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)

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

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!

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
Solve for practice:
Y
ou should be able to solve all problems in 19.1-19.5. Except for 19.3: 18 19.
19.6: 1...16, 19
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, 25Solve 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 Solve for practice:
19.7: 1..8
We discussed Taylor polynomials in two variables
We covered
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:
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 3 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)