Name: _________________________
1.4
1
.4
T
h
e
P
e
ci
se
D
ef
in
it
io
n
o
f a
L
im
it
1.4 The Precise Definition of a Limit
Ticket in the Door
In order to be prepared for class you must watch the module and complete the following
activity. Remember you must submit your work as a PDF file.
Example 1:
Use the given graph of ??(??) = √?? to find a ?? such that if |?? − 4| < ?? then �√?? − 2� < 0.4
Example 2:
Use the given graph of ??(??) = ??2 to find a ?? such that if |?? − 2| < ?? then |??2 − 4| < 0.3
Example 3: Prove the statement using the ??, ?? definition of a limit.
lim
x→1
9 + 3x
4
= 3
??
?? = ??2
??
1.4 The Precise Definition of a Limit
Name: _________________________
2.3
2.
3
T
ig
on
om
et
i
c F
un
ct
io
ns
Unit 2.3 Trigonometric Functions
Ticket in the Door
In order to be prepared for class you must watch the module and complete the following
activity. Remember you must submit your work as a PDF file.
Pre-Calculus Review: Rewrite cos (x+h) using the SUM/DIFFERENCES IDENTITIES
Example 1. As we did in the video, using the limiting definition to prove ????
????
[cos x] = -sin x.
Example 2: Use the quotient rule to find the derivative of
f(x) =
2-5cos x
3x2 + 5
Example 3: Given ??′′(??) = −4??????(??) and ??′(0) = −3 and ??(0) = 3. Find f �π
3
�
Unit 2.3 Trigonometric Functions
Name: _________________________
2.4
Un
it
2
–
Le
ss
on
2
.4
C
om
po
si
tio
n
of
F
un
ct
io
ns
P
a
t 1
Unit 2 – Lesson 2.4 Composition of Functions Part 1
Ticket in the Door
In order to be prepared for class you must watch the module and complete the following
activity. Remember you must submit your work as a PDF file.
Example #1 Consider y = √3??2 − ??
a. What is the “inner” function? _______ What is the derivative of this function? ___________
. What is the “outer” function? _______ What is the derivative of this function? ____________
c. Find ??
????
√3??2 − ??
Example #2 Evaluate
??
????
??????(5??3 + 12??)
Example #3 Evaluate the integral.
�(2 + x)5 ????
XXXXXXXXXXLet u = _______
________ du = dx
Example #4 Evaluate the integral.
�2x sin(4 + x2)????
XXXXXXXXXXLet u = _______
________ du = dx
Unit 2 – Lesson 2.4 Composition of Functions Part 1
Name: _________________________
2.7
Un
it
2
–
Le
ss
on
2
.7
D
e
iv
at
iv
es
o
f H
yp
e
o
lic
F
un
ct
io
ns
Unit 2 – Lesson 2.7 Derivatives of Hype
olic Functions
Ticket in the Door
In order to be prepared for class you must watch the module and complete the following
activity. Remember you must submit your work as a PDF file.
Pre-Calculus Review: Given the definitions of the hype
olic sine, cosine and tangent
functions are
??????ℎ ?? = ??
??−??−??
2
??????ℎ ?? = ??
??+??−??
2
??????ℎ ?? = ??????ℎ ??
??????ℎ ??
Find the numerical value of each expression
(a) ??????ℎ 0 (b) ??????ℎ 1 (c) ??????ℎ (???? 3)
Example 1: Find the first derivative of ??(??) = sinh−1(??2 − 1).
Example 2: Find the first derivative of ??(??) = ln[cosh(5??2 − 2??)]
Unit 2 – Lesson 2.7 Derivatives of Hype
olic Functions
Name: ________________
Section 3.1: Related Rates
TICKET-IN-THE-DOOR
In order to be prepared for class, you must complete the following activity and turn it in online.
Guidelines for solving related rates problems
1. Make a ___________ and ___________ the quantities.
2. Read the problem and identify all quantities as: ___________ , ___________ and
___________with the appropriate information.
3. Write an ___________ involving the variables whose ___________ of ___________ either are
given or are to be determined.
4. Using the _________ _________, implicitly _________ both sides of the equation with respect to
time, _________.
5. _________ into the resulting equation all known values for the variables and their rates of change.
Then _________ for the required rate of change.
Example: The Spreading Circle
An oil rig springs a leak, and the oil spreads in a circular patch around the rig. If the radius of the oil patch
increases at a rate of 30 m / hr, how fast is the area are of the patch increasing when the patch has a radius
of 100 meters?
Draw and label a sketch
Know
Given
Find