Math 3, Fall 2015
Scott Pauls
Department of Mathematics
Dartmouth College
Class 5 - 9/25/2015
Seating
1: the W𝜙ΣS
3
6
9
2
5
8
11
1
4
7
10
2: Don't drink and derive
3: Apple 𝜋
4: The Dirty Derivatives
5: Too cool to function
6: Gee I'm a tree
Front of the classroom
7: Push it to the limit
8: ∞ and beyond
9: Gettin' triggy with it
10: Shapes on a plane
11: Kool by induction
xhour - 9/24/2015
Facilitator
Scribe
Presenter/Timer
Checker
Wildcard
Wildcard
Facilitator
Scribe
Presenter/Timer
Checker
Intuition vs. proof
Use the definition of the limit to show that
lim 3𝑥 + 1 = 7
𝑥→2
The limit of 𝑓(𝑥) as 𝑥 approaches 𝑎 equals 𝐿 if for every 𝜖 >
0 there exists a 𝛿 > 0 so that if 𝑥 − 𝑎 < 𝛿 then 𝑓 𝑥 − 𝐿 < 𝜖.
Class 5 - 9/25/2015
Intuition vs. proof
Use the definition of the limit to show that
lim 3𝑥 + 1 = 7
𝑥→2
Given an 𝜖 > 0, we must find a 𝛿 > 0 so that if 𝑥 − 2 < 𝛿
then 3𝑥 + 1 − 7 < 𝜖.
Class 5 - 9/25/2015
Intuition vs. proof
Use the definition of the limit to show that
lim 3𝑥 + 1 = 7
𝑥→2
Given an 𝜖 > 0, we must find a 𝛿 > 0 so that if
𝑥 − 2 < 𝛿 then 3𝑥 + 1 − 7 < 𝜖.
First note that 3𝑥 + 1 − 7 = 3𝑥 − 6 = 3 𝑥 − 2 .
Class 5 - 9/25/2015
Intuition vs. proof
Use the definition of the limit to show that
lim 3𝑥 + 1 = 7
𝑥→2
Given an 𝜖 > 0, we must find a 𝛿 > 0 so that if
𝑥 − 2 < 𝛿 then 3𝑥 + 1 − 7 < 𝜖.
First note that 3𝑥 + 1 − 7 = 3𝑥 − 6 = 3 𝑥 − 2 .
As a consequence, if 𝑥 − 2 < 𝛿 then
3𝑥 + 1 − 7 = 3 𝑥 − 2 < 3𝛿.
Class 5 - 9/25/2015
Intuition vs. proof
Use the definition of the limit to show that
lim 3𝑥 + 1 = 7
𝑥→2
Given an 𝜖 > 0, we must find a 𝛿 > 0 so that if
𝑥 − 2 < 𝛿 then 3𝑥 + 1 − 7 < 𝜖.
First note that 3𝑥 + 1 − 7 = 3𝑥 − 6 = 3 𝑥 − 2 .
As a consequence, if 𝑥 − 2 < 𝛿 then
3𝑥 + 1 − 7 = 3 𝑥 − 2 < 3𝛿.
So, to
make this less than 𝜖 we need to choose
𝜖
𝛿 < . With this choice of 𝛿, the definition is
3
satisfied.
Class 5 - 9/25/2015
Continuity
Continuity is our first application of limits
- a function is continuous at a point if the
limit of the function exists at that point is
equal to the value of the function at the
point. Functions can fail to be continuous
is a number of ways - jump discontinuities,
removable discontinuities, vertical
asymptotes, and so on.
Definition: A function 𝑓(𝑥) is continuous
at 𝑥 = 𝑎 if lim 𝑓 𝑥 = 𝑓(𝑎).
𝑥→𝑎
Class 5 - 9/25/2015
Group work
1. For each of the points given, assign one to
each person (for groups of 4, one will be left
over).
2. (2 minutes) Each person triages the problem
– for your point, why is the function likely
continuous or not at that point?
3. (8 minutes) Each person presents to the
group and discusses the solution to the
problem. The Checker must make sure
everyone understands the solutions.
4. (10 minutes) Presentation (2 minutes each
or 5 groups)
Class 5 - 9/25/2015
For a∈ {−4, −2,0,1,3},
determine if the
function shown is
continuous at 𝑥 = 𝑎.
If the function is
discontinuous,
classify the
discontinuity as a
jump discontinuity, a
removable
discontinuity, or a
vertical asymptote.
Class 5 - 9/25/2015
at x=-4
Class 5 - 9/25/2015
Coming up next…
Introducing the idea of derivatives using
our first example – finding the slope of
a tangent line.
Class 5 - 9/25/2015
Main idea
To find the slope at 𝑥0 , look a small bit
away from that point, 𝑥0 + ℎ. What is
Δ𝑦
?
Δ𝑥
Class 5 - 9/25/2015