Midterm Review One (Partial) Solutions
Instructor: M. Tychonievich
Math 1161, Autumn 2012
1. (18 Points) Find the following limits:
a)
(2 + 2h)4 16
h0
h
lim
Set f (x) := (2 + 2x)4 . Then f (0) = 16, so
(2 + 2h)4 16
f (0 + h) f (0)
= lim
= 4(2 + 0)3 2 = 48
Calculus for Scientists and Engineers: Multivariable
By Briggs, Cochran, Gillett
First Printing Errata List
Chapter 13
p. 895, In the first line of Section 13.2, Chapter 11 should read Chapter 12.
p. 968, In the last sentence of the first paragraph on the
Here is a solution to the bonus problem on the sample nal. Please remember that as
a bonus problem, it is not only worth fewer points than a standard problem, but it is also
much more dicult than a standard problem. You should skip such problems unless yo
Math 2162.01 & 02
Accelerated Calculus II
Name:
name.#:
Recitation Instructor:
Recitation Time:
W RITTEN H OMEWORK 3
Directions:
Show your work. Your solution should be easily understood by someone who is familiar with
mathematics, but has not seen the a
Math 2162.02 Midterm 3 Review Problems
Our third midterm will be held on Monday, April 11 in class. It will cover all that weve
done in Chapters 13.7 through 15.4. There will be approximately six questions, and you
will have 55 minutes to complete the exa
Math 2162.01 & 02
Accelerated Calculus II
Name:
name.#:
Recitation Instructor:
Recitation Time:
N OT-H OMEWORK S TAND - IN FOR W RITTEN H OMEWORK 5
Directions:
Below are a few interesting examples and other ideas that might have made up a fifth written
h
Math 2162.01 & 02
Accelerated Calculus II
Name:
name.#:
Recitation Instructor:
Recitation Time:
W RITTEN H OMEWORK 4
Directions:
Show your work. Your solution should be easily understood by someone who is familiar with
mathematics, but has not seen the a
MIDTERM 1 SOLUTIONS
Question 1. (11 points)
i) (7 points) Let f (x) = 2 2 cos(tan(x) x. Find the value of the real number a if: a = 2f 0 (0) + 1.
Sol. Using the chain rule, we have: f 0 (x) = 2 2 sin(tan(x) sec2 (x) 1, so f 0 (0) = 1, since tan(0) = 0
and
APPLICATION OF ROLLES THEOREM-SOLUTIONS
Problem 1. Show that the equation:
(a) 4ax3 + 3bx2 + 2cx = a + b + c has at least one root on (0, 1).
Sol. Let F (x) = ax4 + bx3 + cx2 (a + b + c)x, for x on [0, 1]. F is continuous on [0, 1] and differentiable
on (
APPLICATION OF MEAN VALUE THEOREM
Problem 0.1. Show that f (x) = 2x + sin x 5 has at most one solution.
Proof. We prove this by contradiction. Assume that f (x) has more than one solutions, and suppose
x1 , x2 are two solutions of f (x). Then we have f (x
Midterm Review One
Name:
Instructor: C. Ross
Math 1161, Autumn 2012
48-minute exam
All work must be shown to receive maximum points. Justify your responses; answers
without supporting work will receive no credit.
Exact answers only; do not use decimal app
Math 1161, Autumn 2012
Dr. C. Ross
Personal Information
Email: cross@math.ohio-state.edu
Oce: Math Tower (MW) 349
Hours: Tuesday & Thursday at 12:00-1:30 (tentative), and by appointment
Course Information
Content: Dierential and integral calculus
Text: Ca
Math 1161, Autumn 2012
Michael A. Tychonievich
Personal Information
Email: tycho@math.ohio-state.edu
Oce: Math Tower (MW) 629
Hours: Monday, Wednesday, and Friday 12:30PM 1:30PM, and by appointment
Course Information
Content: Dierential and integral calcu
Final Review
Read textbook and class notes. Refer to the book or class notes if you forget any
concept or formula.
1. Limits
1) Techniques for computing limits
1) Using continuity and limit laws: (What are they? When are they applicable?)
a) Suppose F (x)
Final Exam Review
Name:
Instructors: Ross, Tychonievich
Math 1161, Autumn 2012
Exam duration: 1.8 hours
All work must be shown to receive maximum points. Justify your responses; answers
without supporting work will receive no credit.
Exact answers only; d
FIRST DAY OF CLASS
Registration Instructions
Access Code
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math 1161 AU12
Math 1161.01=Math1161.02: Autumn Syllabus and Calendar: 5 credit hours
Textbook = Calculus forScientists and Engineers by William Briggs
Warning: Although Briggs' motivation for the differential equations in Section 8.1 is excellent,
his ch
Midterm Review One
Name:
Math 1161, Autumn 2012
55-minute sample exam
All work must be shown to receive full credit. Justify your responses; answers without
supporting work receive no credit.
Exact answers only; do not use decimal approximations unless to
Midterm Review Three
Name:
Instructors: Ross, Tychonievich
Math 1161, Autumn 2012
48-minute exam
All work must be shown to receive maximum points. Justify your responses; answers
without supporting work will receive no credit.
Exact answers only; do not u
Midterm Review Two
Name:
Instructors: Ross, Tychonievich
Math 1161, Autumn 2012
48-minute exam
All work must be shown to receive maximum points. Justify your responses; answers
without supporting work will receive no credit.
Exact answers only; do not use
Midterm Review Two
Hints and Partial Solutions
Math 1161, Autumn 2012
1. (20 Points) Find the following limits:
a)
1
n n3
2n
(k + 1)(k 1)
lim
k=1
You can evaluate this using summation formulas, or notice that it is almost a Riemann
sum. There are 2n terms
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Problem 1. (Asymptotes via Limits):
(1) Determine the vertical asymptotes of
f (x) =
24x3 6x2 + 8x 2
.
4x2 9x + 2
(2) For each vertical asymptote x = a, describe the behavior of f (x) at a by calculating limxa f (x) and limxa+ f (x).
(3) Determine any hor