AP Calculus AB
Day 5
Section 1.4
3/19/2016
Perkins
Continuity
f(x) will be continuous at x = c unless one of the following occurs:
c. lim f ( x )  f (c )
b. lim f ( x ) does not exist
a. f(c) does not exist
x c
x c
lim f ( x )  lim f ( x )
x c 
6
6
4
4
2
2
x c
6
4
2
c
c
5
-2
5
c
-2
-2
Removable Discontinuity
A graph with a “hole” in it
Non-removable Discontinuity
Any other type
5
Discuss the continuity of each.
1
1. f ( x ) 
x
Not continuous at
x = 0 (V.A.)
Non-removable
x2  1
2. g ( x ) 
x 1
4
4
2
2
-5
5
-5
-2
-4
-4
Continuous function
x  1 x  1


x 1
 x 1
5
-2
3. h( x )  sin x
Not continuous at x = 1
Hole in graph at (1,2)
Removable
3 x 2 for x  2
4. f ( x )  
ax for x  2
Find a so that f ( x ) is a continuous function.
If x < 2, the function is a parabola. (continuous)
If x > 2, the function is a line. (continuous)
To be continuous, the two sides must also meet when x = 2.
lim  f ( x )  lim  3 x 2   12
x 2
lim  f ( x ) 
x 2
x 2
D.S.
lim  ax   2a
x 2
D.S.
2a  12
a6
Intermediate Value Theorem
If f is continuous on [a,b] and k is any number between f(a) and f(b),
then there exists a number c in [a,b] such that f(c) = k.
4
2
a
-5
b
c
c
-2
k
5
The red graph has 1 c-value.
Orange has 1 c-value.
-4
Blue has 5 c-values.
Translation:
If you connect two dots with a continuous function, you must hit every y-value
between them at least once.
AP Calculus AB
Day 5
Section 1.4
Perkins
Continuity
f(x) will be continuous at x = c unless one of the following occurs:
c. lim f ( x )  f (c )
b. lim f ( x ) does not exist
a. f(c) does not exist
x c
x c
lim f ( x )  lim f ( x )
x c 
6
6
4
4
2
2
x c
6
4
2
5
5
-2
Removable Discontinuity
Non-removable Discontinuity
5
-2
-2
Discuss the continuity of each.
x2  1
2. g ( x ) 
x 1
1
1. f ( x ) 
x
4
4
2
2
-5
5
-5
5
-2
-2
-4
-4
3. h( x )  sin x
3 x 2 for x  2
4. f ( x )  
ax for x  2
Find a so that f ( x ) is a continuous function.
Intermediate Value Theorem
If f is continuous on [a,b] and k is any number between f(a) and f(b),
then there exists a number c in [a,b] such that f(c) = k.
4
2
a
-5
b
c
c
-2
k
5
The red graph has 1 c-value.
Orange has 1 c-value.
-4
Blue has 5 c-values.
Translation:
If you connect two dots with a continuous function, you must hit every y-value
between them at least once.
Intermediate Value Theorem
If f is continuous on [a,b] and k is any number between f(a) and f(b),
then there exists a number c in [a,b] such that f(c) = k.
4
2
-5
5
-2
-4