Analytic Geometry and Calculus II
Math. 114, Sec. 003, Spring 2018
Homework and Announcements
Here I will post important announcements to the class.
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.
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".
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. :
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.
- Review: 3, 15, 16, 17, 21, 22, 25, 29, 30, 38, 50, 60.
- 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.
- Lectures and Homework (HW) from Week 2.
HW for RCT February 5. :
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.
- 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
- Sec-6: 7, 8, 15, 16, 18, 51, 52.
- Lectures and Homework (HW) from Week 3.
HW for RCT February 12. :
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
- 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.
- Lectures and Homework (HW) from Week 4.
HW for RCT February 19. :
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.
- 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.
- Sec 1: 8, 9, 23, 25, 27, 28, 29, 30, 32, 33, 35, 36, 37, 42.
- Lectures and Homework (HW) from Week 5.
Exam Info (Quizzes, Midterm and Final)
Here below are the handwritten solutions to
the quizzes so far.