Calculus I
Math 150
Spring 2010
Instructor: Prof. Dr. rer. nat. Henri Schurz
Time: MWThF : 11 - 12 / Location: Neckers 156
Office Hours: MWF 12:00-1:50pm
(Last Update: 01/26/10)
Sample midterm exams and solutions are available (See below)!
Textbook:
Essential Calculus: Early Transcendentals, 1^{st} Edition.
by J. Stewart, Brooks/Cole Publishing Company, Pacific Grove, 2007.
Read this
Without Tears:
WHAT IS EXPECTED OF YOU
(From "Teaching at the University Level" by Stephen Zucker, Notices Amer.
Math. Soc. 43 (1996), p. 863):
1. You are no longer in high school. The great majority of you, not having
done so already, will
have to discard high school notions of teaching and learning and replace
them by university-level
notions. This may be difficult, but it must happen sooner or later, so
sooner is better. Our goal is
more than just getting you to reproduce what was told you in the classroom.
2. Expect to have material covered at two to three time the pace of high
school. Above that, we
aim for greater command of the material, especially the ability to apply
what you have learned to
new situations (when relevant).
3. Lecture time is at a premium, so it must be used efficiently. You cannot
be "taught" everything
in the classroom. It is YOUR responsibility to learn the material. Most of
this learning must
take place outside the classroom. You should be willing to put in two hours
outside the classroom
for each hour of class.
4. The instructor's job is primarily to provide a framework, with some of
the particulars, to guide
you in doing your learning of the concepts and methods that comprise the
material of the course. It
is not to "program" you with isolated facts and problem types nor to monitor
your progress.
5. You are expected to read the textbook for comprehension. It gives the
detailed account of the
material of the course. It also contains many examples of problems worked
out, and these should
be used to supplement those you see in the lecture. The textbook is not a
novel, so the reading
must often be slow-going and careful. However, there is the clear advantage
that you can read it
at your own pace. Use pencil and paper to work through the material and to
fill in omitted steps.
(till here: extracted from Joe Mashburn (Uni Dayton)/
joe.mashburn@udayton.edu)
6. Do not expect that exam problems are exactly the same as in homework
problems. Homework problems should be understood to strengthen and to
improve
your knowledge on subject related issues.
7. Work with your TA's as much as you can and form study groups if
necessary, ask subject related questions, and be active.
8. Read further additional literature (Do not stick to course material
completely!). The professors can always give you more literature hints.
9. Make always written notices what the instructor tells you about the
subject, work through your course notes after the lectures continuously.
10. We all expect mutual respect, polite, correct, honest and
sincere personal behaviour on student's and instructor's side as it
should be common among human beings.
(These are my main 10 principles for success in academic studies!)
Course
Description: This course is the first part of a two
semester course on Calculus, meant to be an introduction to basic
aspects of functions, series, limits, differentiation and
some elements of integration.
Calculus is a very large field, and we will certainly
not be able to cover all of the important techniques in a
one or two-semester course. A preliminary list of topics covered
includes differentiation and integration, tangent and secants,
limits, l'Hospital's rule, basic techniques of integration,
integration by parts, trig substitution, volumes or revolving areas
(I am afraid of skipping any other very important issues).
Prerequisites
and Development of Contents:
This course should be accessible to any student with a $C-$ in
Precalculus / College Algebra or placement exam. However, I strongly advice you to review
your knowledge which you should know from high school math in your
previous carrier. I will always assume that you profoundly know the
facts from that part, including standard trigonometric formulas.
The content of this course itself should very nearly coincide with that of
Stewart's book, running from chapter 2 until 7.2.
I am not perfect. However, be sure that I
will do my very best to please you and your expectations according
to course requirements.
Readings,
Problem Sets, Exams:
Readings and problem sets will be from the text and my manuscript, and
it will be assigned in classes and perhaps additionally published at
my homepage. Exams will cover all material covered in the lectures,
recitation classes and/or the readings. James Stewart seems to
have put a fair effort into their presentation from very practically
oriented point of view, and their approach is probably quite different
from what you've seen before (hopefully not). Thus, I really encourage
you to read the book as lectures advances.
Exam
Dates:
- Midterm I: Friday, February 23, 2007 (in class)
- Midterm II: Thursday, March 29, 2007 (in class)
- Midterm II: Friday, April 27, 2007 (in class)
- Final: Monday, May 7, 2007 (10:10am-12:10pm)
Grading Strategies - Grade Distribution:
- 25 % Homeworks & Quizzes, 50 % Midterms, 25 % Final Exam
- Deviation departmental guideline: Except for very
well-documented cases, there is only such an overall grade
possible which differs from the grade result of final exam
at most about one grade.
- Your scores from homeworks & quizzes are recorded by your TA.
The score of the final exam is also evaluated according to the
gradeline meeting on total grade distribution of all students after the final
date. Thus, your results in the midterm and final exams will rule over the
remaining portion of quizzes and homeworks.
Course
Syllabus (Last Update: 01/15/07, Need to be Updated) - You will need an utility
like ghostview to read it!
Please, note you need ghostview to read the postscript files after
downloading! See http://www.cs.wisc.edu/~ghost/index.html for more software
product information.
Remarks for the Prerequisities and First Week:
Please, review problems from section 1 and appendix B. You should know the
point-slope form of equation of a line, and strong grasp what is a function.
Remarks for the 1st Midterm Exam:
Please, review your notes and sections 1.1 - 3.3. The midterm exams are
not of multiple choice. Midterm I will contain 4 problems (no word
problems). The main goal is to check your capability of learnt calculus
techniques. In particular, you should be sure in limits, tangents,
secants, derivatives, sum-, product- and quotient-rules ...
No books, no notes, no
tables, no calculators at all, no notebooks are permitted (pocket
calculators wouldn't help you much anyway).
I don't expect that you know all formulas, but you should be able to
understand and work with them. Do the sample midterm exam I below and
you will discover all formulas you need to solve the midterm problems.
Sample Exam Midterm I:
(1) Calculate f'(x), g'(t) where (a) f(x) = (x^2+6x-1)/(x^3-7x+10) and
(b) g(t) = 2 (t^2+t+1+squareroot(t)) e^t
Solution: (a): f'(x) = (-x^4-12x^3-4x^2+20x-53)/[(x^3-7x+10)^2]
(b): g'(t) = 2 (t^2+3t+2+squareroot(t)+1/[2 squareroot(t)]) e^t.
(2) Find limits
(a) lim_{x to -\infty} (x^3+1)/(x^9+10x-1)
(b) lim_{x to 1-} (x^3-x)/(x+1)
Solution: (a): 0 and (b): 0
(3) Find tangent line to f(x)=x^3+x^2+x+1 at all real x=a. Is there a
horizontal tangent?
Solution: f'(a) = 3a^2+2a+1, y = m (x - a) + b = (3a^2+2a+1) (x - a) + a^3+a^2+a+1
There are horizontal tangents whenever f'(a)=3a^2+2a+1=0. However, there is
no real root satisfying f'(a)=0. Therefore there are no horizontal tangents
in the real plane.
(4) Find constants m,b such that
f(x) defined by f (x) = 10 x^2 + 1 if x <= 0 and
f(x) = m x + b if x > 0
is differentiable everywhere.
Solution:
(i) f is differentiable at least at all x <> 0 due to differentiability of
all polynomials.
(ii) f must be continuous at x = 0, i.e.
1 = f(0) = lim_{x to 0+} f(x) = b.
(iii) f must be differentiable at x = 0, i.e.
m = f'_+(0) = f'_-(0) = 0.
Remarks for the 2nd Midterm Exam:
Please, review your notes and sections 3.4 - 4.3. The midterm exams are
not of multiple choice. Midterm II will contain 4 problems (one is a word
problem). The main goal
is to check your capability of learnt calculus techniques. In particular,
you should be sure in the implicit and logarithmic differentiation,
differentiation of exponential, logarithmic, hyperbolic and
trigonometric functions, elementary trigonometric and hyperbolic identities,
linear approximations, differential notations, extreme values, closed
interval method, concave upward and downward functions, critical numbers,
necessary and sufficient conditions for maximum and minimum functions,
intermediate value theorem, and mean value theorem.
Limits and the rule of L'Hospital as well as
inverse hyperbolic functions will
not occur in any midterm II problem. No books, no notes, no tables, no
cheating sheets, no calculators, no notebooks are permitted (pocket
calculators won't help you much anyway, you would only spoil your time
by using them). To clarify which formulas
you need to memorize, solve the example exam II which I will give (gave) on
Monday of the exam's week.
Sample Exam Midterm II:
Explain your mathematical approach in detail!
(1) Use the Mean Value Theorem to find the smallest constant L > 0 such that
|e^(-x^2/2) - e^(-y^2/2)| <= L |x-y| for all real numbers x,y.
Solution: Recall MVT: f(b) - f(a) = f'(c) (b-a).
Take b = x, a = y, f(x) =
e^(-x^2/2). Calculate f'(x) = - x e^(-x^2/2).
|f(x)-f(y)| = |e^(-x^2/2)-e^(-y^2/2)| = |f'(c)| |x-y|
<= max { |f'(z)| : z real } * |x-y| } .
Hence L = max { |f'(z)| : z real } . Now, for its computation, take
f''(x) = (x^2 - 1) e^(-x^2/2) = 0 <=> (x = + 1 or x = - 1)
Note, f' decreasing if -1 < x < +1, and f' increasing
elsewhere.
Hence, by first derivative test, f' has maximum value e^(-1/2) at x = -1,
and f' has minimum value e^(-1/2) at x = +1
Using piecewise monotonicity of f', one finds L = max { |f'(z)| : z real } =
e^(-1/2)
(2) Differentiate and find explicit form of derivatives with respect to x:
f(x) = sin [x^2 + x^(1/2) + cos(x) + tan(x) + sinh(x) - cosh(x) + ln(x)]
and F(x,y) = x^4/2 - y^4/4 = 10
Solution: Using chain rule
f'(x) = (2 x + 0.5 x^(-1/2) - sin(x) + sec^2(x) + cosh(x) - sinh(x) + 1/x)
cos [x^2 + x^(1/2) + cos(x) + tan(x) + sinh(x) - cosh(x) + ln(x)] and
y'(x) = +/- 2 x^3 / (2 x^4 - 40)^(3/4) = 2 (x/y)^3 by implicit
differentiation.
(3) A kite 10 m above the ground moves horizontally at constant speed of 2
m/s. At what rate is the angle theta(t) between the string and the plane ground
increasing when 20 m of string have been let out straight and constantly
at all times?
Solution: Assume that the string always forms a straight line, kite is
identified by one single mass point and the place where the string is fixed
on the ground is at the same height as the plane ground is. Now, using
implicit differentiation and trigonometric identities, assuming
z(t) = 20 m = constant, we have
cos(theta(t)) = adjacent / hypothenuse = s(t) / 20 m, sin(theta(t)) =
opposite / hypothenuse = h(t) / 20 m ,
d cos(theta(t)) / dt = - sin(theta(t)) theta'(t) = s'(t) / 20 m, s'(t*) = -
2 m / s,
hence theta'(t*) = 1/5 rad/s at the moment t* when h(t*) = 10 m.
Hint: Use the elementary trigonometric relations in triangle with
right angle (that is that for sin and cos, or tan, depending on your
approach) to find a relation between the angle theta(t), height h(t),
position s(t) and string length z(t).
(4) Use the Closed Interval Method (CIM) to find the absolute extreme values
of f(x) = x^4 - 4 x^3/3 + 1 on [-2,+2]. Find the regions of concavity (i.e.
where f is concave upward and concave downward), and the regions of
increasing and decreasing behavior.
Solution: Calculate f'(x) = 4 x^2 (x-1) and f''(x) = 4x (3x-2).
Then, the absolute maximum f(-2) = 83/3 is attained at x = -2, the absolute
minimum
f(1) = 2/3 at x = 1 since f(0) = 1, f(2) = 19/3. We also find that f is
concave
upward in (-2,0) and in (2/3,2), concave downward in (0,2/3) by concavity
test
using second derivatives (compute the chart diagram as in class). Hence, the
points of inflection are given at x = 2/3 and x = 0. Furthermore, f is
increasing in (1,2), and decreasing otherwise with critical numbers x = 0
and x = 1.
Remarks for the 3rd Midterm Exam:
Please, review your notes and sections 4.4 - 6.3. The midterm exams are
not of multiple choice. Midterm III will contain 4 problems (no word
problems). The main goal is to check your capability of learnt calculus
techniques.
In particular, you should be sure in the .....
No books, no notes, no tables, no graphing calculators, no notebooks are
permitted (pocket calculators won't help you much).
I don't expect that you know all formulas, but I do expect that you
are able to derive, understand and apply the basic ones
(you should know at least the most
important ones). Do the sample exam and then you will discover all
formulas and related concepts you need to know.
Sample Exam Midterm III:
Explain your mathematical approach in detail!
(1) Compute the left-hand Riemann sum and its limit to evaluate the
integral \int^b_a (c0 + c1 x + c2 x^2 + c3 x^3 + c4 e^(p x) ) dx
where a < b, c0, c1, c2, c3, c4, p are real constants.
(2) What are the dimensions of the largest rectangle inscribed in
the ellipse (x/a)^2 + (y/b)^2 = r^2 where a, b, r > 0 are real constants.
(3) Calculate the area between cos(x) and sin(x) on [0,n2\pi] where
n is any integer.
(4) Compute the volume of the solid obtained by revolving the area
between x^n e^(x^(n+1)) and the x-axis from 0 to b (b>0) about the x-axis.
Remarks for the Final Exam:
The final exam is a fairly comprehensive exam for 2 hours.
The topics run from chapter 1 till 6. We exclude the mean
value theorems, Newton method, rule of L'Hospital, hyperbolic
functions, linear approximations.
The final involves problems on computation of limits and
derivatives, definite (FTC) and indefinite integrals
(antiderivatives), optimization, absolute min/max (CIM),
area and volume computations, implicit differentiation,
concavity, monotonicity and tangent lines.
Scientific calculators are allowed. Other calculators,
notebooks, notes, cheating sheets and books are not permitted.
Bring your I.D. with photo with you.
Current Course Outline (Last Update:
01/26/10):
Week 1-2 : Introduction, 1.1.-1.6.
Week 3-4 : 2.1.-2.9., 3.1.-3.2.
Week 5-6 : 3.2.-3.5., Exam I
Week 9-10 : 4.1.-4.7. (omit 4.6), Exam II
Week 11-12 : 4.9.-4.10. (omit 4.8), 5.1.-5.4.
Week 13-14 : 5.5.-5.6, Exam III
Week 15 : 7.1.-7.2., Review
Remarks for Homeworks, Quizzes and Recitation Classes:
The homeworks and quizzes are assigned in classes (as on syllabus and
webpage) and are due as announced in class (see also below).
If time permits, then some recitation classes are done to strengthen your
knowledge in related areas touched in lectures. In any case, your active
participation is required.
The homeworks and quizzes play an essential role in forming your grade
according to the presented grade distribution. There is
no make-up quizz and no make-up homework, unless you are hospitalized.
Homework Assignments
(due as announced in class):
Week XIV: 6.2.: 4, 6, 8, 10, 34 | 6.3.: 8, 14, 18, 20 | 6.5.: 10 |
5.4.: 38, 42 | 5.5.: 4, 6, 18, 32, 66, 82 | 6.1.: 2, 4
Week XIII: 5.2.: 36, 46, 48, 64 | 5.3.: 16, 20, 52, 58 | 5.4.: 4, 28
Week XII: 5.1.: 2, 6, 16, 18, 20 | 5.2.: 6, 12, 16, 26, 32
Week XI: 4.7.: 6, 10, 18, 20, 38, 54 | 4.9.: 10, 16 | 4.10.: 14, 36
Week X: 4.4.: 6, 8, 10, 52, 58, 72, 78 | 4.5.: 4, 10, 16
Week IX: 4.1.: 4, 54, 77 | 4.2.: 4, 14, 18, 27, 29, 36 | 4.3.: 12, 30, 44
Week VIII: 3.9.: 8, 34, 49, 50, 52 | 3.10.: 29, 32 | 3.11.: 8, 36, 48
Week VII: 3.6.: 18, 28, 46, 56, 67 | 3.7.: 55, 63 b | 3.8.: 18, 24, 36
Week VI: 3.3.: 7, 16, 27 | 3.4.: 9, 10, 16, 38 | 3.5.: 17, 24, 40
Week V: 2.9.: 23, 24, 42 | 3.1.: 17, 27, 58 | 3.2.: 10, 18, 24
Week IV: 1.4.: 22, 24, 32, 44, 46, 48 | 1.5.: 4, 17, 20, 33 (pages 44-45, 54-55)
(due Friday, February 12)
Week III: 1.3.: 6, 8, 18, 22a+b, 34, 40 | 1.4.: 4, 6, 12, 28 (no graphing)
(pages 33-35, 44-45) (due Friday, February 5)
Week II: 1.2.: 17c, 17d, 36, 38, 41, 44, 46, 47, 50, 52e (pages 22-23) (due
Friday, January 29)
Week I: 1.1.: 4, 6, 22, 26, 28, 34, 40, 44, 58, 60 (pages 8-10)
due on Fridays, always, 11am, turned in during class
hours
On-line Exercises:
For those who like to work with internet access and check their
knowledge electronically, there are on-line exercises available.
See
On-line Web-Drill