By request: Consider the function cfw_x if x is rational, f(x) = cfw_ cfw_ x if x is irrational Where is it continuous? Even worse: the function cfw_ x2 if x is rational, g(x) =cfw_ cfw_ x2 if x is irrational is differentiable at 0 but nowhere else! Secti
Time Sheet for 92.241 Assignment #1 Name: Minutes spent on 10.1.4: Minutes spent on 10.1.6: Minutes spent on 10.1.10: Minutes spent on 10.1.17(a): Minutes spent on 10.1.18: Minutes spent on 10.1.20: Minutes spent on 10.1.30: Minutes spent on 10.1.36: Minu
Time Sheet for 92.241 Assignment #2 Name: Minutes spent on 10.2.2: Minutes spent on 10.2.6: Minutes spent on 10.2.18: Minutes spent on 10.2.20: Minutes spent on 10.2.26: Minutes spent on 10.2.33: Minutes spent on 10.3.9: Minutes spent on 10.3.10: Minutes
Time Sheet for 92.241 Assignment #3 Name: Minutes spent on 10.4.9: Minutes spent on 10.4.10: Minutes spent on 10.4.11: Minutes spent on 10.4.12: Minutes spent on 10.4.18: Minutes spent on 10.4.36: Minutes spent on 10.4.38: Minutes spent on 10.4.45: Minute
Time Sheet for 92.241 Assignment #4 Name: Minutes spent on 10.6.2: Minutes spent on 10.6.4: Minutes spent on 10.6.12: Minutes spent on 10.6.22: Minutes spent on 10.6.32: Minutes spent on 10.6.34: Minutes spent on 10.7.2: Minutes spent on 10.7.6: Minutes s
Time Sheet for 92.241 Assignment #5 Name: Minutes spent on 10.8.10: Minutes spent on 10.8.18: Minutes spent on 10.8.24: Minutes spent on 10.8.30: Minutes spent on 10.8.44: Minutes spent on 10.9.18: Minutes spent on 10.9.22: Minutes spent on 10.9.32: Minut
Time Sheet for 92.241 Assignment #6 Name: Minutes spent on 11.1.20: Minutes spent on 11.1.22: Minutes spent on 11.1.24: Minutes spent on 11.1.26: Minutes spent on 11.1.28: Minutes spent on 11.1.32: Minutes spent on 11.1.4146: Minutes spent on 11.2.6: Minu
Questions about group work from last time? Note: Friday's exam will have four questions rather than three. (Not sure about Monday's yet.) If there are more questions, they'll be easier individually. Concept Check for chapter 12: questions 3, 7, 9 [Hand ou
[Hand out Concept Check 13 sheet] [Hand out cover sheet for exam and read it aloud] The exam will focus on chapters 12 and 13. You may use a five-page "cheat-sheet", but it must be created by you; you cannot photocopy reference books or print things out f
Math 241, Problem Set #1 (due in class Fri., 9/9/11) Stewart, section 10.1, problems 4, 6, 10, 17(a), 18, 20, 30, 36. For the last of these problems, you may use the formula proved in Exercise 7.2.27 (which you don't need to prove!). For some of these pro
Math 241, Problem Set #1 (due in class Fri., 9/16/11) Stewart, section 10.2, problems 2, 6, 18, 20, 26, 33. Stewart, section 10.3, problems 9, 10, 18, 21, 27, 30, 34, 38, 40. Also: A. Find all unit vectors that are orthogonal to both i + j and j + k. B. (
Math 241, Problem Set #3 (due in class Fri., 9/23/11) Stewart, section 10.4, problems 9, 10, 11, 12, 18, 36, 38, 45. Stewart, section 10.5, problems 1, 10, 18, 24, 30, 32, 44, 52, 54. Also: A. Given points P = Q in R3 , describe geometrically the set of a
Math 241, Problem Set #4 (due in class Fri., 9/30/11) Stewart, section 10.6, problems 2, 4, 12, 22, 32, 34. Stewart, section 10.7, problems 2, 6, 10, 14, 28 (hint: use trig functions), 34, 44, 46, 50, 58, 64, 70, 78. Also: A. Show that for the helix h(t)
Math 241, Problem Set #5 (due in class Fri., 10/7/11) Stewart, section 10.8, problems 10, 18, 24, 30, 44. Stewart, section 10.9, problems 18, 22, 32, 34. Also: A. Recall that in the metric system of measurement, position has units of meters and time has u
Time Sheet for 92.241 Assignment #7 Name: Minutes spent on 11.5.6: Minutes spent on 11.5.28: Minutes spent on 11.5.41: Minutes spent on 11.5.47: Minutes spent on 11.6.4: Minutes spent on 11.6.8: Minutes spent on 11.6.12: Minutes spent on 11.6.34: Minutes
Time Sheet for 92.241 Assignment #8 Name: Minutes spent on 12.1.8: Minutes spent on 12.1.16: Minutes spent on 12.1.26: Minutes spent on 12.1.28: Minutes spent on 12.1.42: Minutes spent on 12.2.12: Minutes spent on 12.2.16: Minutes spent on 12.2.20: Minute
[Collect section notes on 10.6.]
Section 10.6: Cylinders and quadric surfaces
Main ideas?
.?.
Main ideas:
Cylinders are three-dimensional extensions of curves
in two variables
Quadric surfaces are the graphs of second-degree
polynomial equations in x,
Section 10.6: Cylinders and quadric surfaces (concluded)
Use traces (and other kinds of mathematical reasoning) to
describe the following surfaces:
z = x2 + y2 + 1
z2 = x2 + y2
z = sqrt(1 x2 y2)
[Show http:/jamespropp.org/241/09.19.pdf when done.]
[Hav
Section 10.7: Vector functions and space curves
Main ideas of pp. 580584?
.?.
.?.
The connection between space curves and ranges of
vector functions
Matching vector equations with the curves they
determine
Parametrizations of curves in space are not un
Whos looked at the solutions to the first assignment?
.?.
.?.
Was it helpful?
[Collect section summaries for 10.5.]
Section 10.5: Equations of lines and planes
Main ideas:
.?.
Three ways to describe a line:
o Vector (parametric) equations:
r = [OP0] + t
Section 10.3: The dot product (concluded)
Paradox I asked you to think about for today: Let u be a
vector such that |u|=1. Choose a vector v such that |v| = 3
and uv = 5. Now we have
|uv|2 = (uv)(uv)
= uu 2(uv) + vv
= 1 2(5) + 9
= 0.
Hence u=v. But u and
[Collect section summaries for 10.1.]
Chapter 10: Vectors and the Geometry of Space
Section 10.1: Three-dimensional coordinate systems
Main ideas?
.?.
.?.
Right-hand rule
Coordinates (example?)
Coordinate axes and coordinate planes (example?)
Projecti
Section 10.3: The dot product (continued)
Suppose a b = a c 0, with a 0. Can we conclude
that b = c?
.?.
.?.
No; e.g., i (i+j) = i (i+k) (both dot-products equal 1)
but i+j i+k.
If a b = a c, with a 0, there is a conclusion we can
draw about equality of t