Sample Problems for Chapters 2 and 3 (Final-Exam Review)
MAC2311, Section 17, Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
Problem 1 (similar to 9 of Section 2.3 ) Find the limit, if it exists. Otherwise, write DNE. limx-7 1-|x| 1+x Problem 2 (similar
NEW Oce Hours and Location for MAC2311 (Section 17)
Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
OFFICE HOURS:
Monday: 3:00 pm - 4:00 pm
Tuesday: 2:00 pm - 3:00 pm
Thursday: 3:00 pm - 4:00 pm
LOCATION:
Due to the increasing number of students coming to
Chapter 6 (Final-Exam Review)
MAC2311, Section 17, Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
From today's review of Sections 6.3 and 6.2, I hope you see the differences between the methods of Disk/Washer (6.2) and Cylindrical Shell (6.3) to compute
Student's Name:
Make-up Test 3 (Sections 3.9, 4.1, 4.2, 4.3, 4.4, and 4.7)
MAC2311, Section 17, Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
Problem 1 (15pts) Two bikes start moving from the same point. One travels south at 3 mi/h and the other travels
Section 6.3: Volumes by Cylindrical Shells
Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
Some volume problems are very dicult to handle by the methods discussed in Section 6.2.
The Volume Problem
Let S be the solid obtained by rotating about the y-axis
Section 6.2: Volumes
Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
The Volume Problem
Find the volume of a solid S that lies between x = a and x = b.
The method
We start by intersecting S with a plane and obtaining a plane region that is called a cross-
Students Name:
Pop-quiz on Section 4.7 (Time: 3 minutes)
Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
Show all your work to get full credit.
Find a positive number such that the sum of the number and twice its
reciprocal is as small as possible.
1
Sample Problems for Chapters 4 and 5 (Final-Exam Review)
MAC2311, Section 17, Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
Problem 1 (similar to 8 of Section 4.1 ) Find the critical numbers of the function f (x) = x2 9. Then nd the domain of this funct
THE DERIVATIVE AS A FUNCTION Example 1 (2.8)
The graph of a function f is given in
the figure.
Use it to sketch the graph of the
derivative f.
THE DERIVATIVE AS A FUNCTION Example 1
For instance, for x = 5, we draw the
tangent at P in the figure and estim
TANGENTS
Let Q approach P along the curve C by letting x approach a.
If mPQ approaches a number m, then we define the tangent t to be the line through P with slope m. This m amounts to saying that the tangent line is the limiting position of the secant li
Students Name:
Pop-quiz on Section 2.6 (Time: 3 minutes)
Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
Show all your work to get full credit.
2
3
Drill question: Compute limx x31+x 2+2x 5
5x 3x
1
HORIZONTAL ASYMPTOTES
An example of a curve with two
horizontal asymptotes is y = tan-1x.
1
1
lim tan x =
lim tan x =
x
x
2
2
So, both the lines y =
2
are
and y =
2
horizontal asymptotes.
This follows from the fact
that the lines x =
2
are vertical
CONTINUITY
Example 2 (2.5)
Where are each of the following functions discontinuous? x2 x 2 a. f ( x) = b. c.
x2 1 if x 0 2 f ( x) = x 1 if x = 0
x2 x 2 if x 2 f ( x) = x 2 1 if x = 2
Why?
CONTINUITY
The figure shows the graphs of the functions in Examp
Students Name:
Pop-quiz on Section 3.1 (Time: 3 minutes)
Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
Show all your work to get full credit.
Drill Question: Let g (x) =
1
3x
+ ex + xe + e3 . What is g (x)?
1
USING THE LIMIT LAWS
Example 1 (2.3)
Use the Limit Laws and the graphs
of f and g in the figure to evaluate the
following limits, if they exist.
a. xlim2 [ f ( x) + 5 g ( x)]
b. lim [ f ( x) g ( x)]
x 1
f ( x)
c. lim
x 2 g ( x)
USING THE LIMIT LAWS
2
Exam
Student's Name:
Pop-quiz on Section 3.2 (Time: 3 minutes)
Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
Show all your work to get full credit.
Drill Question: Find the equation of the line tangent to f (x) = x = 1.
x2 +1 ex
at
1
THE LIMIT OF A FUNCTION
Guess the value of
Example 1 (2.2)
x 1
lim 2
x 1 x 1
.
Is f(x) defined when x = 1?
Does this affect the limit of f(x) as x approaches
1?
THE LIMIT OF A FUNCTION
The tables give values
of f(x) (correct to six
decimal places) for
val
Student's Name:
Pop-quiz on Section 3.3 (Time: 3 minutes)
Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
Show all your work to get full credit.
tan( +h)-tan( ) 4 4 ? h
Drill Question: What is limh0 (A) 2 (B) - 22 (C) 0 (D) 1 (E) Does not exist.
1
Section 6.1: Areas between curves
Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu)
The Area Problem Special Case: Find the area of the region S that lies between two curves y = f (x) and y = g(x) from a to b (f , g are continuous functions and f (x) g(x) f