Homework MATH
2182H: Honors Calculus II
Instructor: Rodica D. Costin
Will be updated
after each class!
Koch flake
(picture produced by Prof. Ed Overman, with angles a bit
modified, to look nicer)
Cassini Mission to Saturn
Learn
Mathematica! Here is a first
plot in polar coordinates
Here is the rose with 8 leaves: the
file used (same, but printed as a pdf
file)
Learn to solve equations and to integrate: the file (its pdf print)
Learn
Maple! here is a first plot (pdf file)
There rose with 8 leaves: the code printed
as a pdf file
Learn to solve
equations and to integrate (pdf
file)
Now you try:
Plot r=2*sin(t). Try each of the intervals
t=0..Pi, t=-Pi..0, t=0..Pi/2
Solve for practice:
15.2: 1, 2, 3 and plot the parabolas in 3, 9
15.3: 1, 3, 4
15.4: 1,3,5,7, 17, 18, 19, 24a
16.1: 1, 5, 9b
Regarding Ex 4: We plotted two leaves, on one period of
sin(2theta). But we should still plot the picture in the whole
plane, meaning that we should cover with theta an interval of
length 2Pi. But using the periodicity Pi in r=f(theta), in the
rest of the plane we obtain the picture by rotating our two leaves
by an angle Pi.
16.2: 1, 2ab, 4 aegh, 5 abcdef, 6abcd
16.3 1,2,7,13,15,17
16.5: 5,7
17.1: 1, 2, 3
New files to help you learn Mathematica and/or Maple are posted
above.
Write up and turn in for grading on T Jan. 17:
15.2: 1a,f; plot
the parabolas 3b,e; 9
15.3: 3
15.4: 18, 24a
16.1: 1j,l, 4, 6a 16.2: 5 def, 6 bcd
Write up and turn in for grading on T Jan. 24:
16.3: 2,7,17
16.4: 10, 3ab, 4,12 ALSO: 5
16.5: 2,4,6
17.1: 4, 6, 13
Files to help you learn Mathematica and/or Maple are posted above.
See the Cycloid.
Another cycloid (with Maple
code)
See Hypocycloids: a=3, b=1
a=4, b=1
a=5, b=1
a=5, b=2
a=7, b=3
a=15, b=4
a=Pi, b=2
Review integration Sec. 10.3,
10.4. Learn trig formulas by heart.
Solve for practice:
17.2: 1, 7,
11
17.3: you should be able to
solve all problems here
17.4: you
should be able to solve all problems here
Write up and
turn in for grading on T Jan. 31 (date of our first
midterm!):
17.2: 2,
6, 8
17.3: 10 and 4, 6, 8
17.4: 2, 3, 4, 6
Midterm info:
You are allowed to use one page with your own notes.
The use of other notes, textbooks, electronic devices etc. is
strictly prohibited on this exam.
Topics covered:
Everything up to 17.4 (excluding from 17.4 centripetal force and
arc length parametrization).
Solve for practice:
17.5:
you should be able to solve all
problems there.
17.6: you should be able to solve all problems there.
17.7:
Maple
code for graphing evolutes, and their history
Write up
and turn in for grading on T Feb 8:
17.4: 12
17.5: 2cd, 8, 13 and 6
(which shows on an example that approximations by circles is
like approximation by Taylor polynomials of degree 2, as we discussed in class)
17.6: 2, 4, 8, 12,
10
17.7: A point is in uniform circular motion: R(t)=a
cos(omega t)i+a sin(omega t)j. Find the radial and
angular components of the velocity and of acceleration. Find a
physical intuition for your results.
Saturday Feb. 25! Rasor-Bareis-Gordon Math Competition! Fun and cash prizes!
Every year one or
two of my students placed high!
See more information here
Solve for practice:
You should be able to solve all
problems in 18.1-18.4 (but we will review 18.3, 18.4 and discuss
work later)
Write up and
turn in for grading on T Feb 14:
17.7: 2
18.1: 4c, 6cd, 8,
10a, 12, 14a, 16a
18.2: 2, 4b, 6, 8a,
10
Sign
up for the RBG competition here!
Solve for practice:
You
should be able to solve all problems in 19.1-19.5. Except for
19.3: 18 19.
19.6: 1...16, 19
Comments: 19.1: 14, 15 will be revisited. 19.3: 11, 12,
13, 18 neglect the name of the surfaces, we will revisit.
Write up and
turn in for grading on T Feb 21:
19.1: 8, 10, 12, 24
19.2: 10, 16, 20, 22, 27, 30d
19.3: 8, 12 I
deleted 16 and 18; they will be reassigned later.
19.5: 4, 6, 10
19.6: 8, 10, 14, 16
We solved in class, for practice:
18.2: 14, like 13, 15, 17
18:3: 2,3,15, distance from point to line, distance between 2 skew
lines (problem 11a)
Write up and turn in
for grading on T Feb 28:
18.2: 18, 21, 22 New!
18.3: 2, 4, 6, 8, 10, 12
18.4: 4, 6, 8, 10a,
14a, 20, 22, 24
19.3: 16 New!
19.6: 13a, 15 New!
And:18.4: 16
And: 18.5:
2, 4, 6, 12, 16
18.6: 2, 4, 6, 12, 16,
20, 24 (And solve for practice
1-16, 19-21, 24, 25) Here is the intersection of #19.
Solve for practice:
18.7:
1, 2, 5-11, 13-18
19.2: calculate partial derivatives for
practice
Review problems 1,2, 4 in the handout.
19.2: 19a, 29d
Write up and turn in
for grading on T March 7 (day of Midterm II):
18.7:
6, 8,10,16,18
19.1: 13, and find and plot the domain in 3,5
19.2: 19, 21b, 29a, 30c
and
19.2: 32c
19.3: 13
19.6: 4, 6, 9, 19c
Also: in the partial differential
equation a dw/dx=dw/dy (partial detivatives here) change variables
to u=x+ay, v=x-ay, then solve.
19.10: 6, 10, 12, 14b(But: do not use the formula, rather
use implicit differentiation).
AND: We noted that the gradient of f(x,y) is orthogonal to
the level curves of z=f(x,y). Use this to find a vector
orthogonal to the Folium of Decartes in problem 12 at the point
(3a/2, 3a/2).
Second Midterm info:
You are allowed to use one page with your own notes.
The use of other notes, textbooks, electronic devices etc. is
strictly prohibited on this exam.
Topics covered:
Everything starting with 17.4 up to 19.6 AND whatever we cover
this Friday and Monday, namely 19.10. (We will do 19.7,8,9 after
the midterm exam).
Just
because we are curious how formulas for solving
cubic and quartic equations by radicals look like: The cubic formula Quartic equations. Abel proved (1823) that here are no
formulas in terms of radicals for solving general higher degree
polynomial equations.
Thursday, March
9: You may enjoy this lecture on Neutrinos,
Dark Matter and the Nature of the Universe
Solve for practice:
19.7: 1..8
We discussed Taylor
polynomials in two variables
We covered Absolute
extrema (see
also here,
p. 84-90)
Write up and turn in
for grading on T March 21st:
19.7: 8, 12
and 19.7: 29 and
A. Find the absolute max and min of f(x,y)=1+4x-5y on the
closed triangular region with vertices (0,0), (2,0) and (0,3).
B. Find the
absolute max and min of f(x,y)=x^4+y^4-4xy+2 on the domain
D={(x,y)|0<x<3,0<y<2} where here < means "less or
equal".
and, now due next
week: 19.8: 2, 4a, 8,10
Have a Great Spring Break!
Solve for
practice:
20.1:
1, 3-28
20.2:
you should be able to solve all problem
20.4:
1-23, 39
Change
of variables in double integrals
20.3: you should be able to
solve all problems there
Write up and turn
in for grading on T March 28th:
19.8: 2,
4a, 8,10
20.1:
10, 12, 18, 28
20.2:
6, 10, 12, 14, 16
20.3: 8
20.4:
2,
6, 10. [Will be due the following week 14, 39a,e]
Solve for
practice:
20.5: be sure
you can solve: 1...20 by Friday Apr 31st, 21-30 by Monday
Apr 3rd
We did: change
of variable in triple integrals
Write up and turn in for grading on T April
4th:
20.4: 14, 39a,e
20.5: 10, 12, 18, 20 and
22,24,26,28,30
Also: use change of variable in triple integrals to
prove the following: Suppose the solid E is symmetric about
the xy plane, in the sense that a point (x,y,z) is in E if
and only if the point (x,y,-z) is in E. Show that the
integral on E of a function f(x,y,z) which is odd in z (that
is, f(x,y,-z)=-f(x,y,z)) is zero.
20.6: 6, 8, 10, 22
Solve for
practice:
20.7: 1, 3, 5, 9, 11,
13 For Wednesday: See Bumpy Spheres and
calculation of the volume
21.1 you
should be able to solve all problems
On Monday Apr. 10 we
will do and you will need to know the Theorem
in
Conservative vector fields
Write up and turn in for grading on T April
11th:
20.7:
4, 6, 8, 10
and
20.6: 24
20.7: 12, 13
21.1: 6, 8, 10,
20, 22
21.2: 4, 10, 12
Solve for practice:
21.3: 1, 2,3, 5, 7...11,
13...20, 23...28
20.8: you should be able to solve all 1..19
Write up and turn in for grading on T April 18th:
21.3: 4, 6, 8,
14, 20, 24, 30
20.8: 10, 18
21.4: 2, 4, 6, 10 and 12c, 14, 18
Solve for practice:
You need to know about conservative vector fields and simply
connected domains in 3d (see here).
See also the Informal Discussion here.
21.5: 1, 5, 7, 9, 11, 13, 15 see Klein bottle and Mobius strip
Write up and turn in for grading on MONDAY
April 24th:
Solve: Explain why the vector field F=yz i+(xz+z)
j+(xy+y+1) k is conservative and find its
potential. Is the potential unique?
21.5: 4, 6, 12,
14, 16
Special office
hours before the final exam: Thursday Apr 27, 11-12:30 am. (by
appointment between 4:30-5:30 pm)
Topics not to miss when
reviewing for your final exam:
Conic sections (be
able to recognize, plot and use)
Polar coordinates (be
able to use when needed)
Parametric equations
for lines, curves, surfaces (be able to parametrize the objects
you need and use in calculations)
Partial derivatives,
the gradient vector, the tangent plane, find normal vectors
Use the chain rule
when needed, use implicit differentiation
Use dot product, find
orthogonal projections of vectors and angles between vectors
Use cross product,
find area of parallelograms, find normal vectors to surfaces
Cylindrical and
spherical coordinates (be able to use them when needed)
Extrema (local,
absolute, Lagrange multipliers)
Multiple integrals (be
able to set up, evaluate, use appropriate coordinates, also
calculate masses, centroids, volumes, area)
Line integrals:
calculation, conservative fields (or not), find potential of F
(or argue it does not exist)
Green's Theorem (be
able to state and use)
Gauss's Theorem (be
able to state and use)
Stokes' Theorem (be
able to state and use)