is an odd function since f (x) = - f (- x) 7. Is the function f (x) = x - 5 even, odd, or neither? Ans : function f (x) = x - 5 is neither neither even nor odd. 8. What is the inverse of the function y = , and is it a function? Ans : The inverse is y = x2 +1 and It is a function. 9. A piecewise function is defined this way: f (x) = - x for x < 0 , f (x) = x2 for 0 ≤ x ≤ 3 , f (x) = 3x for x > 3 . What is f (- 4) + f (3) + f (7) ? Ans : -4 + 9...
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...Jump to: navigation, search This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). | It has been suggested that Infinitesimal calculus be merged into this article or section. (Discuss) Proposed since May 2011. | Topics in Calculus | Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus | Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities:Power rule, Product rule, Quotient rule, Chain rule | [show]Integral calculus | IntegralLists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order | [show]Vector calculus | Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem | [show]Multivariable calculus | Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian | | Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change,[1] in the same way that geometry is...
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...Samantha Poteet ENGWR 480 Professor Kiefer 31 October 2013 Little is known about Jakob von Uexküll, the author of A Foray into the Worlds of Animals and Humans: With a Theory of Meaning, other than his credentials and contributions. Uexküll studied zoology at the University of Tartu, known as Dorpat at the time, in Estonia from 1884 to 1889 (SUNY Press; Jakob von Uexküll Centre). Later Uexküll worked at the Institute of Physiology of the University of Heidelberg and at the Zoological Center in Naples, focusing on the behavior and interaction of living beings (SUNY Press; Jakob von Uexküll Centre). His written works were dedicated to the question of how living beings subjectively perceive their environment and how their perception determines their behavior, which is a dominant theme throughout A Foray in the Worlds (Jakob von Uexküll Centre). Uexküll argues that the spider builds its web before it has ever met a physical fly, therefore it is a representation of the spider’s primal image of the fly (159). Uexküll’s claim depends on three assumptions: (1) the fly is the prey of the spider, (2) the spider has a primal image of the fly, and (3) the web is the end product of the spider’s primal image. Though Uexküll’s claim may well have some merit, he presents an incomplete argument as his assumptions are insufficient and based solely on the evidence he offers, his argument cannot be accepted as valid. Uexküll’s first assumption is correct but also insufficient as it does...
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...Introduction Brand image is that the current outlook of the shoppers about a brand. It is characterized as a exclusive bundle of associations at gaps the minds of target customers. It signifies what the emblem presently stands for. it is a assembly of convictions order about a exact brand. In short, it is not anything but the consumers’ perception about the product. it is the kind inside which a particular emblem is positioned in the market. Emblem likeness expresses emotional worth and not just a mental representation. Emblem image is not anything but aide degree organization’s feature. it is aide degree accumulation of contact and fact by individuals external to an association. It ought to focus aide degree organization’s operation and dream to any or all. The major parts of positive emblem image are- exclusive emblem reflective organization’s likeness, saying recounting organization’s business in short and emblem identifier carrying the key values. Brand image is that the overall impression in consumers’ mind that's formed from all sources. shoppers develop various associations with the brand. supported these associations, they form brand image. an image is made about the brand on the idea of subjective perceptions of associations bundle that the shoppers have about the brand. Volvo is associated with safety. Toyota is associated with reliableness Image of a brand has a great impact on consumer’s preferences, because it’s the current view of a product according to the...
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...systems in which there is change and a way to deduce the predictions of such models; Calculus provides a way for us to construct relatively simple quantitative models of change and to deduce their consequence. By studying this, you can learn how to control the system to do make it do what you want it to do. CHAPTER 1: FUNCTIONS AND LIMITS FUNCTIONS * A bunch of ordered pairs of things with property that the first members of the pairs are all different from one another. Ex [ {1,1,}, {2,1}, {3,2} ] Arguments – first number of the pair Domain – whole set Values – Second number of the pair Range – set of values Classification of functions 1. Linear Functions – “steepness of the line” w/c can go uphill or downhill. y = mx + b 2. Quadratic Functions – it has a degree and forms a parabolic path. The highest (or lowest point) of the parabola is called the vertex. At has a form of (standard form of quadratic equation) F(x) = Ax2 + Bx + C where A, B,C are constant. Vertex form of Quadratic F(x) = a (x-h)2 + K Quadratic Formula 3. Polynomial Functions – a quadratic, a cubic, a quartic and so on involving only non-negative...
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...solutions in simplest radical form. a) b) 4. Find the maximum or minimum value of the function and the value of x when it occurs. a) b) 5. Write a quadratic equation, in standard form, with the roots a) and and that passes through the point (3, 1). b) and and that passes through the point (-1, 4). 6. The sum of two numbers is 20. What is the least possible sum of their squares? 7. Two numbers have a sum of 22 and their product is 103. What are the numbers ,in simplest radical form. Unit 3 1. Determine which of the following equations represent functions. Explain. Include a graph. a) b) c) d) 2. State the domain and range for each relation in question 1. 3. If and , determine the following: a) b) 4. Let . Determine the values of x for which a) b) Recall the base graphs. 5. Graph . State the domain and range. Describe how the graph can be obtained from the graph of . Also Try! a) Graph b) 6. Given , a) determine the equation of the inverse b) graph f(x) and its inverse c) Is the inverse a function? Explain. If not, restrict the domain of f(x) so that the inverse is a function. Unit 4 1....
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...Tom Gause 4/22/13 Unit 4 research paper: Port Expander “A port expander is a hardware device designed to utilize more than one device on a single port at one time.” In layman’s terms, this can also be called a “splitter.” There are many different types of port expanders, but they can be narrowed down to two major groups; internal, and external. They are generic devices that will function no matter where they’re installed. Internal port expanders attach to your motherboard, and the user will only see the back plate. From a hardware standpoint, there is no difference in the type of computer. If the card fits, it will work. The first image in the appendix is an example of a port expander. External port expanders are not attached to the motherboard. They plug into a slot on the computer, and allow multiple devices to attach to that slot. The second image in the appendix is an example of an external port expander. One of the biggest advantages to using a port splitter is during multitasking. For example, most user interface devices are USB devices, mice, keyboards, printers, and even cell phone chargers can all be plugged into a USB slot. Therefore, the more ports you have, the more peripheral attachments you can use. On the other hand, with so many devices plugged into a single port the power source could get overwhelmed. Without the proper safety measures, an overloaded power source could “burn out” and cease functioning. Appendix Image 1: Internal Port Expander ...
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...FOCAL LENGTH OF A LENS AIM: The aim of this experiment is to determine the focal length (f) of a convex lens by two methods. YOU WILL NEED: A 10 cm focal length bi-convex lens, lens holder, screen (a wooden block with a white painted side is ideal), ruler, light source (mounted clear bulb), power supply suitable for the lamp. An optical bench is ideal if one is available. WHAT TO DO: (a) Minimum distance method Set up the lamp, lens and screen so that a clear image of the lamp filament is formed on the screen. Measure the object and image distances (u and v). Repeat the experiment for a series of values over the range u+v = 45 cm to 100 cm. ANALYSIS AND CONCLUSION: Plot a graph of u+v against u. The minimum point on the graph is at a point u+v = 4f, u = 2f (b) Two position method Set up the lamp, lens and screen 0.6m apart. Find the TWO positions where a clearly focused image of the lamp may be obtained. (one of these will give a large image and the other a small image). Measure the separation of lamp and screen (d) and the distance between the two positions of the lens (a) Repeat the procedure for other values of d between 0. 5m and 1m ANALYSIS AND CONCLUSION: The focal length of the lens may be calculated from the formula: f = [d2 – a2]/4d ----------------------- a v2 d u1 2...
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...failure of an earlier ERP project, a new project called BMIS was envisioned. Highlights of BMIS project: • Vision: Functional councils of Methods, quality, Work and material planning, procurement created for role and direction determination. • Team: Best employees from various functions along with BTS (Business transformation services) with an employee to consultant ratio of 10:1. • Implementation strategy: Progressive and phased approach, beginning with Mirabel plant near Montréal. • Procurement function restructure: improvement of inventory visibility and anticipated substantial savings in product costs($ 22 million) and procurement overhead($7.1 million) • First Rollout challenges: lack of documentation, system functionality, substandard training material, tight training schedules and lack of role clarity during training. • Go Live: Power users and super users supported the move initially. Later on, if support staff was not able to resolve issues, the legacy systems were used to conduct business. • Stabilization: The project did not disrupt production schedules and the company delivered one more plane than planned during the period. There was overall saving of $ 1.2 billion. Procurement functions changed even though lots of others were de-scoped. • Next milestones: Two additional programs were implemented eventually with better rate of success at...
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...Name: Bekenov Kuandyk Assignment 1 1. Prove that fx=2x+5 is one-to-one function. (10 points) Let make an assumption tha fx1=fx2t and then prove that x1=x2. fx1=fx2=2x1+5=2x2+5→x1=x2 , Then fx=2x+5 is one-to-one function 2. B B A A Let f: A→B, as given below. Is f a one-to-one function? Please explain why or why not. (10 points) f f 5 5 5 5 1 1 3 3 1 1 6 6 2 2 2 2 4 4 6 6 7 7 4 4 8 8 3 3 F is not one-to-one function, because f(1)=1 and f(2)=1. 3. The modulo function (a mod n or a modulo n) maps every positive integer number to the remainder of the division of a/n. For example, the expression 22 mod 5 would evaluate to 2 since 22 divided by 5 is 4 with a remainder of 2. The expression 10 mod 5 would resolve to 0 since 10 is divisible by 5 and there is not a remainder. a. If n is fixed as 5, is this function one-to-one? (5 points) F(n mod 5) is not one-to-one function, because f(n mod 5)= between 0 and 4. It means that we have the same image of F for different n (integer). b. List five numbers that have the exact same image. (5 points) 12, 17, 22, 27, 32 4. Find limn→∞n2-1n3+n (6 points) limn→∞n2-1n3+n=limn→∞n2n3-1n3n3n3+nn3=0-01+0=0 5. Find limx→∞x1001-xx1000+x (6 points) limx→∞x1001-xx1000+x=limx→∞1-1x10001x+1x1000=1-00+0=∞ 6. Find limx→∞8x6+4x4-3x2x6+x5-7 (6 points) limx→∞8x6+4x4-3x2x6+x5-7=limx→∞8+4x2-3x52+1x-7x6=8+0-02+0-0=4 ...
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