Analytic Geometry and Calculus II
Math. 114, Sec. 003, Spring 2018
Homework and Announcements
Important Stuff
I have posted a review sheet for the final exam
at Final Exam Info here below. I intend to
go over that in fair detail for what remains in this course.
2nd Computer Assignment
Please note that I have posted the second (and last!)
MATHEMATICA assignment here below at
Mathematica Info
The due date is Monday April 30
in your RCT session.
Both midterm exams
I still have a big pile of both the first and second
midtem exams that many of you have not gotten back.
You should get them back from me for your own records.!
General Info
For general information regarding this course, please
read the Syllabus here below.
It should answer all practical questions, like when tests are, what
the grading policies are etc.
NB!
All handouts for this class, listed below, will be labeled with
an itemized icon.
Your teaching assistant (TA) is Long Nguyen. He will see
you during the recitation sessions (RCT) on Mondays and
discuss homework (HW) exercises and give you quizzes. His email is:
"lnguye33 the-at-sign masonlive dot gmu dot edu".
Homework Info
Below are the homework assignments (HW) so far in the course.
All HW is from the text book of this course.
They includes exercises to be discussed in your
recitation sessions by your TA. Be sure to
attempt them before going to the sessions.
- Lectures and Homework (HW) from Week 1.
HW for RCT January 29. :
- Chapter-5:
- Review: 3, 15, 16, 17, 21, 22, 25, 29, 30, 38, 50, 60.
- Chapter-6:
- Sec-2: 5, 6. 7, 11, 12, 15, 17, 23, 25, 31, 33, 34, 40, 45, 47, 51, 55, 57, 69.
- Sec-3: 7, 8, 9, 10, 12, 17, 19, 20, 21, 25.
Be sure that you know how to differentiate any function, be
it a product, quotient or composite of functions. In theory,
you should be able to find the derivative of any function
that is handed to you! This is the reason why differentiation
is easier that integration, since it is not always possible
to find a closed form of an indefinite integral. Review how
to integrate the basic functions, like polynomials, exponential
functions, logarithmic functions and the trig functions. --
The main thing about slicing method is that the total
volume is always the integral of the area A(x)
of the slice, where x goes from one bounding value
to the other.
- Lectures and Homework (HW) from Week 2.
HW for RCT February 5. :
- Chapter-6:
- Sec-3: 27, 28, 29, 45, 46, 47.
- Sec-4: 5, 6, 7, 15, 19, 20, 21, 24, 41, 42, 43, 55, 64, 70, 71.
- Sec-5: 10, 11, 12, 13, 28, 29, 39, 41
- Chapter-13:
- Sec-6: 7, 8, 15, 16, 18, 51, 52.
The main thing to remember is that the total volume
V is always the integral of dV,
whether dV is (1) a small circular disc with
thickness dx (or dy), or (2) a cylindrical
shell of height f(x), thickness dx
and circumference 2 times pi times the radius of the cylinder.
Similarly, the total arc length is always the integral of
ds, whether we view it in terms of x
and dx, or in terms of y
and dy. Using some general principles of centers
of mass, we can compute the CM of various regions and curves.
We derived some of such formulas in class that can be applied
to a variety of examples.
- Lectures and Homework (HW) from Week 3.
HW for RCT February 12. :
- Chapter-13:
- Chapter-6:
- Sec-6: 7, 8, 9, 10, 17, 20, 32, 33, 34.
- Sec-8: 7, 8, 9, 10, 15, 16, 17, 18, 19, 21, 23, 25, 43, 44, 45, 50, 74, 75.
As we have now seen, we can use integration to compute
volumes, arc lengths, center of masses (CM) and surface
areas of regions obtained when rotating curves around
either x-axis or y axis.
The method is pretty much the same, that we integrate
some designated "small piece", that is dM,
xdM or ydM. Hence, we always use
the same principle, as the various formulas for uniform regions
or uniform strings indicate. Note! If the plates or strings are not
uniform, then we cannot cancel out the density delta in the formulas.
Also, keep an eye open for various geometrical symmetries when
computing CM, since that can simplify many of your calculations. -- When
computing CM for uniform plates or strings, you can
cancel out the density and use the nice formulas that I presented
in class. Never forget to keep an eye open for various symmetries
of the geometrical figures! -- The precise definition of the natural
logarithm is given by an integral. From this definition alone we
are able to derive all the properties of the logarithm and also
the exponential funtion as well. Note that this gives us a way
to define the number e as the inverse of the natural
logarithm at 1 (that is, the exponential function at 1). Also
we have that the limit of (1+x) to the power of 1/x
goes to e when x goes to zero. This is an important
limit!
- Lectures and Homework (HW) from Week 4.
HW for RCT February 19. :
- Chapter-6:
- Sec-9: 11, 14, 15, 36, 37, 45.
- Sec-10: 11, 12, 14, 15, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 39, 41, 45, 47, 48, 52, 53, 54, 55, 56, 63, 64, 65, 75, 77, 81, 84, 87, 103, 104, 107, 111.
- Review: 66.
- Chapter-7:
- Sec 1: 8, 9, 23, 25, 27, 28, 29, 30, 32, 33, 35, 36, 37, 42.
A consequence of the limit of (1+x) to the power of 1/x
when x goes to zero, is that continuously compounded interest
is always more profitable than any discrete one, no matter how often per
yeat the discrete one is performed. -- Hyperbolic functions are defined
in terms of the exponential function. They have properties that very
closely resemble those of the trig functions. Many identities are
almost the same as for trig functions (usually only differ in sign
somewhere...) This can be very useful when integrating.
The hyperbolic functions are yet
another group of new functions that can be helpful for
integration. The main hyperbolic functions
are the hyperbolic cosine, the hyperbolic sine and
the hyperbolic tangent. Hence, we have added some new functions to our
tool bag of functions, the hyperbolic functions,
that can be helpful for certain substitutions as we
discussed in class. -- When we face an integral
involving a square root of a second degree polynomial
it is always a good rule of thumb to complete the square
and then use the corresponding substitution.
- Lectures and Homework (HW) from Week 5.
HW for RCT February 26. :
- Chapter-7:
- Sec 1: 44, 47, 57, 61.
- Sec 2: 7, 8, 9, 10, 11, 17, 23, 26, 29, 32, 35, 39, 44, 47, 48, 51, 55, 57, 59, 66, 74, 75.
- Sec 3: 9, 10, 11, 12, 16.
- Sec 5: 5, 7, 8, 13, 14, 15, 16, 27, 28, 31, 33, 39, 40, 43, 46, 62, 63, 64, 71, 73, 76, 79, 81, 83, 89.
When integrating by parts, look at which function you
like to get rid of (which you label as "N")
and which function you feel like integrating
(that you label as "D"). -- When you want to integrate
a rational function P(x)/Q(x) where the denumerator
is linear (of degree 0 or 1), then you can always use substitution.
It gets harder when the denumerator is a quadratic polynomial.
In general, to integrate a rational function P(x)/Q(x) one must use
the method of partial fractions (PF). This methods decomposes
the rational function into a sum of basic rational functions that
we then integrate term by term. The general theme is as follows:
(i) Check if P(x) is of lower degree than Q(x).
If that is not the case, then use long division. (ii) Factor
Q(x) completely into linear and quadratic factors
(where each quadratic factor cannot be factored further).
(iii) Then use the partial fraction decomposition to
determine the constants A, B, C, D, ... etc.
This can be done in various ways: (a) The usual way of
comparing the coefficients to each power of x.
(b) Assigning specific values to x. (c) Differentiate
with respect to x. (d) Use the Heaviside cover-up method
if Q(x) is a product of distinct linear factors.
(iv) Finally integrate each summand in the partial fraction
decomposition. -- To integrate trigonometric functions must be treated
on a case by case basis. However, the general theme is to find
and isolate a term that is the derivative of another trig
function and then use substitution to solve the integral or
at least reduce it to lower powers.
- Lectures and Homework (HW) from Week 6.
HW for RCT March 5. :
- Chapter-7:
- Sec-4: 17, 18, 19. 20, 27, 28, 35, 39, 40, 51, 52, 59, 67, 69, 79 (you can use Pappus).
- Sec-8: 5, 6, 7, 8, 9, 14, 15, 17, 18.
There are three basic
trigonometric substitutions, each getting rid of a square root of
a specific type. Be sure to draw the triangle and the
angle theta! Everything can then be read from this
triangle. Trigonometric substituitions are important to get rid of a square
root, but they can also be used to compute integrals where we
need to integrate rational functions where the numberator
is a constant and the denumerator is some positive power
of a non-factorable quadratic polynomial. The we proceed
just like when there was a square root, namely (i) complete
the square of the quadratic polynomial and then (ii) use
the substitution involving tangent of an angle (the new variable).
This will yield an integral involving a power of a trig
function that we have already discussed. -- Be sure to write
the final outcome/answer in terms of x, the original
variable. This can all be read from the triangle you wrote down
initially. -- An improper integral
is defined as a limit of a proper definite integral, like the ones we
have been dealing with so far, and for simple cases we can proceed
computing those limits.
- Lectures and Homework (HW) from Week 7.
HW for RCT March 19. :
- Chapter-7:
- Sec-8: 19, 20, 21, 22, 23, 24, 28, 29, 31, 35,
37, 41, 44, 47, 51, 59, 64.
The first thing to notice for an improper
integral, is where exactly is it improper? It is clear that when
the limits include an infinity (plus or minus), then that contributes
to the improper
part, but sometimes it is also improper at a number, say at x = 0,
when the function f(x) we want to integrate is not defined
at x = 0. So a good plan of attack is as follows:
(1) Find out where the "improperness" of the integral is
(2) Express the improper integral as a limit
(3) Evaluate the integral within the limit by usual methods
(4) Take the limit and check it it/they exist. Then the improper integral
is convergent if, and only if, all the corresponding limits exist as
finite numbers. Sometimes, when we cannot evaluate the integral
as a limit, we still need to know if it is convergent or not.
One thing we can use then is the comparison test (CT). -- Have a
nice Spring Break! See you the week after that.
- Lectures and Homework (HW) from Week 8.
HW for RCT March 26. :
- Chapter-7:
- Sec-8: 70, 71, 72, 74, 75, 76.
- Review: 35, 90.
- Chapter-8:
- Sec-1: 9, 10, 11, 12, 17, 18, 20, 21, 22, 23, 24, 25, 26, 31, 32, 33. 34, 35,
37, 47, 49, 50, 53, 55, 56, 57, 82.
- Sec-2: 9, 10. 13, 14, 15, 16, 19, 20, 21, 25, 29, 30, 34, 35, 39. 40, 43, 45, 46, 53, 57, 63, 64, 67, 73, 77, 78, 81, 85, 94, 97, 101.
To determine whether or not an improper integral is convergent we
must first determine where the problem is: At a discontinuity or
at infinity? For integrals where the problem is at infinity
there are several ways to determine the convergence without
directly evaluating the limit. (i) By Comparison Test (CT)
or (ii) By Limit Form of Comparison Test (LFCT). When using LFCT be sure
that the limit is actually a number not equal to zero. Remember,
you can only conclude convergence if (a) the integral is less
than a known convergent integral or (b) the integral is greater
than a known divergent integral. If (c) the integral is less
than a known divergent integral or (d) the integral is greater
than a known convergent integral, then there is NO CONCLUSION. --
A sequence is just a function from the natural numbers into the
set of real numbers. Two sequences are equal if, and only if,
the n-th term is the same in both sequences for
all natural numbers n.
Sequences will play an important role in series, something we will discuss
soon. Just like real functions, a sequence is convergent when n
tends to infinity, when the n-th term gets closer and closer
to some number. Otherwise the sequence is divergent. There are many ways
to see if a sequence is convergent or not. If the corresponding function
of x instead of n has a limit when x
tends to infinty, then the sequence is convergent. Most sequences
are defined by a formula or by some rule in terms of the previous
terms (like the Fibonacci numbers, where each term is the sum
of the previous two terms.) Many similar facts hold
for sequences when n tends
to infinity, as for regular real functions when we take the limit as
x goes to infinity. In particular, we can use L'Hospital's
rule on the corresponding real function and deduce from that what
happens to the sequence if the sequence is given by a formula/expression
we are familiar with in terms of a real function. Other facts holds
as well, like the Sandwich Theorem. To prove that a sequence
is divergent it is very often convenient to show that it has two
subsequences where one converges to L and the other
converges to L' distinct from L.
- Lectures and Homework (HW) from Week 9.
HW for RCT April 2. :
- Chapter-8:
- Sec-3: 7, 15, 19 21, 22, 27, 28, 39, 41, 47, 54, 55, 59, 61, 63, 65, 68, 71, 73, 77, 87, 89.
- Sec-4: 9, 17, 19, 21, 24, 28, 29, 30, 34, 43, 45, 50, 52, 56, 57, 58, 60, 64, 69, 70.
- Sec-5: 27, 28, 31, 37, 38, 44, 46, 47, 51, 53, 60.
A fundamental fact for convergent sequences is that
(1) An increasing sequence bounded from above is convergent and
(2) A decreasing sequence bounded from below is convergent.
Pickard's Method is an appealing general method that works to solve
an equation of the form g(x) = x when |g'(x)| < 1
on a chosen interval that includes the root/solution
x = r. To show that a sequence is divergent
it is sufficient to find two subsequences that tend to different
limits as n tends to infinity. -- Every sequence yields
a series, which then is convergent or divergent. A series is convergent
if, and only if, the partial sums of it is convergent. One of the most frequent
and easy series is the geometric series. We know exactly when
it converges and what its value is, when it converges. Another
type of series is the telescoping series. It converges
if, and only if, the corresponding function f(n) has
a limit when n tends to infinity.
One of the most important test for divergence is the
n-th term test: If the n-th term does not
go to 0 then the series is divergent. Also, the sum of two convergent series
is a convergent series. The sum of a convergent series and a divergent one is
divergent. The sum of two divergent series can be either convergent
or divergent. -- To check for convergence in series
with positive terms, there are basically
five tests: (1) The integral test. To be used if the n-th term
can be integrated when n has been replaced by x.
(2) The direct comparison test. To be used if you know that each term
is strictly less than a known term, the seris of which is convergent,
or if each term is greater than a known term, the series of which
is divergent (Remember Less-Conv, or Great-Div, or LCGD (ala ACDC..!))
(3) The limit form of comparison test is basically a refinement of
the direct comparison test. -- The two remaining test (ratio and root
tests) will be discussed next time.
- Lectures and Homework (HW) from Week 10.
HW for RCT April 9. :
- Chapter-8:
- Sec-5: 9, 10, 11, 12, 15, 17, 19, 20, 21, 22, 23, 26, 39, 41, 42, 43, 52,56, 57, 58, 59, 63, 69, 70, 71, 73, 75, 78, 80.
The ratio and the root tests are nice since they are purey
mechanical; just compute the limit and conclude (unless the limit
is one.) The ration test should be used when
the n-th term involves a lot of factorial stuff.
The root test should be used if the n-th term
has the form of something to the power of n.
Recall that when comparing with another series, it is not
always clear what to compare with, kinda like when integrating
it is not always clear what substitution to make. When using
the ratio or root test however, you just have to be careful
to make the computations correct, kinda like when differentiating:
You just do the calculations. We have seen that there are two kinds of convergence:
absolute convergence and conditional convergence. If a series
is absolutely convergent, then it is convergent, and more over
it is OK to do all kinds of manipulations on them, like rearrange
the terms at will and the corresponding series would also be
absolutely convergent and with the same sum as the original one. --
If an alternating series is not absolutely convergent, then
on uses Leibniz Test to test for conditional convergence.
- Lectures and Homework (HW) from Week 11.
HW for RCT April 16. :
- Chapter-8:
- Sec-6: 11, 12, 13, 17, 18, 23, 27, 39, 45, 46, 47, 49, 55, 63.
- Review: 31, 37, 53, 54, 59, 71.
- Chapter-9:
- Sec-2: 9, 10, 11, 15, 16, 18, 19, 21, 22, 29, 30, 35, 37, 38, 41, 46, 47, 48, 51, 52, 62, 63, 66, 68, 76.
Power series allow us to write a function as a series on a given
interval with a given center (the radius of the convergence
is half the intervals length). Such functions can be differentiated
infinitely often, term by term, and also integrated (as often as you
like) term by term. This is the best one could have hoped for regarding
these functions! They behave exactly like we want them to.
- Lectures and Homework (HW) from Week 12.
HW for RCT April 23. :
- Chapter-9:
- Sec-1: 3, 15, 16, 17, 18, 23, 77, 79, 81, 82.
- Sec-3: 9, 10, 11, 12, 15, 21, 29, 30, 32, 37, 39, 40, 45, 46, 50, 51, 52,
62, 63, 64, 66, 80, 81, 87.
- Sec-4: 7, 8, 9, 13, 17, 19, 37, 55, 56, 61, 71, 73, 78.
- Review: 47, 50, 51.
- D1: (see D1 for the online chapter, it is not in the book.)
- D1-1: 15, 16, 17.
- D1-3: 7, 8, 11, 12, 19, 25.
- D1-4: 5, 6, 7, 11, 12, 45, 46, 47.
To express a given function as a power series is to find
the Taylor series (or Maclaurin series) for the function.
We have so far been able to find the Taylor series of
some key functions at x = 0. These series
we can then differentiate and integrate, and further multiply
by x or any power of it, and also substitute in for
x to obtain Taylor series for other functions as
well. This is a powerful method to obtain Taylor series
for known functions. The applications of Taylor
series is nice, since they give us a way to see
the hidden behaviours of them for small x
and we can effectively use them to compute variuos
limits that are basically impossible with other
methods like L'Hospital's. The n-th
Taylor polynomial of a function is then the unique
polynomial that approximates the function best
among all n-degree polynomials, just
like the best linear approximation of a function
is given by the tangent line through the point
in question. In this way Taylor polynomials are a
generalization of the best linear approximations
that is covered in math 113. -- The last application
of Taylor series is for differential equations.
But in order to fully understand this applicatoin, we
need some basic info on differential equations in general.
The two types of 1st order differential equations we have
so far briefly discussed are the (i) separable ones,
and then (ii) linear first order differential equations (LFDE).
Next week we will continue our discussion about differential
equation and how Taylor series can be used to solve some of them.
- Lectures and Homework (HW) from Week 12.
- Lectures of week 12, 2nd part (see above.).
Exam Info (Quizzes, Midterm and Final)
QUIZZES:
Here below are the handwritten solutions to
the quizzes so far.
1ST MIDTERM EXAM:
Below is a review sheet for the first midterm. The
midterm is to be held in the classroom,
Tuesday, March 6 in the latter half of the lecture. -- Please
note that the review sheet is not a recipe for the midterm,
but merely a collection of problems some of which could
appear on the midterm. Basically, the topics for this midterm
includes all the lectures from and including last Thursday February 22nd.
This is roughly
Chapter 6, sections 2 -- 6 plus center of mass (from Chapter 13, section 6),
and theorems of Pappus,
sections 8, 9 (finance model), section 10.
Chapter 7, sections 1,2,3,5. -- The midterm
will consist of 5 - 8 problems, none of which are multiple choice.
I will give partial credit if I see the methods are correct, even
if the answer is totally wrong. No calculators nor cheatsheets
are allowed.
The median score for the 1st midterm exam was 51/100.
Many of you are doing well, but a considerable portion still
has troubles with basic algebra. Please review your algebra, as
we will rely on it throughout the course (and beyond!)
Note that the letter grade posted on the Patriot Web is only
for this first midterm exam, and nothing else.
Below are handwritten solutions to the first midterm.
Please print out a hard copy for your records.
Make sure you understand the solutions.
2ND MIDTERM EXAM:
Below is a review sheet for the second midterm. The
2nd midterm is to be held in the classroom,
Tuesday, April 3rd in the latter half of the lecture. -- The format
of the 2nd midterm is the same as for the first midterm. And, again,
please note that the review sheet is not a recipe for the midterm.
Basically, the topics for this 2nd midterm
includes all the lectures from the first midterm to and including
Thursday March 22. This is roughly Chapter 7: sections 3,4,8.
Chapter 8, sections 1,2.
Below are handwritten solutions to the second midterm.
Please print out a hard copy for
your records. Make sure you understand the solutions.
The raw median score (of the elevated "boxed" score you got)
was 43/100.
FINAL EXAM:
Below is a review sheet for the final exam. My plan is to go over most of
these problems in class. However, it is important that you
look them over and do what you can before the classes. -- Please note
that this review sheet is not a recipe for the final exam.
MATHEMATICA Info
IST ASSIGNMENT:
Here below is the first MATHEMATICA assignment for you, along
with a short info-sheet on the basics of MATHEMATICA. Read carefully
the instructions and make note of the due date!
You are to hand in your solutions to your TA in your RCT session
no later than April 9th, 2018. There should be plenty of time so
there will be no acceptance of late turn-ins. Make sure that you
read and answer all the questions, since some of them do not
require any MATHEMATICA outputs whatsoever! You may work in groups
up to 3 people in a group. In case you work in a group, you should
turn in just one solution to your TA with the names of all in the
group clearly printed and signed. Be sure you don't wait until
the last day to do this. This assignment is supposed to be fun
and not an added stress booster.
As the above guide indicates, you can get your
own copy of MATHEMATICA at this
GMU computing site.
Also, this
Wolfram site
contains pretty much all you need to play with MATHEMATICA.
From here you can click on
``Symbolic and Numeric Computation'' and get to
``Mathematical Functions'' or
``Calculus''. From ``Calculus'' you can go to
``Integrate'', then under ``Tutorials'' you can go to
``Simplifying Algebraic Expressions''.
For plots click on
``Visualization and Graphics'' and get to
``Functional Visualization''. From there you can
to to ``ParametricPlot'' for plotting a parametrized curve.
A useful "how to do this and that" in MATHEMATICA can be obtained from
Wolfram How Tos. Please note!
The commands are case sensitive, and all functions are capitalized.
2ND ASSIGNMENT:
Below is the second (and last) MATHEMATICA assignment, along
with an info-sheet on MATHEMATICA you need for this assignment.
Read carefully the instructions and make note of the due
date!
You are to hand in your solutions to your TA in your RCT session
no later than April 30th, 2018.
EOF/EOS