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Lecture 2

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Mathematics & Statistics (Sci)
MATH 140
Ewa Duma

MATH140 - Lecture 2 Notes Linear Models: When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS |||| 25 for the function as: y = mx+b which has a definitive feature that the line grows at a constant rate. A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear ff▯x▯ ▯ 3x ▯ 2and a table of sam- • Note that m is the slope of the linever x increases by 0.1, the f▯x▯increases by 0.3. So f▯x▯ncreases three times as fast as x.Thus the slope y ▯ 3x ▯ 2p, namely 3, can • The valbe interpreted as the rate of change of y with respect to x. y x f ▯x▯ ▯ 3x ▯ 2 y=3x-2 1.0 1.0 28 |||| CHAPTER 1 FUNCTIONS AND MODELS 1.1 1.3 1.2 1.6 0 x 1.3 1.9 _2 We thereforepredict that the C1.2 level wi2.2exceed 400 ppm by the year 2019. This 1.5 2.5 predictionis somewhat risky becauseit involves a time quite remote from our Example:RE 2 observations. M V EXAMPLE 1 POLYNOMIALS (a) As dry air moves upward, it expands and cools. If the ground temperature is 20▯C Polynomials: and the temperature at a height of 1 km is 10▯C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the functionP▯x▯ ▯ a n ▯ aWhan▯1xes t▯ ▯▯▯ ▯ a x 2 a x ▯ 1 0 A function is (c) What is the temperature at a height of 2.5 km? SOLUTION where is a nonnegative integerandthenumbers a0, a1, 2 , ...,naareconstantscalledthe The domain of any polynomial coefficients of the polynomial. The domain of any polynomial is ▯ ▯ ▯▯▯, ▯▯. If the leading coefficient a ▯ 0 , then the degree of the polynomial is . For example, the equal to 0, then the degree of the polynomial iT ▯ mh ▯ b function We are given that T ▯ 20 when h ▯ 0, so 6 4 2 3 Taking the example of f(x) = 4x^4 - x^2 + 3, we see here that the function is a 5 s2 polynomial of degree 4 because the highest power is 4. is a polynomial20 ▯ m ▯ 0 ▯ b ▯ b A polynomialof degree 1 is of the form P▯x▯ ▯ mx ▯ b and so it is a linear function. In other words,A polynomialof degree 2 is of the form P▯x▯ ▯ ax ▯ bx ▯ c and is called a quadratic • A polynomiaWe are also given that T ▯ 10 when h ▯ 1, so = mx + b 2 • A polynomial of degree 2 is of the form f = ax^2 + bx + c and is called a quadratiche parabola y ▯ ax , as we T will see in the 10 ▯ m ▯ 1 ▯ 20e parabolaopens upward if a ▯ 0 and downward if a ▯ 0. function where we get a parabola 20 • A polynThe slope of the line is therefore m ▯ 10 ▯ 20 ▯ ▯10 and the required linear function is T=_10h+20 10 cubic function T ▯ ▯10y ▯ 20 y 0 1 3 Poweh function:b) The graph is sketched in Figure 3. The slope is m ▯ ▯10▯C▯km, and this2represents the rate of change of temperature with 2espect to height. (c) At a height of h ▯ 2.5 km, the temperature is A function of the form f(x) = x^a where a is a constant. We have several cases of a FIGURE 3 power function such as: T ▯ ▯10▯2.5▯ ▯ 20 ▯ ▯5▯C M x 1 If there is no physical law or principle to help us formulate a model, we construct an •The graphs of quadraticl, which is based entirely on collected data. We seek a curve that “fits” the data in the sense that it captures the basic trend of the data points. •fua = 1/n, where n is a positive integer (root function) (b) y=_2▯+3x+1 a = -1 (reciprocal function) • A polynomialof degree 3 is of the form 3 2 P▯x▯ ▯ ax ▯ bx ▯ cx ▯ d ▯a ▯ 0▯ and is calleda cubicfunction. Figure8 shows the graphof a cubicfunctionin part(a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see later why the Volume as a function of pressure 0 P at constant temperature Another instance in which a power function is used to model a physical phenomenon SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS||||x33cise 26. An important property of the sine and cosine functions is that they are periodic func- tions and have per2▯d. This means that, for all vaxues of ,IONAL FUNCTIONS Rational Functions: A rational function f is a ratio of two polynomials: sin▯x ▯y2▯▯ ▯ sin x cos▯x ▯ 2▯▯ ▯ cos x P▯x▯
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