Download - Math 31A October 8 Lesson Plan
Math 31A October 8 Lesson Plan
• Warm-up problem: Consider the function
{ET, ×-4
fix) = 10 , ×-4.
Which of the following best describes the
behavior of f
at ✗ = 4?
A. There is a removable discontinuity. *
B. There is a jump discontinuity.
C. There is an infinite discontinuity.
D- There is an oscillating discontinuity.
E. The function is continuous.
To see why the correct answer is A,
note that we can
factor ✗ 2 -16 = (✗ -4) Cx + 4)
(this is a
difference of squares). So =
X + 4 for
all ✗ #4. Thus, ¥54T Lex =
¥34-Lex> = 4+4=8. However, f (4)
=L 0 #8. So f has a removable
discontinuity at ✗ = 4.
I. Review
• In lecture, you've been discussing
limit laws and continuity
(including the various types of discontinuities].
• Your homework is on evaluating limits
and applying the
limit laws.
• Are there any questions on this
material?
Spend as much time as needed on questions.
I. Problems
(2.4. 6 6) Find the value of the constant c
that makes the function continuous at the
point × =3.
{2x + 9×-1 , for
× =3
fix> =-4x + c , for
✗ > 3.
Sot To check that it is possible to choose
such a value
for c, we must first check that the left-
and right -
hand limits of f at ✗ =3 "For all ✗
<3, we have f (x) = 2x + 9×-1, so
3- fix> = 3-
(2×+9×-1)
= 2 ¥33-✗ + 9
Its-I'
= 2. 3 + 9 - 1/3
= 6 + 3 = 9.
For all ✗ > 3, we have f (x) = -4x
+ e, so
¥3-Aca- ¥3T c- 4×+0
=-4 ⇒ + × +
¥3-c
= -4 -3 + c
= -12 + C.
Now, in order for ¥33 LID to exist,
we need the
left- and right- hand limits to be
equal. So we
need to solve for a such that
9 = -12 + c.
Thus c= 21.
• The floor function is defined as follows:
For
each ×, fix) is the greatest integer less
than or equal to ×. CE. g. , fL3-27)
=3, I C- 1.73=-2.) where do the
discontinuities of this function occur?
Describe them.
Sot The floor function looks like this i
There is a jump discontinuity at each
integer. If n is an integer>
IF-LCD = n-1, while ✗ Ent
fo-n.
•2- 4.97" Show" that f is a
discontinuous
function for all ×, where LCD is
defined as follows:
{is ✗ is rational
fix) = -1, × is irrational
Show that It is continuous for all
✗ .
Sot Since 12 is just the constant
function 1> it is continuous
everywhere.
To see that f is not continuous
at any ✗ , let ✗ be any real
number. Between any two distinct reel
numbers, there exist both a rational
number and an irrational number. Thus, no
matter how close we get to ✗ , we will
meet values ×, and ×, such that
Lex,> = 1 and fly)=-1. So, in fact,
neither the left- no-the right-hand
limit at ✗ exists. (This is a bit
informal. To improve it, we could do
what's called an
epsilon -delta proof, but proofs are
beyond the scope of this course?
• 2.4.71 Draw the graph of a function on
[95] with
the given properties: f is not continuous at ✗
=L, but
✗ "It fix> and I - fix> exist and
are equal.
so We are being asked to draw a graph of
a function with a removable discontinuity at ✗
= 1. There are many possibilities,
one of which is sketched below.
'2.4.71Draw the graph of a function on
[95] with
the given properties: if has a removable
discontinuity at
✗ =L, a jump discontinuity at ✗ =L, and
¥3-Lex> =-• ,
33+11×3=2.
⇒ Again> there are many possibilities. Recall
that we have a:
• removable discontinuity when the left-
and right - hand
limits exist and are equal;
• jump discontinuity when the left- and
right-hand limits exist but are unequal.
•2.4.90: which of the following quantities
would be represented by continuous
functions of time and which would have one
or more discontinuities?
⑨ Velocity of an airplane during a flight
⑥ Temperature in a room under ordinary
conditions
⑨ Value of a bank account with interest
paid yearly
② salary of a teacher
② Population of the world
So Some of these are a little debatable,
but they are fun to think about!
⑨ Continuous. Even in cases of rapid
acceleration or deceleration,
the velocity does not change drastically in a
single instant.
⑥ continuous. We usually conceive of
temperature as
changing gradually-
⑨ Jump discontinuities. At the instant when
the interest is
paid, the value of the account jumps
from one point
to another.
② Jump discontinuities. At the instant
when the salary is
raised, the payment value jumps from
one point to
another.
⑨ Jump discontinuities. At the instant
when someone is born
or dies, the population jumps or falls by
a whole
number.
Graph these if students wish to see
them.
22.4.81 Suppose that f and g are
discontinuous at
✗ =L. Does it follow that ft ng is
discontinuous at ✗ =L?
If not, give a counterexample-
Sot This does not follow in general.
To try and
find a counterexample, we'll consider
additive
cancelations. (This is always a natural
place to look when we're dealing with
addition!) Let's consider
{1. ✗ so
Lex> = -1, ✗ <0 and
{ 'is ✗ so
go = Is ✗ <0.
Notice that, for all ✗ , LCD +
go =D. So Ltg is continuous
everywhere, and in particular at
✗ =D. However, each of f and ng hes
a jump discontinuity
at ✗ =D. So this is a valid
counterexample. (There are many more
counterexamples?