Math 242: Calculus I
1.2 Mathematical Models
1.2.1 Mathematical Models¶
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Definition¶
A mathematical model is a function, equation, or system of equations that describe relationships between measurable physical phenomena.
- The path of a projectile
These are all real-world phenomena that an be described using calculus. The first step is to identify all of the independent and dependent variables that we use to measure a phenomenon. Then we might use existing theory of the phenomenon to express relationships between variables. We might also collect data from existing instances of the physical phenomena. We might plot the data and identify patterns. A graphical representation might suggest a algebraic relationship.
Then we apply mathematical logic such as differential or integral calculus to these equations and draw conclusions about relationships between the variables. We can use these conclusions to deduce predictions about the outcomes of future instances of the phenomena.
Finally, we create experiments or instances of physical system. We measure the independent and dependent variables and compare them to our predictions.
Differences between measurements and our predictions can lead us to revise our mathematical model and repeat this process.
1.2.2 Linear Models¶
Definiton¶
A linear function is a function of a real variable whose graph is a line. A linear function of one real variable is a function of the form
$$ y = f(x) = mx + b $$
This is called the slope-intercept form of the equation of a line. The standard form of the equation of a line is given by
$$ ax + by = c $$
Finally, the point-slope form of the equation of a line is defined by
$$ y - y_0 = m(x-x_0), $$
where $(x_0,y_0)$ is any point on the line.
The characteristic feature of a line is that they change at a constant rate. This rate of change is the slope $m = -\frac{a}{b}$.
Definition¶
A horizontal line has the equation
$$ y = 0\cdot x + y = c $$
For this reason a horizontal line is said to have zero slope, $m = -\frac{0}{b}$.
A verical line is not really a function and has the equation
$$ x = x + 0\cdot y = c $$
This would give us the undefined expression that slope $m = -\frac{1}{0}$. For this reason a vertical line is said to have no slope.
1.2.3 Tabular Data¶
Example 1¶
Everyone can download the recorded data collected by NOAA via the webpage Trends in Atmospheric Carbon Dioxide (CO2
| Year | Mean | Year | Mean | Year | Mean | Year | Mean |
|---|---|---|---|---|---|---|---|
| 1960 | 316.91 | 1980 | 338.76 | 2000 | 369.71 | 2020 | 414.21 |
| 1961 | 317.64 | 1981 | 340.12 | 2001 | 371.32 | 2021 | 416.41 |
| 1962 | 318.45 | 1982 | 341.48 | 2002 | 373.45 | 2022 | 418.53 |
| 1963 | 318.99 | 1983 | 343.15 | 2003 | 375.98 | 2023 | 421.08 |
| 1964 | 319.62 | 1984 | 344.87 | 2004 | 377.7 | ||
| 1965 | 320.04 | 1985 | 346.35 | 2005 | 379.98 | ||
| 1966 | 321.37 | 1986 | 347.61 | 2006 | 382.09 | ||
| 1967 | 322.18 | 1987 | 349.31 | 2007 | 384.02 | ||
| 1968 | 323.05 | 1988 | 351.69 | 2008 | 385.83 | ||
| 1969 | 324.62 | 1989 | 353.2 | 2009 | 387.64 | ||
| 1970 | 325.68 | 1990 | 354.45 | 2010 | 390.1 | ||
| 1971 | 326.32 | 1991 | 355.7 | 2011 | 391.85 | ||
| 1972 | 327.46 | 1992 | 356.54 | 2012 | 394.06 | ||
| 1973 | 329.68 | 1993 | 357.21 | 2013 | 396.74 | ||
| 1974 | 330.19 | 1994 | 358.96 | 2014 | 398.81 | ||
| 1975 | 331.13 | 1995 | 360.97 | 2015 | 401.01 | ||
| 1976 | 332.03 | 1996 | 362.75 | 2016 | 404.41 | ||
| 1977 | 333.84 | 1997 | 363.88 | 2017 | 406.76 | ||
| 1978 | 335.41 | 1998 | 366.84 | 2018 | 408.72 | ||
| 1979 | 336.84 | 1999 | 368.54 | 2019 | 411.65 |
The plot of this data can suggest a linear, parabolic, or exponential mathematical model of the rise in $\text{CO}_2$ in the atmosphere measured on Mount Mauna Loa, Hawaii. If we hypothesize that the data is linear, there are calculus and linear algebra techniques to compute the line that is closest to all of these data points. The line below
The equation for this linear model of $\text{CO}_2$ concentration $C$ as a function of year since 1960 $y$ is a linear function $\text{CO}_2\,:\,[1960,2023]\rightarrow[315,422]$ defined by
$$ c = \text{CO}_2(y) = 1.6415y - 2909 $$
Exercise 1¶
Does the data agree with the hypothesis that $\text{CO}_2$ levels are rising linearly? What better model could be used to accurately forecast $\text{CO}_2$ levels in 1930?
1.2.4 Polynomials¶
Definition¶
A function $P$ defined by
$$ P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0, $$
where $n$ is a positive integer, is called a polynomial. If the leading coefficient $a_n=\neq 0$, then the polynomial has degree $n$. The domain of any polynomial is $\mathbb{R} = (-\infty, \infty)$.
Example 2¶
the polynomial
$$ Q(x) = 8x^7 - 4x^6 -2x^4 + 2x^3 - 5x^2 + 4 $$
is a polynomial with degree $7$.
Example 3¶
A polynomial with degree less than or equal to 1 is a line
$$ L(x) = mx + b $$
Example 4¶
A polynomial with degree 2 is a quadratic function. The graph of a quadratic function is a parabola. Perhaps our NOAA data is better modeled by a quadratic function.
1.2.5 Power Functions¶
The exponent in a function may be any real number.
Definition¶
A function defined by $f(x) = x^a$, there $a\in\mathbb{R}$ is a constant is called a power function.
Example 5¶
Polynomials are also power functions. The graphs of polynomials should have recognizable shapes that you should review!
Example 6¶
A function $f(x) = x^{1/n} = \sqrt[n]{x}$ is a root function. Computing a even root of a negative number is a complex number. For our root functions to be real-valued, the domain of a root function must be limited to non-negative values.
Example 7¶
The reciprocal function $y = \frac{1}{x}$ has domain $(-\infty, 0)\cup(0,\infty) = \mathbb{R}\setminus\left\{0\right\}$ When ever we have an algebraic expression in a denominator, one must make sure that zeros of the denominator are not in the domain of the function!
Example 8¶
Boyle's Law
$$ P = \frac{C}{V} $$
This is an inverse function whose domain is $\mathbb{R}^+ = (0,\infty)$ because negative pressures, or pressure equal to zero make no physical sense.
1.2.6 Rational Functions¶
Definition¶
A rational function $f$ is a ratio of two polynomials
$$ f(x) = \frac{P(x)}{Q(x)} $$
The function $y=\frac{1}{x}$ is also a rational function. The zeros of the polynomial in the denominator are not elements of the domain of a rational function because division by zero is undefined.
Example 9¶
Consider the function defined by
$$ y = \frac{x(x+2)(x-3)}{4(x-1)(x+1)} $$
The domain of this function is $\mathbb{R}\setminus\left\{-1,1\right\}$
1.2.7 Algebraic Functions¶
Definition¶
A function constructed using algebraic operations including function composition of algebraic functions is also and algebraic function.
All of our previous examples are algebraic functions. If we combine these functions, then we have another algebraic function.
Example 10¶
The mass $m$ of a particle with rest mass $m_0$ is a function of its velocity
$$ m = f(v) = \frac{m_0}{\sqrt{1-\frac{v^2}{c}}}, $$
where $c$ is the speed of light.
1.2.8 Exponential Functions¶
Definition¶
A function of the form
$$ f(x) = b^x, $$
where $b\in\mathbb{R}^+$ is a positive constant is called an __exponential function___.
Exponential growth and exponential decay often occur in our mathematical models. They are useful modeling natural processes, as well as populations. The domain of an exponential function is the real line $\mathbb{R}$ and the codomain is $\left(-\infty,\infty\right)$.
Example 11¶
In the Earth's atmosphere interaction of carbon atoms with cosmic rays can result in a radioactive isotope of carbon, $^{14}\text{C}$. This combines with other gases to form radioactive version of molecules that contain the $^{14}\text{C}$. Plants incorporate the radioactive carbon during photosynthesis, and animals acquire it by eating the plants, etc. When an organism dies, it stops exchanging carbon with its environment. $^{14}\text{C}$ is unstable and will release a beta particle(s) to become nitrogen-14, $^{14}\text{N}$.
$^{14}\text{C}$ has a half-life of 8267 years. By measuring the relative amounts of $^{14}\text{C}$ and $^{12}\text{C}$, one can estimate the amount of time that the $^{14}\text{C}$ has been decaying, or the amount of time since the organism died. The equation for $^{14}\text{C}$ decay with initial ammount $^{14}\text{C}_0$ is given by
$$ ^{14}\text{C} = ^{14}\text{C}_0\,e^{-\lambda t} $$
1.2.9 Logarithmic Functions¶
Definition¶
A logarithmic function $f(x) - \log_b\,x$, where the base $b\in\mathbb{R}^+$ is a positive constant, is the inverse of an exponential function.
The domain of a logarithmic function if $(0\infty)$, and the codomain is $\mathbb{R}$.
Example 12
The logarithmic form of Example 11 allows one to compute the number of years since an organism died by measuring the current amount of $^{14}\text{C}$ is a sample.
$$ t = \ln\left( \frac{^{14}\text{C}_0}{^{14}\text{C}} \right)\cdot 8267\ \text{years} $$
1.2.10 Transcendental Functions
Definition¶
Exponential and logarithmic functions with a base $b$ that is a transcendental number such as $e$, and trigonometric functions are transcendental functions.
Example 13¶
The sine function has domain $\mathbb{R}$ and codomain $[-1,1]$. The sine function is defined by
$$ \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} $$
Example 14¶
The tangent function of a ratio of two trigonometric functions.
$$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$
The domain of the tangent function does not include any odd multiples of $\frac{\pi}{2}$,
$$ \text{domain}\left(\tan(x)\right) = \mathbb{R}\setminus\left\{\,\frac{(2k+1)\pi}{2}\,:\,k\in\mathbb{Z}\,\right\} $$
1.2.11¶
The Formula Cards link in the table of contents of your e-book, or the covers of your hard bound textbook cover the prerequisites of our course.
- Reference Page 1
- Reference Page 2
- Reference Page 3
- Reference Page 4 (except the hyperbolic functions)
Every student should review these formula cards and make sure they are familiar with information in them.
Your use of this self-initiated mediated course material is subject to our
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