MATH140 - Lecture 2 Notes
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
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.
x f ▯x▯ ▯ 3x ▯ 2
28 |||| CHAPTER 1 FUNCTIONS AND MODELS 1.1 1.3
0 x 1.3 1.9
_2 We thereforepredict that the C1.2 level wi2.2exceed 400 ppm by the year 2019. This
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
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
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
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
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
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
A rational function f is a ratio of two polynomials:
sin▯x ▯y2▯▯ ▯ sin x cos▯x ▯ 2▯▯ ▯ cos x