12.4. #2-20,30,36,38,44. Bonus: #47,48.
12.5. #2,4,10,14,18,20,22. Bonus: #12.
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.
HW5A. Read: 13.3. Arc Length and Curvature (up to Normal and Binormal Vectors (non-inclusive)).
13.3. #2,4,14,16,18,22,24,28,30,32,46. Bonus: #57 (only first sentence)
HW5B. Read: 13.3. Normal and Binormal Vectors (finish). 13.4. Motion in Space: Velocity and Acceleration (up to Kepler's
Laws, non-inclusive). 14.1. Functions of Several Variables (up to Level Curves, non-inclusive)
13.3. #48,50,56. Bonus: #57 (finish)
13.4. #4,6,12,16,18(a),23,38,40,42. Bonus: #22,28,45
HW6. Read: 14.1. Functions of Several Variables (finish; concentrate on graphs,level curves, and 3-variable functions).
14.2. Limits and Continuity
14.2. #4,6,10,16,26,30,32,34,36,38*. Bonus: #40,41,44.
Notes on limit problems in HW6 and in general: When looking for the limit of z = f(x,y) at (a,b), there are several cases that can occur:
1. The function f(x,y) is continuous at (a,b) (use the CL's, Continuity Laws, and our conclusions about continuous functions from class).
After stating this and why the function is continuous at (a,b), you can plug in (a,b) for (x,y) and the limit will come out easily.
For example, 14.2 #6 in HW6 is about a rational function that is continuous at (2,-1). This type of situation is the easiest that can occur.
2. The function f(x,y) is not continuous at (a,b) because, for example, f(x,y) is not defined at (a,b).
(a) Try several different paths of approach to (a,b) and find the limits of f(x,y) along these paths. If two such limits are different,
then conclude that the overall limit does not exist. See, for example, 14.2. #4,10,44 in HW6. This situation is of "intermediate" difficulty.
(b) Trying different paths of approach to (a,b) produces the same limit of f(x,y) along all these paths! This is the "toughest" situation.
If you conjecture that the limit exists, you need to prove it, using other approaches. For example, substituting r = x^2+y^2 as in 14.2 #40,41 in HW6
simplifies the function and the limit to the single-variable case, and then you can appyl LH to the resulting single-variable function.
If such a substitution does not exist, things get even more complicated. You might be able to apply the Squeeze Theorem as in 14.2 #16 in HW6.
If nothing else works, the final resort is the epsilon-delta definition of limit.
In this course, for the time being, we will concentrate on cases #1 and #2(a), and perhaps, some simple cases of 2(b) where substitution works to reduce
the problem directly to a single-variable function.
Two good examples on non-continuity, which you need to be able to do, involve piecewise-defined functions:
(A) f(x,y) = 2xy/(x^2+2y^2) for (x,y) not equal to (0,0) and f(0,0)=0 (or whatever else). This is like 14.2 #4 from HW6.
The function is NOT continuous at (0,0) because the limit DNE at (0,0), regardless of what we define f(0,0) to be.
(B) f(x,y) = (x y^2+y^2 x)/(x^2-y^2) for (x,y) not equal to (2,-1) and f(2,-1)=1. This is like 14.2 #6 from HW6.
The function is NOT continuous at (2,-1) because the limit at (2,-1) equals -2/3, while the value of f there is 1.
HW7A. Read: 14.3. Partial Derivatives (read on your own the example associated with Table 1 on p.912; read everything else except Partial DE on p.920
and Cobb-Douglas Production Function on p. 922 -- these are optional); 14.4. Tangent Planes and Linear Approximations
14.3. #2,6 (I find #6 tricky because one has to visualize and interpret what the picture is telling us and keep in mind the positive
direction of all axes!), #12 (you can draw the graph by hand), #18,20,22,26,28,34,36,42,56,60,62; Bonus: #30,38,72
14.4. #4,6,12,14,16,17,18,19,20,21(don't graph); Bonus: #46
HW7B. Read: 14.4. Tangent Planes and Linear Approximations (finish; concentrate on differentiable functions: Definition 7 and Theorem 8; read also about Differentials for your general understanding);
14.5. Chain Rule. 14.6 Directional Derivatives(concentrate on directional derivatives up to Example 2, inclusive)
14.5. #1,2,4,6,8,10,14,16,20,22,28,30,34; Bonus: #49,51
14.6. #2 (read Example 1, p. 948), #4,6,8,12,14,20,28. Bonus: #30
HW8A. Read: 14.6. Directional Derivatives (starting from the Gradient Vector); 14.7. Maximum and Minimum Values (concentrate on critical points
and how to find them using the gradient, and on Second Derivatives Test)
14.6. #21,22,23,24,29,34,38,42,44,50,54,56; Bonus: #52,62*
14.7. #2,4,6,8,12,14,16,18,20; Bonus: #22.
Notes on the Gradient and Its Applications from lecture on March 8. The following concepts and theorems were covered today in lecture:
2. How gradient is used in computing directional derivatives
3. How gradient gives the answer to questions like: "In which direction is the rate of change of a function z = f(x,y) largest?"
4. How to locate the gradient on a contour map of z = f(x,y): the gradient points in the direction of largest ascent; the gradient is perpendicular to the tangent
line to any level curve, thus the gradient can be thought of as a "normal" to any level curve.
5. Similarly, how to think of the gradient in regards to a level surface g(x,y,z) = c of a function t = g(x,y,z), the gradient of f points in the direction of largest ascent; the gradient is perpendicular
to the tangent plane to the level surface, and thus the gradient can be thought of as a "normal" to the level surface.
6. The gradient as part of the equation of the tangent line to a level curve f(x,y) = c, and analogously, the gradient as part of the equation of the tangent plane
to a level surface g(x,y,z) = c.
Warning about confusion between the formulas for tangent plane to a surface z = F(x,y) and for tangent plane to the level surface
g(x,y,z) = c. The formulas eventually will give the same tangent plane equation, provided you use the formulas correctly. For example, if you have a surface z = F(x,y)
and a point on the surface P(a,b,z=F(a,b)), you can use our previous formula for the tangent plane: z = F(a,b)+F_x(a,b)(x-a)+F_y(a,b)(y-b), OR you can turn your
function into a level surface F(x,y)-z = 0 and treat this with the gradient approach for the function t = F(x,y)-z = g(x,y,z); the tangent plane will be
g_x(P)(x-a)+g_y(P)(y-b)+g_z(P)(z-f(a,b)) = 0; since g_x = F_x and g_y = F_y (why?), while g_z = -1 (why?), we end up with exactly the same tangent plane equation as above:
F_x(a,b)(x-a)+F_y(a,b)(y-b)-1(z-F(a,b)) = 0, or equivalently, solving for z we get z = F(a,b)+F_x(a,b)(x-a)+F_y(a,b)(y-b). Thus, yes, you can use either approach to find the
equation of the tangent plane at a point on the graph of a FUNCTION z = F(x,y). But you have to be careful with the second approach, where you need to turn your function z = F(x,y)
into the level surface of a "larger" function, namely, the level surface 0 = F(x,y)-z of the function t = F(x,y)-z. On the other hand, if you are already given a level surface (i.e., an equation of the form
g(x,y,z) = c), you should proceed directly with the "gradient" approach, instead of trying to solve for z and use the "function" approach.
7. The gradient used to find critical points of functions z = f(x,y). Critical points yield ONLY POTENTIAL local min/max of a function.
8. And if the above is not enough, :), we also covered a theorem that is NOT using just the gradient, but something "more": the Second Derivatives Test, which
yields the REALIZED (the actual) local min/max.
HW8B. Read: 14.7. Maximum and Minimum Values (finish; concentrate on finding global min/max). 14.8. Lagrange Multipliers
(up to Two Constraints, non-inclusive)
14.7. #32-38,42-52; Bonus: #40 (do not graph),57
14.8. #4-12,15; Bonus: #14,16.
Notes on HW8B from 14.7: The exercises here fall into two major categories:
Type A: problems in which the function and the domain are already given (e.g., section 14.7, #32-38). The algorithm here goes as follows (assuming f is continuous
on D, has continuous partial derivatives on D, and D is closed and bounded):
(1) find the critical points of f inside D (set gradient of f to be the zero vector); evaluate f(x,y) at these critical points.
(2) restrict f on the boundary of D; and each piece of this boundary, find the critical points of f (but now as a function of 1 variable); evaluate f
at these points.
(3) evaluate f at the "endpoints" of each piece of the boundary.
(4) make a list of all these values of f, and determine which is the largest (the global max) and where it occurs; and which is the smallest (the global
min) and where it occurs.
Type B: problems in which we need to first create the function from the situation in the problem (e.g., section 14.7 #42-52), determine the (desired) domain, and only
then optimize the resulting function as in a Type A-problem. Typically, these are harder problems, but they fall into several subcategories: (minimal) distance
(#42,44) problems, (maximal) area or volume problems (#48,50,52), "number" problems (#46). You can get better at these "word" problems only by doing more of them on your
own, studying the HW solutions and redoing the harder-for-you problems again, coming regularly to lectures, and participating actively in sections.
Notes on HW8B from 14.8: The exercises here (section 14.8, #4-12) address ONLY the global min/max "along the boundary"
(i.e., on a constraint curve or constraint surface). They are solved by setting by the system for Lagrange multipliers and solving it. We shall assume in these exercises
that global min/max exist along the givent constraint. [In the next lecture we will finish the story of lagrange multipliers and how they give an alternative method
for finding global min/max on closed+bounded domains and more.]
HW9A. Read: 15.1. Double Integrals over Rectangles. 14.8. Lagrange Multipliers (finish; Bonus: Two Constraints)
14.8. #22,23,24; Bonus: #18,20,49*.
15.1. #4-24,30,34-38,42,48; Bonus: #50,52.
HW9B. Read: 15.2. Double Integrals over General Regions (may skip for now Example 4 and all exercises on volumes in 15.2).
15.3. Double Integrals in Polar Coordinates (concentrate only on notes from lecture, i.e., writing Cartesian regions in polar coordinates)
15.2. #2-22,46-62; Bonus: #64,68.
HW10A. Read: 15.2. Double Integrals over General Regions
(finish, concentrate on Example 4 and exercises on volumes in 15.2). 15.3. Double Integrals in Polar Coordinates (finish, read all examples).
15.4. (Optional) Applications of Double Integrals (concentrate only on Probability and Expected Values)
15.2. #24-30,36,38,66; Bonus: #32,69.
15.3. #8-26,30,32. Bonus: #38,40*.
(Optional, not to be included on exams:) 15.4. #27,28,29,30,33.
Notes on HW10A-10B: Material up to 15.3 and HW10A (inclusive) will be on Midterm 2.
HW10B. Read: 15.5. Surface Area. 15.6. Triple Integrals
15.5. #2,4,6,10,12,21,23; Bonus: #22*,24*.
15.6. #2,6,10,12,14,16,18,20,22,36,38; Bonus: 54,55(a)*.
HW11. Read: 15.8. Triple Integrals in Spherical Coordinates. 15.9. Change of Variables in Multiple Integrals. 16.1. Vector Fields
(start reading as far as you can go in 16.1)
15.8. #2-12,18-26,30; Bonus: #28.
15.9. #2-10,15,16,17; Bonus: #12,14,18.
HW12A. Read: 16.1. Vector Fields. 16.2. Line Integrals, Part I (up to Line Integrals in Space, non-inclusive)
16.1. #2-18,22-26,30-32; Bonus: #34,36.
Notes on Lecture, 4/12/2016:
1. Section 16.1 on Vector Fields. We agreed to write a little vector on top of F for a vector field F(x,y) and F(x,y,z) since we cannot write in
"boldface" as in the textbook. Please, adhere to this notation so that you are reminded at any moment whether
you are working with scalar functions or vector functions/fields.
2. Part of Section 16.2. Line Integrals, as follows:
a. We defined for now informally the line integral of f(x,y) ds over a curve C as the area of a "shower curtain" hanging from a shower rod f|C (f restricted to C) all the way down to the curve C.
b. We have NOT yet defined the line integral using Riemann sums (Definition 2), nor have we proven the Formula 3 (p. 1075).
c. However, we introduced Formula 3 and we did a several examples with it.
d. We discussed the connection between ds and dt (as in the beginning of page 1076) via the arc-length formula.
e. We introduced a short-hand notation for Formula 3 as the integral from a to b of f(r(t))|r'(t)|dt where r(t) is the parametrizing vector function for C, and we recognized |r'(t)| as the speed along C.
f. We have NOT discussed yet that line integrals are generalization of ordinary integrals (third formula on page 1076).
g. We talked about splitting a curve C into pieces when calculating line integrals (bottom of page 1076).
h. We also introduced the special line integrals along C of f(x,y)dx and of f(x,y)dy by Formula 7 (page 1076), and talked about changing dx = x'(t)dt and dy=y'(t)dt.
i. We have NOT talked about these special integrals dx and dy via Riemann sums.
3. On Thursday:
a. We will cover all of the stuff for line integrals of scalar functions that was not covered on Tuesday (basically, everything that uses Riemann sums)
b. We will define and work with line integrals of vector fields.
c. We will interpret line integrals of vector fields as "work".
d. We will cover the statement and proof of the Fundamental Theorem for Line Integrals in 16.3.
HW12B. Read: 16.2. Line Integrals, Part II (finish). 16.3. Fundamental Theorem for Line Integrals (pp. 1087-1088)
16.2. #18-22;28,30,29(a),32(a),40-44; Bonus: #49,50; (Note: Just evaluate the integrals in #28,30. Do not use FTL Integrals in this unit).
16.3. #1,2; Bonus: Read as much as you can on your own in Section 16.3.
HW13A. Read: 16.3. Fundamental Theorem for Line Integrals (up to p. 1091, non-inclusive; optional: read section on "Conservation of Energy").
16.4. Green's Theorem
16.3. #4-24, 26*; Bonus: #28,35.
16.4. #2-14,18,19; Bonus: #21.
HW13B. Read: 16.4. Green's Theorem (finish; especially the proof of GT). 16.3. Fundamental Theorem for Line Integrals (finish; Theorem 6 and pp. 1091-1093; Conservation
of energy is optional). 16.5. Curl and Divergence (up to Example 5, inclusive)
16.3. #31,32,33,34; Bonus: #28,35.
16.4. #27,28,29; Bonus: #30 (redo without looking at your class notes).
HW14A. Read: 16.5. Curl and Divergence (finish; may skip for now Vector Forms of GT). 16.6. Parametric Surfaces and Their Areas
(may skip "Surfaces of Revolution"). 16.7. Surface Integrals (of scalar functions; up to "Oriented Surfaces", non-inclusive).
16.5. #20,22,24,26,28,30; Bonus: #32. (Find in the text what "irrotational" and "incompressible" mean.)
16.6. #2-6, 14-26,34,40-48; Bonus: #50,64*(a)(c).
HW14B. Read: 16.7. Surface Integrals (of vector fields; finish). 16.8. Stokes' Theorem (may skip for now "Proof of a special case of ST" and
"Meaning of curl F).
16.8. #2-10,12(a),14,16*,18*; Bonus: #17,19.
Solutions to HW14.
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 Solution within the next few days.
HW15A (Optional). Read: 16.8 Stokes' Theorem (concentrate on Proof of ST in a special case, and Examples 1 and 2);
16.9. Divergence Theorem (Theorem Statement and Examples 1 and 2; may skip proof of DT)
Write Exercises (Optional):
Make your own summary tables, according to the given handout here
Suggested Tables for a Review.
HW15B (Optional). Read: 16.10. Summary.
Write Exercises (Optional):
Review for Chapter 16: Concept Check. True-False Quiz: #1-13. Exercises: #3,9,10,11,13,15,17,19,21,25,27,29,33,35,36.
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