M140.  Homeworks assigned.  Dr. Leisinger, Fall 2014.
Updated 12/03/2014, 1:00 pm.

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SEE THE BOTTOM OF THIS FILE for practice materials for the final exam.
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==== NOTE: HW NUMBERS BEYOND #20 HAVE CHANGED =====

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TUTORING INFORMATION IS AVAILABLE !!!!!   see "Tutoring.txt"
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Name of this file:   http://math.umb.edu/~aleising/M140/Homeworks.txt

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Syllabus for course:  To be provided by 9/10/2014
Tutoring for course:  see "Tutoring.txt" file in this directory
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Please note that the numbers below in the first column
are the homework # for each assignment !

HW#,
Date	Date	Assignment
Assigned Due
0:8/22	9/3	Read www.math.umb.edu/~aleising/General_Information.txt
		and follow the directions there

1:8/22	9/3	Take the diagnostic tests (A,B,C,D) on this website,
		in the file "DiagnosticTests.pdf".
		Write your work in neat format to be turned in at the
		first class.

2:9/4	9/8	Number Systems, and the Field of Real Numbers
		THIS HOMEWORK IS TO BE DONE IN YOUR HW NOTEBOOK.
		MAKE SURE IT IS IN CORRECT FORMAT.
		Learn the contents of:
		  (a)	NumberSystems_Notes.pdf (in this directory)
		  (b)	Construct_Real_Numbers.pdf (in this directory)
			NOTE: MINOR CORRECTIONS IN THE SECOND DOCUMENT!
		Prove the following four properties of the real numbers,
		using the definitions in Construct_Real_Numbers.pdf :
		Here, capital letters are Dedekind Cuts (i.e., real numbers).
		1. If A<B and B<C, prove A<C
		2. If A<B and C<D, prove A+C<B+D
		3. (Trichotomy) Let Z= {x in Q | x < 0}.
		    Then, Z will be the zero element in R.
		    Prove that, for any Dedekind Cut A, either:
			A < Z  or   A = Z   or Z < A.
		4. Again, Z is as in (3) above.
		   Suppose A,B,P,M are Dedekind Cuts, and suppose Z<P. and M<Z
		(Think of M as a negative real number,
		          Z as zero,
		      and P as a positive real number.)
		   Then prove: (a)   If A < B,  then AP < BP
		   	       (b)   If A < B,  then BM < AM
		(in other words,
		  multiplying an inequality by a positive number keeps the sense
		  multiplying an inequality by a negative number reverses sense.

3:9/4	9/8	HW Notebook Format
		You will receive credit for this assignment when
		your HW notebook is in correct format.

4:9/6	9/8	Section 1.1: Functions (25 problems)
		WORK IS TO BE DONE IN YOUR HW NOTEBOOK!
		Read and study the following examples on pp 11-17:
		Ex #1-4,6-10.
		[ note:   " * " means written work! ]
		* pp 19-22: #1,2,7-10,12,18,28,31,38,42,43,47,68,71,72

		For those still without a textbook, pages 11-22 will be
		found in this directory in text_pp_11-22.pdf

5:9/6	9/10	Section 1.2: Functions (9 problems)
		* pp 33-34: 1,2,3,4,8,9,10,19,22,28.

6:9/6	9/12	Section 1.3  (15 problems)
		* pp42-44 #1,2,9,10,13,14,15,18,28,31,32,33,41,42
		Section 1.4: Read example 1 p. 45.
		* Section 1.4 pp 49. #5

7:9/6	9/12	Section 1.5: Limits (4 problems)
		Read examples pp 51-58 1-10
		Read definitions 1-6 pp 50-57 (in boxes)
		* p59 #1,2,3,4
		Section 1.7:
		Compare the definition of limit on p.73 with the one given
		in class.  Note the in-class use of "For All" and
		"There Exists" quantifiers.  Understand how these two
		definitions mean exactly the same thing.
		Memorize the in-class definition.
		(or, if you missed class, memorize the p.73 definition).

8: 9/6	9/15	Some limit problems. (13 problems)
		* Section 1.5.  p59-61 #5,7,38,39.
		Section 1.6,Limit laws.
		Learn the first 5 limit laws. (see box p. 62)
		Don't forget, these laws are INCORRECT as stated in the text.
		In each case, the statements should read:
		   "WHENEVER THE LIMITS ON BOTH SIDES EXIST".
		Read and study Examples 1-5 pp 63-66.
		* Exercises 1.6,p69; #1 (except part c);#3,4,5,10-14,20.

9: 9/6	9/17	More limit problems. (16 problems)
		Learn the definition of Infinite Limits (box, p56)
		* p70 ALL #21-26,47.
		Study examples #1,2,5 pp74-79.
		* p81 #19,21,22,41,42,43
		Learn the definition of continuity in the box on p.82.
		Study p 82-83 examples 1,2

10: 9/6	9/19	Continuity exercises. (16 problems)
		Note: this is a rather long assignment.
		The "LEARNING" section of the assignment is to be done first.
		It should be completed by 9/22
		The exercises should be started.  Complete them if you can.
		However, they are not due until 9/24.
		[Review the epsilon-delta definitions of limit, continuity at
		 a point; and infinite two-sided and one-sided limits.]
		Learn the definition of continuity on an interval.[Box,p.84]
		Learn: the following functions are continuous on their domains:
			any polynomial with real coefficients.
			any root function (positive integer roots)
			any rational function, that is, a quotient
				of two polynomials over the reals.
			any of the six trigonometric functions
				[see Theorem 7, p. 86]
			log(x) (to any base); b^x (for any legal base b)
		Learn: whenever f(x) and g(x) are continuous, then so are:
			fg(x) (function product); (f+g)(x); (f-g)(x); (f/g)(x);
				[See theorems 4 and 5, pp 84 and 85]
			function composition:  f(g(x)) [see Theorem 8 p.87]
				on any interval on which they are defined.
		==> Learn and understand the Intermediate Value Theorem,
			[Theorem 10, p. 89.]
			Remember that we have or will prove it, using
			the Dedekind Cut definition of the Real Numbers.
		  Understand the example #9 on p. 89.  This is a special
		  	case of the Fundamental Theorem of Algebra.
		  	It can be shown that every polynomial of odd degree
			over the rational numbers has at least one real root.
			That is proved almost exactly as in this example.
		Study the examples in the continuity section.
		* Section 1.8, pp 90-91. ALL: #1,3-8,11,13,15,23,24,28,29,33,35.
		  For #33, do not use a calculator.  Instead, ask yourself:
		  for which values of x does sin(x) = -1?

11:9/6	9/22	The Derivative (7 problems)
		*p112 #27,28
		*p124 #19-21,23,29	

12:9/6	9/24	Derivative:  (12 problems)
		Using the sum, difference, power and product rules
		To reference these rules:
		Sum and difference rules: text, p.129, two boxes.
		Power rule: text, p.134 in the box at the bottom of the page.
		  [we have proved this when the power n = an integer > -3.
		Product rule: text, p.130 in the box at the bottom of the page.
		* section 2.3, p136, #1,3-8,11,15,16,23,26 [ALL]

13:9/6	9/26	Derivative: (29 problems)
		using:  Function composition rule (Chain rule),
		general Power rule, Quotient rule,
		and Derivatives of the six Trig functions, ln(x) and e^x
		* Section 2.4 p146 ODD #1-15
		* Section 2.5 p154 ODD #1-45

14:9/6	9/29	Using the derivative. (10 problems)
		* p137 ODD #31-43. [just some more derivatives to calculate]
		* p137 ODD #51-57.
			Please note: the NORMAL to a curve at a point is a line
			that is PERPENDICULAR to the tangent line at that point.
			Example:  the NORMAL to the circle of radius 1 centered
			at the origin at a point (cos(pi/4),sin(pi/4)) is
			the (extended) radius of the circle there, or the
			line y=x.

15:9/6	10/1	Second and higher derivatives (5 problems)
		$1.2: Read p120-122.
		* p125 #41,43,45,46,47.(in #45,46, don't bother with the graphs)

16:9/6	10/3	Implicit differentiation (19 problems)
		$2.6: Study examples p158-161 [also done in class]
		* $2.6 pp161-162; Ex.#1-4 (all); ODD 5-29 AND #24,28

17:9/6	10/6	Implicit Differentiation, (7 + 1 problems)
		PROBLEMS of NORMAL DIFFICULTY:
		* p173 #1,3,5,9,13,15,21

		DO NOT EXPECT TO GET A COMPLETE ANSWER TO THE NEXT PROBLEM.
		* p164 #2.
		(Don't use a computer graphing system or graphing calculator.)
		  This is a challenge problem.
		  It is not due at any particular time.
		  Work on it for a few hours, if you like.
		  At least, evaluate dy/dx implicitly.
		  Try the values c=0; c=-1; c=1;
		  Draw the graphs on the same sheet of graph paper.
		  Use values of dy/dx to help you visualize the curve.

18:9/6	10/8	Related Rates (6 problems)
		Read the examples in section 2.8, Related Rates.
		* $2.8, p180, #1,3,5,7,9,17.

19:9/6	10/10	Crit.Points; Linear Approx; Differentials; Taylor Polynomial
		(19 problems)
		* $2.8, p181#21,27
		* $2.9, p187 ODD #1-7,11-17
		* $3.1, p205 #15,17,19,20,25,27,31,33,35

20:9/6	10/13	Curve sketching. $3.5.Summary of Curve Sketching (7 problems)
		Read the explanation and examples in $3.5 pp 239-242
		* #2.8 p242. #1,5,9,10,11,13,14
		When doing these problems, use as many techniques
		from the summary pp239-242 as you need.
		Graphs should be done on graph paper.
		Use GraphPaper.pdf.  Graph enough points to get a reasonable
		picture of each curve.
		You should use a separate piece of graph paper for each curve.
		Hold the graph paper vertically.  Use the scale 1 box = 1 unit.
		Label each piece of graph paper with your name, hw#, problem #.

21:9/14	10/15	Prepare for a quiz on curve sketching.
		[ No written work. ]
		Review all of our past lessons.
		Especially:  limits. derivative.
			Intermediate Value Theorem
			Mean Value Theorem
			Fermat Lemma on    f'(x) = 0 and local maxima/minima
			Critical points
			Second derivative test for max/min

22:9/14	10/17	Antiderivatives (19 problems)
		Read carefully: Section 3.9, pp 269-271
		(for now, ignore examples 6 and 7)
		* p273 ODD #1-39.

23:9/14	10/20	Area by Antiderivatives (10 problems)
		Worksheet
		(do problems in your HW notebook;
		put answers on the worksheet if you wish.)
		Area_by_Antiderivatives.pdf (worksheet in this directory)

24:9/14	10/22	Area by lower and upper sums. (6 problems)
		* $4.1, p293 #1 (no calculations; just count rectangles)
		* $4.1, p293 #3
		   --> If you remember your trigonometry and half-angle
		   formulas, you should be able to do this without
		   using a calculator for cos(x).
		   You'll need cos(pi/8),cos(pi/4),cos(3 pi/8) and cos(pi/2)
		   --> After doing it that way, do the problem a second time
		   using a calculator for the cosine.  Compare your results.
		* $4.1, p293 #5
		* $4.1, p293 #19,21,23.
		   For problem #23, "i" is an integer, not sqrt(-1).

25:9/14	10/24	Using the Fundamental Theorem of Calculus. (11 problems)
		NOTE: A proof of the extreme value theorem has been posted.
			See: ExtremeValueTheorem_Proof.pdf
		Read the proof of the Fundamental Theorem I, pp 312-313.
		On pp 313-314, read the examples #2 and #3.
		Read and be able to use the Fundamental Theorem Part 2
		(box, p315).
		On pp 315-317, read the examples #5,6,7,8.
		* $4.3. p318 #5,6, and ODD #7-23.

26:9/14	10/27	Using the Fundamental Theorem of Calculus. (9 problems)
		* $4.3. p318 ODD #25-41.
		Note that #37,39,41 involve analysis of continuity.

27:9/14	10/29	Substitution rule for integration. (10 problems)
		Read p330-333.
		* $4.5  p335  ALL #1-4; ODD 5-17.

28:9/14	10/31	Areas between curves (7 problems)
		* $5.1: p349 #1,3,5,7,13,15,19

29:9/14	11/3	Volumes (6 problems)
		* $5.2: p360 #1,3,5,7,13,39  [and find the volume in #39, also]

30:9/14	11/5	Newton-Raphsen method (5 problems)
		Use the method shown in class to do the following:
		* (a) Calculate square root of 2 to 4 decimal places,
		      using the starting value of 2.
		      Remember, you are finding a root of f(x) = x^2 - 2 .
		      Do calculations exactly, until you need a calculator.
		      Make a table like the one shown in class.
		      Terminate the iterations when your answer agrees with
		      a calculator result for sqrt(2) to 4 significant digits.
		* (b) Calculate square root of 10 to 5 decimal places.
		      use the starting value of 3.
		* (c) Calculate square root of 3 to 5 decimal places.
		      use the starting value of 2.
		* (d) Calculate square root of 3 to 5 decimal places.
		      use the starting value of 5 (not a very good guess!)
		* (e) Discuss the difference between your results in #c,#d.
		      Write something intelligent.


NOTE:   see the following website for practice materials for the final exam:
	www.math.umb.edu/courses/materials/math140/
============================= NO HOMEWORK BELOW THIS LINE ===================
================ MATERIAL BELOW THIS LINE IS UNFIT FOR HUMAN CONSUMPTION ====