Democritus was a Greek philosopher who actually developed the atomic theory, he was also an...
27
ritus was a Greek philosopher who actually develope c theory, he was also an excellent geometer. Democritus of Abdera 460 – 370 B.C.
-
Upload
sharyl-page -
Category
Documents
-
view
225 -
download
2
Transcript of Democritus was a Greek philosopher who actually developed the atomic theory, he was also an...
Democritus was a Greek philosopher who actually developed the atomic theory, he was also an excellent geometer.
Democritus of Abdera460 – 370 B.C.
Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.
Length of Fish Parking Meter Cost
Example 1
2
2
3 2 1a. ( ) ; 6
6
x xf x a
x x
2 3 4 if 4
b. ( ) ; 445 if 4
x xx
f x axx
Example 2
2
2
2 5 1a. ( )
2
x xf x
x
2
2
5 6b. ( )
9
x xf x
x
Examples: 2siny x cosy x
“The composition of two continuous functions is continuous.”
Example 3
3 2
2a. lim 12
xx
2 9
3
3b. lim
x
x
xe
1. ( ) existsf a
2. lim ( ) existsx a
f x
3. lim ( ) ( )x a
f x f a
,x a b
1. ( ) existsf b
2. lim ( ) existsx b
f x
3. lim ( ) ( )x b
f x f b
,x a b
It is continuous at x = 0, x = 3 and x = 4.
Take x = 0, since
1. (0) 1 existsf
02. lim ( ) 1 exists
xf x
03. lim ( ) (0)
xf x f
Hence, by definition f is continuous at x = 0.
For the function below, discuss the integer values where y = f (x) is continuous and explain.
Example 4
1 2 3 4
1
2
x
y
y = f (x)
It is continuous at x = 0, x = 3 and x = 4.
For the function below, discuss the integer values where y = f (x) is continuous and explain.
Example 4
1 2 3 4
1
2
x
y
y = f (x)
Take x = 3, since
1. (3) 2 existsf
32. lim ( ) 2 exists
xf x
33. lim ( ) (3)
xf x f
Hence, by definition f is continuous at x = 3.
It is continuous at x = 0, x = 3 and x = 4.
For the function below, discuss the integer values where y = f (x) is continuous and explain.
Example 4
1 2 3 4
1
2
x
y
y = f (x)
Take x = 4, since
1. (4) 1 existsf
42. lim ( ) 1 exists
xf x
43. lim ( ) (4)
xf x f
Hence, by definition f is continuous at x = 4.
For the function below, discuss the integer values where y = f (x) is not continuous and explain.
Example 5
1 2 3 4
1
2
x
y
y = f (x)
This function has discontinuities at x = 1 and x = 2.
Take x = 1, since
1. (1) 1 existsf
12. lim ( ) DNE
xf x
13. lim ( ) (1)
xf x f
Hence, by definition f is discontinuous at x = 1.
For the function below, discuss the integer values where y = f (x) is not continuous and explain.
Example 5
1 2 3 4
1
2
x
y
y = f (x)
This function has discontinuities at x = 1 and x = 2.
Take x = 2, since
1. (2) 2 existsf
22. lim ( ) 1 exists
xf x
23. lim ( ) (2)
xf x f
Hence, by definition f is discontinuous at x = 2.
Determine the intervals where y = f (x) is continuous.
0,1 1, 2 2, 4x
Example 6
JumpInfinite
Types of Discontinuities:
Undefined Removable
From the graph of f, state the numbers at which f is discontinuousand describe the type of discontinuity.
Example 7
Consider the function
3
2
1,
1
xf x
x
f has discontinuities at .1x
3
21
1lim
1x
x
x
2
1
1 1lim 1 1x
x x xx x
1 1 1
2
3
2
a) What type of discontinuities occur at x = 1 and x = -1.
3A hole at 1, , undefined discontinuity.
2
3
21
1lim
1x
x
x
2
1
1 1lim 1 1x
x x xx x
2
1
1lim 1x
x xx
Solution
By definition, x = -1 is a vertical asymptote, infinite discontinuity.
Example 8
3
2
1, 1, 1
1( )3
, 12
xx
xf x
x
Note: The other discontinuity at x = -1 can not be removed, since it is a vertical asymptote.
b) Write a piece-wise function using f (x) that is continuous at x = 1.
Take x = 1, it follows
31. (1) exists
2f
1
32. lim ( ) exists
2xf x
13. lim ( ) (1)
xf x f
Hence, by definition f is continuous at x = 1.
5 4 3 2 1 0 1 2 3 4 5
5
4
3
2
1
1
2
3
4
5
1
Example 9
Example 10
2 3/2( ) (9 )f x x
Example 11
14Consider the function ( ) ln tan , determine the following:f x x x
1
a. Find the domain of ( ).
b. Find lim ( ), if possible.x
f x
f x
Example 13
