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# Math

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Contents 0. Preface 1. Functions and Models 1.1. Basic concepts of functions 1.2. Classiﬁcation of functions 1.3. New functions from old functions 1 2 2 5 8

0. Preface Instructor: Jonathan WYLIE, mawylie@cityu.edu.hk Tutors: Radu Gogu, rgogu2@student.cityu.edu.hk. Texts: Single Variable Calculus, by James Stewart, 6E. In this semester, we will cover the majority of Chap 1-4, 7, 12. Upon completion of this course, you should be able to understand limit, derivatives, and its applications in mathematical modeling and inﬁnite series.

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1. Functions and Models In this chapter, we will brieﬂy recall functions and its properties covered by high school. 1.1. Basic concepts of functions. Text Sec1.1: 5, 7, 39, 57, 67. Deﬁnition 1.1. A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. Usually, we write a function f : x → f (x) where (1) x ∈ D, i.e. x belongs to a set D , called the Domain; (2) f (x) ∈ E, i.e. f (x) belongs to a set E, called the Range; (3) x is independent variable, (4) f (x) is dependent variable.

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For a function f , its graph is the set of points {(x, f (x)) : x ∈ D} in xy-plane. One can also use a table to represent a function. Example 1.1. Sketch the graph of following two piecewise deﬁned functions. (1) f (x) = |x|. i.e. Absolute value of x. (2) f (x) = [x]. i.e. largest integer not greater than x.

The graph of a function is a curve. But the question is: which curves are graphs of functions? Proposition 1.2 (Vertical Line Test). A curve in the xyplane is the graph of a function if and only if no vertical line intersects the curve more than once. VLT is equivalent to following statements: for any given input x, the output f (x) is determined uniquely. Otherwise, f is not well-deﬁned function.

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Symmetry of a function is an important topic. (1) A function f is even if f (−x) = f (x), ∀x ∈ D. (2) A function f is odd if f (−x) = −f (x), ∀x ∈ D.

Monotonicity of a function is another important topic. (1) A function f is increasing on an interval I, if f (x1) < f (x2), ∀x1 < x2 in I. (2) A function f is decreasing on an interval I, if f (x1) > f (x2), ∀x1 < x2 in I

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1.2. Classiﬁcation of functions. Text Sec1.2: 13, 17 A mathematical model is a mathematical description of a realworld phenomenon, and usually represented by a function. ex. We will later study some math models for population, demand of product, speed of an object, ... Some typical functions used for models are (1) Polynomial function P (x) = anxn + an−1xn−1 + · · · + a1x + a0 where ai are coeﬃcients. If an = 0, then the degree of P (x) is n. (a) Linear model f (x) = mx + b where m is slope, b is y-intercept. ex. The relationship between Fahrenheit (F) and Celsius (C) is F = 9 C + 32. 5 (b) Quadratic model f (x) = ax2 + bx + c. (c) Cubic model A polynomial with degree 3.

(2) Power function f (x) = xa where a is constant. Ex. Sketch the graph of power function if a is (1) positive integer; (2) reciprocal of positive integer; (3) -1;

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(3) Rational function P (x) Q(x) where P, Q are polynomials. ex. ﬁnd domain of f (x) = 2−3x . x2 −4 f (x) =

(4) Algebraic function It is a function constructed by poly√ nomials using algebraic operations (such as +, −, ×, ÷, n ).

ex. ﬁnd domain and symmetry of f (x) =

x2 +1 x3

(5) Trigonometric functions The common trigonometric functions are sin, cos, tan, cot. See more details on Reference Page 2 of the Text.

(6) Exponential functions f (x) = ax where the base a = 1 is a positive constant. ex. Sketch the graph of y = ax when a is a constant satisfying (1) a < 1 (2) a > 1.

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(7) Logarithmic functions f (x) = loga x where a = 1 is a positive constant. ex. Sketch the graph of y = loga x when a is a constant satisfying (1) a < 1 (2) a > 1.

(8) Transcendental functions It is a non-algebraic function, including the trig; inverse of trig; exp.; log; and ... ex. Can you ﬁnd a Transcendental function not mentioned in the above?

Example 1.2. Classify following functions as one of the types we discussed: poly, power, rational, algebraic, Trig, exp, log, transc., (1) f (x) = 5x, (2) g(x) = x5 1+x (3) h(x) = 1−√x (4) u(x) =
1+x 1−x1.5

+ x 2.

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1.3. New functions from old functions. Text Sec1.3: 23, 35, 57, 63, 65 We will discuss two ways of obtaining a new function from old functions: (1) Shifting, stretching, or reﬂecting a given function; (2) Combination/Composition of two given functions Let a > 0 and b > 1. Given a function y = f (x), we can obtain a new function using following transformations (1) y = f (x) + a, by shifting y = f (x) a units upward; i.e. ↑a (2) y = f (x) − a, by ↓a (3) y = f (x − a), by →a (4) y = f (x + a), by ←a (5) y = bf (x) by stretching y = f (x) vertically by a factor of b, i.e. b (6) y = 1 f (x), by compressing y = f (x) vertically by a factor b of b, 1/b (7) y = f (bx), by compressing horizontally, ↔1/b. (8) y = f (x/b), by stretching horizontally, ↔b (9) y = −f (x), by reﬂect y = f (x) about x-axis, i.e. Rx. (10) y = f (−x), by reﬂect y = f (x) about y-axis, i.e. Ry . Example 1.3. Using transformation, graph √ √ f (x) = 2 −x − 1, and g(x) = |2 −x − 1|.

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Given Two functions of f and g, we may have following combinations: using deﬁnition of +, −, ×, ÷ f + g, f − g, f g, f /g Also, composition f ◦ g is deﬁned by (f ◦ g)(x) = f (g(x)). Example 1.4. Given F (x) = cos2(x+9), ﬁnd function f, g, h s.t. F = f ◦ g ◦ h.

Remark 1.3. f ◦ g = g ◦ f in general.

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