-
Notifications
You must be signed in to change notification settings - Fork 0
/
outline_2.1.tex
131 lines (77 loc) · 2.88 KB
/
outline_2.1.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
\documentclass[11pt]{article}
\usepackage[letterpaper, margin=1in]{geometry}
\usepackage{amsmath, amssymb, graphicx, epsfig, fleqn}
\setlength{\parindent}{0pt}
\newcommand{\ud}{\,\mathrm{d}}
\everymath{\displaystyle}
\def\FillInBlank{\rule{2.5in}{.01in} }
\pagestyle{empty}
\begin{document}
\begin{center}
\Large
\rm{Math 111}
\\
\rm{Chapter 2.1: Rates of Change}
\\
\end{center}
\vspace{0.2in}
\fboxsep0.5cm
If a population grows by 4,360 persons over the course of 6 years, how is it changing on average? \\
\vspace{0.2in}
If a hot coffee cools by 8$^{\circ}$C in a period of 12 minutes, how is it changing on average? \\
\vspace{0.2in}
(DEFINITION) The {\bf average rate of change} of a function $f$ over an interval $[x_1,x_2]$ is:
\vspace{1.5in}
(EXAMPLES)
\begin{enumerate}
\item{
What is the average rate of change of $g(x) = \sqrt{x-2}$ on the interval $[4,6]$?\\
\vspace{.5in}
What does this number represent graphically?
}
\vspace{1.5in}
\item{A falling object has position given by $f(t)= 4.9t^2$. What is the objects {\bf average velocity} from time $t=0$ to time $t=3$?
}
\end{enumerate}
\pagebreak
\begin{center}
\Large
\rm{Instantaneous velocity}
\end{center}
(BIG QUESTION) How can we determine the velocity of the falling object at one particular point in time? For example, what is the velocity at $t=3$?
(\emph{This is sometimes called instantaneous velocity.})
\vspace{0.2in}
We might try computing average velocities for small intervals around $t=3$.
\vspace{3.5in}
If the invervals are small, we see that the numbers \emph{approach} a single value.
\vspace{.5in}
To understand why we might look at an interval $[3, 3+ \Delta t]$ and see what happens if $\Delta t $ is small.
\vspace{2.5in}
(NOTATION)
\pagebreak
\begin{center}
\Large
\rm{Tangent problem}
\end{center}
(BIG QUESTION) How can we find the equation for a line that is tangent to curve? \emph{Tangent means that the line touches the curve and has the same slope.}
(EXAMPLE) What is the equation for the line that is tangent to the parabola $y=x^2$ at the point $(2,4)$? \\
\vspace{1.5in}
In order to answer, we need the slope of the curve at $(2,4)$. Let's call $(2,4)$ $P$, and lets choose another point $Q$ on the curve and find the slope of the line that joins $P$ and $Q$.
\vspace{4.5in}
We find that for $Q$ close to $P$, the values of the slope are close to:
\vspace{.5in}
\pagebreak
To understand why, we can look at an arbitrary point $Q$.
\vspace{1.5in}
The equation for the tangent line must then be:
\vspace{1in}
\begin{center}
\Large
\rm{Derivative}
\end{center}
(BIG IDEA) For a function $f$ and a number $a$, the rate of change of the function of $f$ at $a$ is the same as the slope of the graph at $a$.
\vspace{.25in}
(BIG DEFINITION) The {\bf derivative of a function $f$ at a number $a$} is:
\vspace{3.5in}
(EXAMPLE) If $f(x) = 2^x$, estimate the value of $f'(2)$.
\end{document}