Transcript AP Calculus Chapter 1, Section 4

AP Calculus
Chapter 1, Section 4
Continuity and One-Sided Limits
2013 - 2014
What is a continuous graph?
Continuity at a Point and on an Open Interval
𝑓(𝑐) is not defined
lim 𝑓(𝑥)
𝑥→𝑐
does not exist
lim 𝑓(𝑥) ≠ 𝑓(𝑐)
𝑥→𝑐
Definition of Continuity
• Continuity at a point: a function f is continuous at c if the
following three conditions are met.
1. 𝑓(𝑐) is defined.
2. lim 𝑓(𝑥) exists.
𝑥→𝑐
3. lim 𝑓(𝑥) = 𝑓 𝑐 .
𝑥→𝑐
• Continuity on an Open Interval: A function is continuous
on an open interval (a, b) if it is continuous as each point in
the interval. A function that is continuous on the entire
real line (−∞, ∞) is everywhere continuous.
Types of Discontinuity
Removable Discontinuity
Nonremovable discontinuity
Discuss the continuity of each function
1
𝑓 𝑥 =
𝑥
𝑥2 − 1
𝑔 𝑥 =
𝑥−1
𝑥 + 1,
ℎ 𝑥 = 2
𝑥 + 1,
𝑦 = sin 𝑥
𝑥≤0
𝑥>0
One-sided limits & continuity on a
closed interval
• One-sided limits come in handy when
evaluating limits on closed intervals.
– Limit from the right:
lim+ 𝑓 𝑥 = 𝐿
𝑥→𝑐
– Limit from the left:
lim− 𝑓 𝑥 = 𝐿
𝑥→𝑐
Find the limit of 𝑓 𝑥 = 4 − 𝑥 2 as x
approaches -2 from the right.
Definition of Continuity on a Closed
Interval
• A function f is continuous on the closed interval
[a, b] if it is continuous on the open interval (a, b)
and
lim+ 𝑓 𝑥 = 𝑓(𝑎)
𝑥→𝑎
and
lim− 𝑓 𝑥 = 𝑓(𝑏)
𝑥→𝑏
• The function f is continuous from the right at a
and continuous from the left at b.
Discuss the continuity of 𝑓 𝑥 = 1 − 𝑥 2
Properties of Continuity
• If b is a real number and f and g are
continuous at x=c, then the following
functions are also continuous at c.
1. Scalar Multiple: bf
2. Sum and Difference: f ± g
3. Product: fg
4. Quotient:
𝑓
,
𝑔
if g(c) ≠ 0
The following types of functions are
continuous at every point in their domains.
1. Polynomial functions:
𝑝 𝑥 = 𝑎𝑛 𝑥 2 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
2. Rational Functions: 𝑟 𝑥 =
𝑝(𝑥)
,
𝑞(𝑥)
𝑛
3. Radical Functions: 𝑓 𝑥 = 𝑥
4. Trigonometric Functions:
sin 𝑥, cos 𝑥, tan 𝑥 , sec 𝑥 , csc 𝑥
𝑞(𝑥) ≠ 0
By using these properties, you can see that each
of the following functions are continuous at
every element of its domain.
• 𝑓 𝑥 = 𝑥 + sin 𝑥
• 𝑓 𝑥 = 3 tan 𝑥
• 𝑓 𝑥 =
𝑥 2 +1
cos 𝑥
Continuity of a Composite Function
• If g is continuous at c and f is continuous at
g(c), then the composite function given by
(𝑓○ 𝑔) 𝑥 = 𝑓(𝑔 𝑥 ) is continuous at c.
Describe the interval(s) on which each
function is continuous.
𝑓 𝑥 = tan 𝑥
1
g x = sin 𝑥 ,
0,
𝑥≠0
𝑥=0
Intermediate Value Theorem
• If a function is continuous on a closed interval
[a, b], and if k is between f(a) and f(b), then
there must exist a value c in [a, b] such that
𝑓 𝑐 = 𝑘.
Height
Age
Applying IVT
• Use the Intermediate Value Theorem to show that the
polynomial function
𝑓 𝑥 = 𝑥 3 + 2𝑥 − 1
has a zero in the interval [0, 1].
Homework
• Pg. 78 – 81: 1 – 61 every other odd, 71, 75, 95