version of the first six chapters

244 Pages·2012·1.78 MB·English

Calculus
Late Transcendentals
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this work or a derivative, include the history of the document.
This text was initially written by David Guichard. The single vari-
able material in chapters 1–9 is a modification and expansion of notes
written by Neal Koblitz at the University of Washington, who gener-
ously gave permission to use, modify, and distribute his work. New
material has been added, and old material has been modified, so
some portions now bear little resemblance to the original.
The book includes some exercises and examples from Elementary
Calculus: An Approach Using Infinitesimals, by H. Jerome Keisler,
available at http://www.math.wisc.edu/~keisler/calc.html under
a Creative Commons license. In addition, the chapter on differential
equations and the section on numerical integration are largely derived
from the corresponding portions of Keisler’s book. Albert Schueller,
Barry Balof, and Mike Wills have contributed additional material.
This copy of the text was compiled from source at 11:59 on 9/27/2012.
I will be glad to receive corrections and suggestions for improvement
at
For Kathleen,
without whose encouragement
this book would not have
been written.
Introduction
The emphasis in this course is on problems—doing calcula-
tions and story problems. To master problem solving one
needs a tremendous amount of practice doing problems.
The more problems you do the better you will be at doing
them, as patterns will start to emerge in both the prob-
lems and in successful approaches to them. You will learn
fastest and best if you devote some time to doing problems
every day.
Typically the most difficult problems are story prob-
lems, since they require some effort before you can begin
calculating. Here are some pointers for doing story prob-
lems:
1. Carefully read each problem twice before writing
anything.
2. Assign letters to quantities that are described only
in words; draw a diagram if appropriate.
3. Decide which letters are constants and which are
variables. A letter stands for a constant if its value
remains the same throughout the problem.
4. Using mathematical notation, write down what
you know and then write down what you want to
find.
5. Decide what category of problem it is (this might
be obvious if the problem comes at the end of a
particular chapter, but will not necessarily be so
obvious if it comes on an exam covering several
chapters).
6. Double check each step as you go along; don’t wait
until the end to check your work.
7. Use common sense; if an answer is out of the range
of practical possibilities, then check your work to
see where you went wrong.
Suggestions for Using This Text
1. Read the example problems carefully, filling in any
steps that are left out (ask someone for help if you
can’t follow the solution to a worked example).
2. Later use the worked examples to study by cover-
ing the solutions, and seeing if you can solve the
problems on your own.
3. Most exercises have answers in Appendix A; the
a v a i l a b i l i t y o f a n a n s w⇒e ”r iast mt ha er k e d b y “
e n d o f t h e e x e r c i s e . I n t h e p d f v e r s i o n o f t h e f u l l
t e x t , c l i c k i n g o n t h e a r r o w w i l l t a k e y o u t o t h e a n -
s w e r . T h e a n s w e r s s h o u l d b e u s e d o n l y a s a fi n a l
c h e c k o n y o u r w o r k , n o t a s a c r u t c h . K e e p i n m i n d
t h a t s o m e t i m e s a n a n s w e r c o u l d b e e x p r e s s e d i n
v a r i o u s w a y s t h a t a r e a l g e b r a i c a l l y e q u i v a l e n t , s o
d o n ’ t a s s u m e t h a t y o u r a n s w e r i s w r o n g j u s t b e -
c a u s e i t d o e s n ’ t h a v e e x a c t l y t h e s a m e f o r m a s t h e
a n s w e r i n t h e b a c k .
4 .A f e w fi g u r e s i n t h e b o o k a r e m a r k e d w i t h “ ( A P ) ”
a t t h e e n d o f t h e c a p t i o n . C l i c k i n g o n t h i s s h o u l d
o p e n a r e l a t e d i n t e r a c t i v e a p p l e t o r S a g e w o r k -
s h e e t i n y o u r w e b b r o w s e r . O c c a s i o n a l l y a n o t h e r
l i n k w i l l d o t h e s a m te hti hs i en xg a, ml ipkl e .
( N o t e t o u s e r s o f a p r i n t e d t e x t : t h e w o r d s “ t h i s
e x a m p l e ” i n t h e p d f fi l e a r e b l u e , a n d a r e a l i n k
t o a S a g e w o r k s h e e t . )
1
Analytic Geometry
Much of the mathematics in this chapter will be review
for you. However, the examples will be oriented toward
applications and so will take some thought.
In the (x, y) coordinate system we normally write the
x-axis horizontally, with positive numbers to the right of
the origin, and the y-axis vertically, with positive numbers
above the origin. That is, unless stated otherwise, we take
“rightward” to be the positive x-direction and “upward”
to be the positive y-direction. In a purely mathematical
situation, we normally choose the same scale for the x-
and y-axes. For example, the line joining the origin to the
◦
point (a, a) makes an angle of 45 with the x-axis (and also
with the y-axis).
In applications, often letters other than x and y are
used, and often different scales are chosen in the horizontal
and vertical directions. For example, suppose you drop
something from a window, and you want to study how its
height above the ground changes from second to second. It
is natural to let the letter t denote the time (the number of
seconds since the object was released) and to let the letter h
denote the height. For each t (say, at one-second intervals)
you have a corresponding height h. This information can
be tabulated, and then plotted on the (t, h) coordinate
plane, as shown in figure 1.1.
We use the word “quadrant” for each of the four re-
gions into which the plane is divided by the axes: the first
quadrant is where points have both coordinates positive,
or the “northeast” portion of the plot, and the second,
third, and fourth quadrants are counted off counterclock-
wise, so the second quadrant is the northwest, the third is
the southwest, and the fourth is the southeast.
Suppose we have two points A and B in the (x, y)-
plane. We often want to know the change in x-coordinate
(also called the “horizontal distance”) in going from A to
B. This is often written ∆x, where the meaning of ∆ (a
capital delta in the Greek alphabet) is “change in”. (Thus,
∆x can be read as “change in x” although it usually is
read as “delta x”. The point is that ∆x denotes a single
number, and should not be interpreted as “delta times x”.)
For example, if A = (2, 1 ) a nBd = (3, 3 ) ,∆x = 3−2 = 1 .
S i m i l a r l y , t h e “ cyh”ains gweriint∆tye.nI n o u r e x a m p l e ,
∆y = 3 − 1 = 2 , t hffeerdenice between the y-coordinates
of the two points. It is the vertical distance you have to
move in going from A to B. The general formulas for the
change in x and the change in y between a point (x1, y1)
a n d a p o i nxt2,(y2) a r e :
∆x = x2 − x1, ∆y = y2 − y1.