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Workbook for Stewart’s "Calculus"

Cory Wilson

Section 1.2 Mathematical Models: A Catalog of Essential Functions

Subsection 1.2.1 Before Class

https://mymedia.ou.edu/media/1.2-1/1_s05beatp
Figure 3. Pre-Class Video 1

Subsubsection 1.2.1.1 Linear Functions

Remember that a linear function requires two pieces of information- a starting value (\(b\text{,}\) the \(y\)-intercept), and an amount of incremental change in the independent variable (\(m\text{,}\) the slope of the function). This gives us three ways to describe a linear function:
  • Verbally: A function with a constant rate of change
  • Graphically: Lines can look like one of four cases:
    A line with positive slope, passing through the origin
    A line with negative slope, passing through the origin
    A line with slope 0 (horizontal line)
    A line with undefined slope (vertical line)
  • Algebraically: \(f(x) = mx+b\)
Question 1.2.1.
Given two points \((x_1,y_1)\) and \((x_2,y_2)\text{,}\) how can we find the slope of the line between them?
Solution.
\(m =\dfrac{y_2-y_1}{x_2-x_1}\)
Example 1.2.2.
The following table gives the percentage of new companies which remained open \(t\) years after beginning business.
Years After Opening 5 6 7 8 9 10
Companies Still Open (in %) 50 47 44 41 38 35
  1. Plot the points on the graph below
    A blank graph to plot the points
  2. Use the data to find an equation for the line between these points.
  3. Give an interpretation for the slope of \(C(t)\)
Solution.
  1. The graph from part (a) with the points plotted
  2. \(\displaystyle C(t) = -3t + 65\)
  3. Each year, the percentage of companies still open decreases by 3 percent per year.

Subsubsection 1.2.1.2 Polynomials

Definition 1.2.3. Polynomial.
A function \(P\) is called a polynomial if
\begin{equation*} P(x) = a_nx^n + a_{n-1}x^{n-1}+\cdots + a_2x^2 + a_1x + a_0 \end{equation*}
Common Polynomials
Polynomial Equation Graph (\(a_n \gt 0\)) Graph (\(a_n \lt 0\))
Linear \(a_1x + a_0\)
The graph of a linear function with positive leading coefficient. The line goes up left to right.
The graph of a linear function with negative leading coefficient. The line goes down left to right.
Quadratic \(a_2x^2+a_1x + a_0\)
The graph of a quadratic function with positive leading coefficient. The curve goes down until it reaches a vertex, then turns back upward.
The graph of a quadratic function with negative leading coefficient. The curve goes up until it reaches a vertex, then turns back downward.
Cubic \(a_3x^3+a_2x^2+a_1x +a_0\)
The graph of a cubic function with positive leading coefficient. The curve begins going upward, slows down to be flat for a moment, then turns back upward.
The graph of a cubic function with negative leading coefficient. The curve begins going downward, slows down to be flat for a moment, then turns back downward.
Even Degree \(a_nx^n +\cdots a_1x+a_0 \) (\(n\) even)
The graph of an even function with positive leading coefficient. The curve goes begins moving upward and ends moving downward; the behavior in the middle depends on the function.
The graph of an even function with negative leading coefficient. The curve goes begins moving downward and ends moving upward; the behavior in the middle depends on the function.
Odd Degree \(a_nx^n +\cdots a_1x+a_0 \) (\(n\) odd)
The graph of an odd function with positive leading coefficient. The curve goes begins moving upward and ends moving upward; the behavior in the middle depends on the function.
The graph of an odd function with negative leading coefficient. The curve goes begins moving downward and ends moving downward; the behavior in the middle depends on the function.

Subsubsection 1.2.1.3 Other Functions

Definition 1.2.4. Exponential/Logarithmic Function.
An exponential function is a function of the form \(f(x)=b^x\text{,}\) where the base \(b\) is a positive constant.
A logarithmic function is a function of the form \(f(x)=\log_b x\text{,}\) where the base \(b\) is a positive constant.
Exponential and logarithmic functions have these properties:
Domain Range Asymptotes?
\(f(x)=b^x\) \((-\infty,\infty)\) \((0,\infty)\) \(y = 0\)
\(f(x)=\log_b (x)\) \((0,\infty)\) \((-\infty,\infty)\) \(x = 0\)

Subsection 1.2.2 Pre-Class Activities

Example 1.2.5.

  1. Find an equation for the family of linear functions with slope \(-1\text{,}\) and sketch a few members of the family.
  2. Find an equation for the family of linear functions such that \(f(2) = 1\text{,}\) and sketch a few members of the family.
  3. Which function belongs to both families?
Solution.
  1. The family is given by \(y = -x+b\text{.}\) Answers will vary on the sketch.
  2. The family is given by \(1=2m + b\text{.}\) Answers will vary on the sketch.
  3. \(\displaystyle y = -x+3\)

Example 1.2.6.

Find an expression for a quadratic function \(f\) if \(f(-2) = 2\text{,}\) \(f(0) = 1\text{,}\) and \(f(1) = -2.5\)
Solution.
\(f(x) = -x^2 -2.5x + 1\)

Example 1.2.7.

The monthly cost of driving a car depends on the number of miles driven. Casey found that in April, it cost her $350 to drive 450 miles, and in June, it cost her $460 to drive 800 miles. Express your answer exactly (no decimals).
  1. Express the monthly cost \(C\) as a function of the distance driven (\(d\)), assuming that a linear relationship gives a suitable model.
  2. Use (a) to predict the cost of driving 1500 miles per month.
  3. Sketch the graph of the linear function. What does the slope represent? Include units.
  4. What does the \(C-\)intercept represent?
  5. Why does a linear function give a suitable model in this situation?
Solution.
  1. \(\displaystyle C(d) = \dfrac{11}{35}x + \dfrac{1460}{7}\)
  2. 680 dollars
  3. The graph of \(C(d) = \dfrac{11}{35}x + \dfrac{1460}{7}\)
    The slope represents the rate of change of cost (dollars per month)
  4. The base cost of owning/maintaining the car without driving it.
  5. Answers vary.

Subsection 1.2.3 In Class

Subsubsection 1.2.3.1 Power Functions

Definition 1.2.8. Power Function.
A power function is any function of the form \(x^a\text{,}\) where \(a\) is a constant.
Example 1.2.9.
Describe the difference between a power function and a polynomial.
Example 1.2.10.
We will look at three special power functions.
  1. If \(a\) is a positive integer, what kind of functions do we see? Sketch a few examples.
  2. If \(a=\dfrac{1}{n}\) (where \(n\) is a positive integer), rewrite the power function using rules of exponents.
  3. The functions in part (b) are called root functions. Sketch the graph of the power function if \(a=\dfrac{1}{2}\) and \(= \dfrac{1}{3}\text{.}\)
  4. When \(a=-1\text{,}\) we call the power function the reciprocal function. Sketch the graph.
Solution.
  1. We see the standard polynomial functions. Graphs will vary.
  2. \(\displaystyle x^{a} = x^{1/n} = \sqrt[n]{x}\)
  3. The graph of \(\sqrt{x}\) on the interval \([-1,1]\)
    The graph of \(\sqrt[3]{x}\) on the interval \([-1,1]\)
  4. The graph of \(1/x\) on the interval \([-2,2]\)

Subsubsection 1.2.3.2 Rational Functions

Definition 1.2.11. Rational Function.
A rational function is a ratio of two polynomials, written as \(f(x) = \dfrac{P(x)}{Q(x)}\text{,}\) where \(P(x)\) and \(Q(x)\) are polynomials.
The domain of a rational function consists of all values \(x\) such that \(Q(x)\neq 0\text{.}\)
Example 1.2.12.
Write the domain of the following functions, in interval notation.
  1. \(\displaystyle f(x) = \dfrac{1}{x}\)
  2. \(\displaystyle g(x) = \dfrac{2x^3 - 1}{x^5 + 1}\)
  3. \(\displaystyle h(x) = \dfrac{6x^4 + x^3 - 7x^2 + 6.5029}{x^2 - 9}\)
Solution.
  1. \(\displaystyle (-\infty,0)\cup (0,\infty)\)
  2. \(\displaystyle (-\infty,-1)\cup (-1,\infty)\)
  3. \(\displaystyle (-\infty,-3)\cup (-3,3)\cup (3,\infty)\)

Subsubsection 1.2.3.3 Algebraic Functions

Definition 1.2.13. Algebraic Function.
A function \(f\) is called an algebraic function if it can be constructed using algebraic operations, such as addition, subtraction, multiplication, division, and taking roots.
Example 1.2.14.
Write an example of an algebraic function using at least one of each type of function we have covered so far.
Solution.
Answers vary

Subsubsection 1.2.3.4 Trigonometric Functions

Basic Trigonometric Functions
\(\sin x = \dfrac{opp}{hyp}\) \(\cos x = \dfrac{adj}{hyp}\) \(\tan x = \dfrac{opp}{adj}\)
\(\csc x = \dfrac{hyp}{opp}\) \(\sec x = \dfrac{hyp}{adj}\) \(\cot x = \dfrac{adj}{opp}\)
A right triangle with angle \(x\text{,}\) with hypotenuse labeled “hyp”, opposite side labeled “opp”, and adjacent labeled “adj”.
Here are some useful properties of trigonometric functions
\(\sin x\) \(\cos x\) \(\tan x\) \(\cot x\) \(\sec x\) \(\csc x\)
Domain \((-\infty,\infty)\) \((-\infty,\infty)\) \(x \neq \dfrac{\pi}{2} + \pi k\text{,}\) \(k\in \Z\) \(x\neq \pi k\text{,}\) \(k\in \Z\) \(x \neq \dfrac{\pi}{2} + \pi k\text{,}\) \(k\in \Z\) \(x\neq \pi k\text{,}\) \(k\in \Z\)
Range \([-1,1]\) \([-1,1]\) \((-\infty,\infty)\) \((-\infty,\infty)\) \((-\infty,1]\cup [1,\infty)\) \((-\infty,1]\cup [1,\infty)\)
Period \(2\pi\) \(2\pi\) \(\pi\) \(\pi\) \(2\pi\) \(2\pi\)
Knowing the values of the unit circle will make your life much easier; fill it out below.
The unit circle with special angles marked, as well as trig values for the angles
Common Trig Identities.
\(\sin^2\theta +\cos^2\theta=1\) \(\sin(2\theta) = 2\sin\theta\cos\theta\)
\(\cos(2\theta) = \cos^2\theta - \sin^2\theta\) \(\cos(2\theta) = 2\cos^2\theta - 1\)
\(\cos(2\theta) = 1-2\sin^2\theta\)
Example 1.2.15.
Sketch the graph of each trig function over one period. If necessary, sketch any asymptotes the graph has.
Solution.
The graph of \(\sin x\) over one period
The graph of \(\cos x\) over one period
The graph of \(\tan x\) over one period
The graph of \(\csc x\) over one period
The graph of \(\sec x\) over one period
The graph of \(\cot x\) over one period
Example 1.2.16.
Find the domain of the function \(f(x) = \dfrac{3}{2\sin x + 1}\text{,}\) first on the interval \([0,2\pi)\text{,}\) then in general.
Solution.
On \([0,2pi)\text{,}\) the domain is \(\left[0,\dfrac{7\pi}{6}\right)\cup \left(\dfrac{7\pi}{6},\dfrac{11\pi}{6}\right)\text{.}\) The general domain is \(x\neq \dfrac{7\pi}{6} + 2\pi k\) and \(x\neq \dfrac{11\pi}{6}+2\pi k\text{,}\) where \(k\) is an integer.

Subsection 1.2.4 After Class Activities

Example 1.2.17.

Classify each function as one of the types of functions discussed in this section.
  1. \(\displaystyle k(t) = t-5+t^2\)
  2. \(\displaystyle \ell(x) = (0.25)^x\)
  3. \(\displaystyle m(y) = \dfrac{1-x}{\sqrt{1-x^2}}\)
  4. \(\displaystyle n(\theta) = \theta^{0.25}\)
  5. \(\displaystyle o(r) = \dfrac{r^3-5r^2 + r^4-1}{r^2 + r+10}\)
Solution.
  1. Polynomial, algebraic
  2. Exponential
  3. Algebraic
  4. Power, algebraic
  5. Rational, algebraic

Example 1.2.18.

Ecologists have modeled the relationship between the area of a region and the number of species inhabiting the region. In particular, the number of species \(S\) of bats living in and around Austin has been related to the surface area \(A\) of the structure by the equation \(S = 0.3A^{0.5}\text{.}\)
  1. A structure near I-35 has a surface area \(A = 10000\) m\(^2\text{;}\) how many species of bat do you expect to find on the structure?
  2. If an overpass has nine species of bats, estimate the area of overpass.
Solution.
  1. 30 species
  2. 900m\(^2\)

Example 1.2.19.

Find the domain of the following functions.
  1. \(\displaystyle t(k) = \sqrt{k^2-6}\)
  2. \(\displaystyle y(b) = \sec\lrpar{\dfrac{\pi}{2}b}\)
  3. \(\displaystyle h(x) = \dfrac{x+3}{2\sqrt{5-x}}\)
Solution.
  1. \(\displaystyle (-\infty,-\sqrt{6}]\cup [\sqrt{6},\infty)\)
  2. \(b\neq 2k+1\) for \(k\in \Z\)
  3. \(\displaystyle (-\infty,5)\)

Subsection 1.2.5 Section 1.2 Resources

Subsubsection 1.2.5.1 Essential Functions

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