Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

i

i
“125-76608_Fernandez_CalculusSimplified_5P” — 2019/4/3 — 14:18 — page 1 — #1

i

i
© Copyright, Princeton University Press
...


Princeton University
Department of Mathematics

1

The Fast Track Introduction to Calculus
Chapter Preview
...
This
chapter provides you with a working understanding of the calculus mindset, core concepts of calculus, and the sorts of problems they help solve
...
After reading this chapter, you will have an intuitive understanding of calculus that will ground your subsequent studies of the subject
...
1 What Is Calculus?
Here’s my two-part answer to that question:
Calculus is a mindset—a dynamics mindset
...


Calculus as a Way of Thinking
The mathematics that precedes calculus—often called “pre-calculus,” which includes algebra and geometry—largely focuses on static problems: problems devoid
of change
...

Example:


...


What’s the perimeter of a square of side length 2 feet? ←− Pre-calculus
problem
...


This statics versus dynamics distinction between pre-calculus and calculus runs
even deeper—change is the mindset of calculus
...
Example:


...


Find the volume of a sphere of radius r
...
1(a))
...
Calculus mindset: Slice the sphere
into a gazillion disks of tiny thickness and then add up their volumes
(Figure 1
...
When the disks’ thickness is made “infinitesimally small” this
approach reproduces the 43 πr3 formula
...
)

For general queries, contact webmaster@press
...
edu

i

i
i

i

i

i
“125-76608_Fernandez_CalculusSimplified_5P” — 2019/4/3 — 14:18 — page 2 — #2

i

i
© Copyright, Princeton University Press
...

•

Introduction to Calculus

Interactive Figure

2

(a)

(b)

Figure 1
...


There’s that mysterious word again—infinitesimal—and I’ve just given you a clue
of what it might mean
...
Right now, let me pause to address a thought
you might have just had: “Why the slice-and-dice approach? Why not just use the
4
3
3 πr formula?” The answer: had I asked for the volume of some random blob in
space instead, that static pre-calculus mindset wouldn’t have cut it (there is no formula for the volume of a blob)
...

That volume example illustrates the power of the dynamics mindset of calculus
...
That was the dominant mindset in your mathematics courses prior
to calculus mathematics courses, so you’re accustomed to thinking that way about
math
...
Let’s continue the adventure
by returning to what I’ve been promising: insight into infinitesimals
...
Here’s
a rough definition:
“Infinitesimal change” means: as close to zero change as you can imagine,
but not zero change
...
490–430 BC), a Greek philosopher
who devised a set of paradoxes arguing that motion is not possible
...
) One such paradox—the Dichotomy Paradox—
can be stated as follows:
To travel a certain distance you must first traverse half of it
...
2 illustrates this
...
But
because of Zeno’s mindset, with his first step he walks only half the distance: 1 foot

For general queries, contact webmaster@press
...
edu

i

i
i

i

i

i
“125-76608_Fernandez_CalculusSimplified_5P” — 2019/4/3 — 14:18 — page 3 — #3

i

i
© Copyright, Princeton University Press
...


1
...
75

∆d = 0
...
5

∆d = 1

•

∆d = 0
...
2: Zeno trying to walk a distance of 2 feet by traversing half the remaining distance with
each step
...
2(b))
...
5
foot (Figure 1
...
Table 1
...

Table 1
...

d
d

1
1

0
...
5

0
...
75

0
...
875

0
...
9375

0
...
96875

0
...
984375

0
...
9921875

···
···

Each change d in Zeno’s distance is half the previous one
...
If we checked back in with Zeno after he’s taken an infinite
amount of steps, the change d resulting from his next step would be
...
an infinitesimal change—as close to zero as you can imagine but not equal
to zero
...
First, it illustrates the dynamics mindset of calculus
...
(Calculus is full
of action verbs!) Second, the example challenges us
...

But as Table 1
...
How do we describe this fact with
an equation? (That’s the challenge
...
We need a
new concept that quantifies our very dynamic conclusion
...


1
...
1
...
Specifically:
d + d = 2,

or equivalently,

d = 2 − d
...
1)

For general queries, contact webmaster@press
...
edu

i

i
i

i

i

i
“125-76608_Fernandez_CalculusSimplified_5P” — 2019/4/3 — 14:18 — page 4 — #4

i

i
© Copyright, Princeton University Press
...


4

•

Introduction to Calculus

This equation relates each d value to its corresponding d value in Table 1
...
Great
...
The table clearly shows that the distance d traveled by Zeno
approaches 2 as d approaches zero
...

(We are using “→” here as a stand-in for “approaches
...
” To express this notion, we write
lim d = 2,

d→0

(1
...
”
Equation (1
...
It expresses the intuitive
idea that the 2-foot mark is the limiting value of the distance d Zeno’s traversing
...
2)
...
2), therefore, is a statement
about the dynamics of Zeno’s walk, in contrast to (1
...
Moreover, the Equation (1
...
The same idea holds for d:
it is always approaching 0 yet never arrives at 0
...

We will learn much more
lim Y
∆x→0
about limits in Chapter 2 (including that (1
...
But the
Finite change ∆x in Y
Infinitesimal change in Y
Zeno example is sufficient to
(Not a calculus concept)
(Calculus concept)
give you a sense of what the
Figure 1
...

calculus concept of limit is and
how it arises
...
Figure 1
...
e
...
Working
through this process—like we just did with the Zeno example, and like you can now
recognize in Figure 1
...
This is what I
meant earlier when I said that calculus is the mathematics of infinitesimal change—
contentwise, calculus is the collection of what results when we apply the workflow
in Figure 1
...


For general queries, contact webmaster@press
...
edu

i

i
i

i

i

i
“125-76608_Fernandez_CalculusSimplified_5P” — 2019/4/3 — 14:18 — page 5 — #5

i

i
© Copyright, Princeton University Press
...


1
...
In the next section
we’ll preview how the calculus workflow in Figure 1
...

(We’ll fill in the details in Chapters 3–5
...
3 The Three Difficult Problems That Led to the Invention
of Calculus
Calculus developed out of a need to solve three Big Problems (refer to Figure 1
...
The instantaneous speed problem: Calculate the speed of a falling object at a
particular instant during its fall
...
4(a)
...
The tangent line problem: Given a curve and a point P on it, calculate the slope
of the line “tangent” to the curve at P
...
4(b)
...
The area under the curve problem: Calculate the area under the graph of a
function and bounded by two x-values
...
4(c)
...
4 already gives you a sense of why these problems were so difficult to
solve—the standard approach suggested by the problem itself just doesn’t work
...

The tangent line problem asks you to calculate the slope of a line using just one
point (point P in Figure 1
...
Similarly, we think of speed as “change in distance
divided by change in time
...

Speed at this instant = ?

Slope of this line = ?

Area of the shaded region = ?

y

y
y = f (x)

y = f (x)

P
x

x
a

(a)

(b)

b
(c)

Figure 1
...

1 These may not seem like important problems
...


For general queries, contact webmaster@press
...
edu

i

i
i

i

i

i
“125-76608_Fernandez_CalculusSimplified_5P” — 2019/4/3 — 14:18 — page 6 — #6

i

i
© Copyright, Princeton University Press
...


6

•

Introduction to Calculus

Limiting
picture

Calculus workflow

Calculus
result

∆d

∆d
∆d

Instantaneous
speed:
∆d
lim —
∆t→0 ∆t

∆t → 0

Large ∆t
y

Smaller ∆t
y

y = f (x)

P

Even smaller ∆t

Infinitesimally small ∆t
y

y = f (x)

y = f (x)
Slope of the
tangent line:
∆y
lim —
∆x→0 ∆x

∆x → 0

∆y

P

∆x
x

a
Large ∆x

∆y
∆x

P
x

a
Smaller ∆x

Area is swept out …

… up to ∆x past b …

y

y
y = f (x)

x
a
Infinitesimally small ∆x

y
y = f (x)

y = f (x)

Shaded area
under the
curve:
lim A∆x

∆x → 0

a

b

{

∆x→0

x

a

b

∆x

… with resulting area
denoted A∆x

x

a

b

x

Infinitesimally small ∆x

Figure 1
...
3) applied to the three Big Problems
...

Nothing about Figure 1
...
” Every image is a static snapshot of something (e
...
, an area)
...
(Yep, I’m encouraging you to think
of calculus as a verb
...
5 illustrates the application of the calculus workflow (from Figure 1
...
Each row uses a dynamics mindset to recast the problem as

For general queries, contact webmaster@press
...
edu

i

i
i

i

i

i
“125-76608_Fernandez_CalculusSimplified_5P” — 2019/4/3 — 14:18 — page 7 — #7

i

i
© Copyright, Princeton University Press
...


1
...
g
...

Specifically:


...


...

Row #2: The slope of the tangent line is realized as the limit of the slopes of
y
the secant lines x (the gray lines in the figure) as x → 0
...


The limit obtained in the second row of the figure is called the derivative of f (x)
at x = a, the x-value of point P
...
Derivatives and integrals round out the three most important concepts in calculus (limits are the third)
...
5
...
Looking back now at Figures 1
...
2, and 1
...
We will employ both throughout the book
...
See you in the next chapter
...
princeton

---