Notebook
1
Learning Target
Table of content
Page
1)
1-1 A Preview of
Calculus
1
2) 1-2 Finding limits graphically and
numerically
3) 1-3 Evaluating limits analytically
Section
4) 1-4 Continuity
1.1
p.47: 4-6, 9
1.2
p. 55
1-27 odd, 57-60 all, 66-70 all
1.3
p. 67; 37,39,4761,115,116,118,119
1.4
p. 68; 80-82, 89,90,
p.79; 3-14,17-20, 35-47
odd,51, 61-66
HW Assignment
Completed?
Quiz
Score
1-4 Continuity
Continuous : Goes on forever with no breaks , no holes, jumps,
asymptote
Discontinuity :
1. Hole Removable or non-removable discontinuity
2. Jump: Step Discontinuity (Piecewise function)
3. Asymptote (no bounds)
#
0
 
 
Definition of continuity at any point:
1. f(c) exist
(y- coordinate exist)
Eliminates: functions w/ holes
f ( x) exist
2. lim
x c
The y-coordinate we are approaching
Eliminates: jumps & asymptote
f ( x ) =f ( c )
3. lim
x c
AP standard : Make sure we know the definition of continuous
 x 1
f ( x)   2
x 1
x0
All polynomials are
continuous
x0
Prove f(x) is continuous for all values?
1. f(0) = 0+1 = 1
(0,1)
2. One-side limits
lim
x 0
𝑥+1
1
lim x 2  1
x 0

lim f ( x)  1
x 0
3. f (0)  lim f ( x) , therefore f(x) is continuous
x 0
Doesn't work: therefore the function is not continuous
1
You try:
 2x

f ( x)   3
x 1

x  1
x  1
x  1
f (1)  3
1. f(c) exist
f ( x) exist
2. lim
x c
lim f ( x)  2
x 1
lim f ( x)  2
x 1
lim f ( x)  2
x 1
f ( x ) =f ( c )
3. lim
x c
f (1)  lim f ( x) , therefore f(x) is not continuous
x 1
AP TEST (Free response)
Find the k value such that f(x) is continuous for
all value
 x3  4 x  2
f ( x)  
2
x2
 kx
Guess
AP TEST (Free response)
Find the k value such that f(x) is continuous for
all value
 x3  4 x  2
f ( x)  
2
x2
 kx
1.
f (2) =5
x 3  4  5
2. xlim
2

lim kx 2 
x2
k  2  5
2
k
5
4
What does this say ?
Guess
More examples
find the value of the constant (a, b, or c) that makes
the function continuous.