Weijiu Liu

1
Calculus

• Weijiu Liu
• Department of Mathematics
• University of Central Arkansas

2
Overview
3
What is calculus?
Calculus is a subject about the study of limiting
processes and it consist of three basic
concepts limit, differentiation, and integration.
4
Who founded calculus?
Isaac Newton (1643 1727) was the greatest
English mathematician of his generation. He laid
the foundation for differential and integral
calculus. His work on optics and gravitation
make him one of the greatest scientists the
world has known.
http//www-groups.dcs.st-and.ac.uk/history/PictDi
splay/Newton.html
5
Gottfried Wilhelm von Leibniz (1646 1716) was a
German mathematician who developed the present
day notation for the differential and integral
calculus though he never thought of the
derivative as a limit. His philosophy is also
important and he invented an early calculating
machine.
http//www-groups.dcs.st-and.ac.uk/history/PictDi
splay/Leibniz.html
For a history of the calculus, see
http//www-groups.dcs.st-and.ac.uk/history/HistT
opics/The_rise_of_calculus.html
6
Why is calculus needed?

• Geometrical problems.

Slope of a sceant line
yf(x)
Tangent line
Q(x,f(x))
Q(x,f(x))
Q(x,f(x))
Slope of a tangent line
7
The area A of a range under the graph of a
function
y
y
y f(x)
y f(x)
x
x
a
b
a
b
8

• Physical problems.

s
Average velocity
Instantaneous velocity
9
m
a
x
b
Work
m
a
x
b
Work
10

• More advanced problems.

Description of particle motion
11
Description of evolution of chemical
concentration partial differential equation

Convection Diffusion Equation
Convection
Diffusoin
12
Description of the string vibration the wave
equation
13
How to study calculus?

• Read your textbook
• Understand concepts clearly
• Do your homework on time
• Ask your instructor and classmates around you
  whenever you have a question
• Learn from your classmate
• Form a group to discuss

14
Tentative schedule

• Chapter 1 Limits, 3 weeks
• Chapter 2 Differentiation, 3 weeks
• Chapter 3 Application of differentiation, 3
  weeks
• Chapter 4 Integration, 3 weeks
• Chapter 5 Application of definite integrals, 2
  weeks

15
Chapter 1-- Limits
16
Problem of Limit
Tangent line problem
Slope of a sceant line
yf(x)
Tangent line
Q(x,f(x))
Q(x,f(x))
Q(x,f(x))
Slope of a tangent line
Problem of Limit As x gets closer and close to
, to what number is a function g(x) like
getting
closer and closer to even though g(x) is not
well defined at ?
17
Find limits of a function graphically
Consider the function
As xgt0 gets closer and closer to 0 f(x) is
getting closer and closer to 1.
As xlt0 gets closer and closer to 0 f(x) is
getting closer and closer to 1.
We say the limit of f(x) as x approaches 0 from
the right is 1, written
We say the limit of f(x) as x approaches 0 from
the left is 1, written
One-sided limits
We say the limit of f(x) as x approaches 0 is 1,
written
18
Find limits of a function numerically
x sin x / x
-0.10000000000000 0.99833416646828
-0.01000000000000 0.99998333341667
-0.00100000000000 0.99999983333334
-0.00010000000000 0.99999999833333
-0.00001000000000 0.99999999998333
-0.00000100000000 0.99999999999983
-0.00000010000000 1.00000000000000
-0.00000001000000 1.00000000000000
-0.00000000100000 1.00000000000000
-0.00000000010000 1.00000000000000
1
0
19
x sin x / x
0.10000000000000 0.99833416646828
0.01000000000000 0.99998333341667
0.00100000000000 0.99999983333334
0.00010000000000 0.99999999833333
0.00001000000000 0.99999999998333
0.00000100000000 0.99999999999983
0.00000010000000 1.00000000000000
0.00000001000000 1.00000000000000
0.00000000100000 1.00000000000000
0.00000000010000 1.00000000000000
0
1
20
In general, if f(x) is getting closer and closer
to L as x gets closer and closer to a, we say
that the limit of f(x) as x approaches a is L,
written
Theorem.
21
Example.
So does not exist!
22
Example.
So does not exist!
23
Graph of problem 4 of Exercises 1.2
24
Intermediate Value Theorem
a b f(a)
f(b) Midp f(midp) 2.0000 3.0000
-2.0000 13.0000 2.5000 3.6250
2.0000 2.5000 -2.0000 3.6250 2.2500
0.3906 2.0000 2.2500 -2.0000 0.3906
2.1200 -0.9519 2.1200 2.2500 -0.9519
0.3906 2.1800 -0.3598 2.1800 2.2500
-0.3598 0.3906 2.2100 -0.0461
2.2100 2.2500 -0.0461 0.3906 2.2300
0.1696 2.2100 2.2300 -0.0461 0.1696
2.2200 0.0610 2.2100 2.2200 -0.0461
0.0610 2.2100 -0.0461 2.2100 2.2200
-0.0461 0.0610 2.2100 -0.0461
2.2100 2.2200 -0.0461 0.0610 2.2100
-0.0461
25
a b f(a)
f(b) Midp f(midp) -1.0000
0 1.0000 -2.0000 -0.5000 -0.1250
-1.0000 -0.5000 1.0000 -0.1250 -0.7500
0.5781 -0.7500 -0.5000 0.5781 -0.1250
-0.6300 0.2700 -0.6300 -0.5000 0.2700
-0.1250 -0.5700 0.0948 -0.5700
-0.5000 0.0948 -0.1250 -0.5400 0.0025
-0.5400 -0.5000 0.0025 -0.1250 -0.5200
-0.0606 -0.5400 -0.5200 0.0025 -0.0606
-0.5300 -0.0289 -0.5400 -0.5300
0.0025 -0.0289 -0.5400 0.0025 -0.5400
-0.5300 0.0025 -0.0289 -0.5400 0.0025
-0.5400 -0.5300 0.0025 -0.0289 -0.5400
0.0025