4. Regarding using LH to verify that the cycloid has vertical tangents at theta = 0, 2 pi, 4 pi, etc.: the formula for the tangent slope is sin(theta)/(1-cos(theta)).
When theta = 2 pi, for example, the ratio becomes 0/0. Thus, we take the limit of sin(theta)/(1-cos(theta)) as theta approaches 2 pi (from the left and separately from the right),
apply LH to replace top and bottom by their derivatives: cos(theta)/sin(theta) (we still have the limit at this stage), and finally, we plug in theta = 2 pi to get
1/0 (0^+ if approaching 2 pi from the right, and 0^- if approaching 2 pi from the left). In any case, we get the tangent slope to be plus or minus infinity, which indicates
a vertical tangent. Try to repeat the calculation from class that did NOT involve LH, but simplified first the tangent slope and then showed directly that where the tangent slope
was infinity (or undefined).
5. The example r = 4 sin (theta): it is a single circle x^2 + (y-2)^2 = 4, as we concluded in class. It is not two circles on top of each other, although it is
good to confirm this. In class we saw what happens when theta was between 0 and pi. When theta becomes larger than pi, e.g., theta = 5 pi/4, then r is negative: r = - 2 (square root(2)).
When we make r positive, we need to subtract pi from theta, so we land on the point (r = 2 (square root(2)), theta = pi/4), which is one of the points on our circle that we
already saw earlier. Thus, when theta becomes larger than pi, we start going over the same circle a second time, etc.
HW2A. Read: 10.2. Calculus with Parametric Curves (Review and finish). 10.4. Areas and Lengths in Polar Coordinates
10.2. #32,36,38,42,48,58 (in #38,58: just set up the integral), #62,66. Bonus: #73,74.
10.4. #4,6,8,12,18,28,32,38,46,48. Bonus: #36,44.
General notes on bonus HW problems: Bonus HW problems will be included in the weekly HW solutions. Ordinarily, they won't be included in quiz for that week, but they may be included
in exams (or even later quizzes, if suitable), so you need to study their solutions carefully. Keep in mind that a "bonus" problem earlier in the semester may not look
as hard later on in the semester.
HW2B. Read: 12.2.Vectors
12.2. #2,4,6,8,14,16,20,22,24,26,28,30,32,42,44,51. Bonus: #40; Prove that the three medians in a triangle intersect in one point (called the centroid of the triangle) and that this point
divides each median in ratio 2:1 counted from the vertices of the triangle. (A median connects a vertex of a triangle with the midpoint of the opposite side.)
Note: Problem 10.2. #36(c) was assigned unintentionally. We will not be testing on exams about centroids of general figures. The only centroid for now that you should be
aware of and be able to solve problems with is the centroid of a triangle (as we did in class). The first video below ("Triangle has a Magic Highway") and its extra footage
will tell you more on that topic.
HW3A. Read: 12.3. The Dot Product (including Ex.7 on work)
12.3. #6-14,18,22-30,36,38,42,50,56,58*. Bonus: #61,62.
HW3B. Read: 12.4. The Cross Product. 12.5. Equations of Lines and Planes (only the first section on Lines, including Example 3 on p. 826)
12.4. #2-20,30,36,38,44. Bonus: #47,48.
12.5. #2,4,10,14,18,20,22. Bonus: #12.
Solutions to HW3.
The HW solutions will be taken off the web in a week and will not be distributed in any form later
on, so make sure you download and study the HW within the next few days.
HW4A. Read: 12.5. Equations of Lines and Planes (finish, concentrating on Planes and Distances, pp. 827-830). 12.6. Cylinders and Quadric Surfaces
12.5. #26,28,38,40,46,56,66,70,72,74,76 (in #76: use formula from #75). Bonus: #62,75.
12.6. #2,6,10,18,34,36,44,46,48. Bonus: #52.
HW4B. Read: 13.1. Vector Functions and Space Curves. 13.2. Derivatives and Integrals of Vector Functions
13.1. #2-12,16,18,28-32,42,44. Bonus: #46,48,50.
13.2. #2-10,18-22,26,28,34,36,40,42. Bonus: #16 (in two ways!),55.
General note on HW problems: Ordinarily, only even-numbered problems are assigned, unless an odd-numbered problem is explicitly included in HW. Thus,
for example, #2-12 means #2,4,6,8,10.
Game and Geometry Puzzles from the Numberphile Channel with Zvezdelina Stankova, directed by Brady Haran, containing
some proof and problem-solving ideas that come handly in a variety of math areas:
Berkeley Math Circle