Homework Assignments for MATH 2421 Calculus III (Spring 2007)
Lynn S. Bennethum                        Mon. and Wed., 5:30-7:20pm, SO 229

Date	What is Due	Notes
Saturday May 5	UNIFORM FINAL 9am-noon  NC 1130	No notes, No Technology.  Cheat sheet will remain the same.
Wed. May 2	Magnetic Field Project due. No homework is to be turned in but please do problems from Sections 12.6 and 12.7.
Wed. April 25	EXAM 2  Section 10.7 through 12.5	Section 12.5 will not be turned in - I will hand out solutions.  For the exam:  No Notes.  No Technology.  Cheat sheet will remain the same.
Wed. April 18	12.3 12.4 12.5  number 3 only	Pracice exam will be handed out.
Wed. April 11	11.7 remaining problems (beginning with 13) 12.1 12.2
Wed. April 4	11.4 Listed Problems 11.5  Listed Problems 11.6  Listed Problems 11.7   1, 6, 10	We will omit Section 11.8
Wed. March 28	11.1 Listed Problems 11.2 Listed Problems 11.3 Listed Problems	Section 11.4 problems will be due April 4.
Wed. March 14	10.7 Listed Problems 10.8  Listed Problems
Wed. March 7	EXAM 1 Through Section 10.6	No Notes.  No Technology. Cheat sheet will be handed out. Solution to Problems from 10.6 will be handed out on Monday.
Wed. Feb. 28	10.3 Problems 51 and 53 10.4  Listed problems 10.5 Listed problems	Practice exams and cheat sheet will be handed out.
Wed. Feb. 21	Remaining problems from 8.6 Problems from 10.1, 10.2 Problems from 10.3 (up to and including 14)
Wed. Feb. 14	Remaining problems from 9.3, Problems from Sections 9.4 and  9.5, and Problems 1, 3, and 6 from Section 8.6
Wed. Feb. 7	Remaining problems from 8.5, Problems from 9.1, 9.2, and 3, 9,  and 14 from 9.3
Wed. Jan. 31	Remaining problems from 8.2, Problems from 8.3, 8.4, and problems from  8.5: 3, 5, 7, 10
Wed. Jan 24	Problems listed below from 8.1, and problems from 8.2 up to and including 35

Section 8.1:  

• 1, 5, 11
• 12  (explain please).  Also for 12:  Give a unit vector parallel to the vector a=<1,-2>
• 19
• 20 Find BOTH the component form and the polar form.
• 23, 31
• 40 

Section 8.3

• 3,10, 11, 18, 
• 26 - Also find the vector w so that a= proj_b a + w.
• 32
• 33a b c e
• 49


Section 8.4

• 5, 9, 12, 
• 27, 30, 65, 66


Section 8.5

• 3, 5, 7, 10,  (find both symmetric and parametric form - parametric form is by far more important)
• 11, 
• 18  Also, if the two lines intersect, where do they intersect?
• 23, 26, 42
• 48  (We will derive the formula in class)


Section 8.6  

• 1, 3, 6, 14, 22  (all sketches should be by hand - you may check using technology)
• 49


Section 9.1  

• 7, 9 (by hand only - be sure to indicate orientation),
• 6  modified:  a)  Show how to eliminate the variable t to get an equation for a conic section in the xy-plane.  b)  From the x-y equation, graph.  c) On this graph, carefully show the of the orientation as t increases.
• 21, 25  Using technology only (Derive5 is good).
• 23 Modified:  a) Sketch the curve using technology.   b)  Find an equation in terms of z^2 = f(x,y).  You should be able to eliminate the parameter t and the sine and cosine functions.
• 31  (this is not easy!  Answers are in back, but you must justify the answers in a sentence or two).
• 37 (use technology)
• 34 (use technology)


Section 9.2

• 11, 13, 14, 15
• 21, 20 (you should have a curve and for each point, two vectors with the tail at the corresponding point)
• 27, 32
• 35, 36
• 41

Section 9.3

• 3, 9, 14
• 25, 37, 39, 40


Section 9.4

• Arc length practice problems from Mike Kawai's Arc Length handout sheet.  Do By HAND (you may check using technology)
1. Find the arc length r(t) = <2sin(t), 5t, 2cos(t)>, -10<t<10  Hint:  ||v(t)|| is a constant.
2. Find the arc length r(t) = <t^2, sin(t)-t cos(t), cos(t)+t sin(t)>, 0 < t < pi  Hint:  ||v(t)|| = sqrt(5t^2)
3. Find the arc length r(t) = <sqrt(2) t, e^t, e^(-t) > , 0 < t < 1  (Hint:  It's a perfect square)
• 5, 6   Do by HAND (you may check using technology)
• 15, 16    Do 15 using technology, do 16 by HAND using cross-product formulation of curvature.

Section 9.5

• 2  Do by HAND
• 13  Do by Hand (only need to do for t=1).  When t=1, is the speed increasing or decreasing?
• 29, 30, 33  Can use technology.

Extra Credit Problem (10 pts):  Suppose a car is going around a turn. Is it better to speed up as one goes around the corner, stay at constant velocity, or slow down? As far as not slipping, this means we want to minimize the normal component of acceleration.   So consider two cars:  car A and car B.  Car A has position vector:
r(t) = <cos(t^2/2), sin(t^2/2)>, and car B has position vector r(t)=<sin(2t-.5t^2),cos(2t-.5t^2)>.
(i) Calculate the speeds of car A and B for 0< t <2.  Which car is speeding up, and which one is slowing down?
(ii) Calculate the principal unit normal vector, N(t), for each car, and plot the vectors on the graph of the curve each vehicle is driving at t=0, 1, 2.  Are both unit normal vectors pointing in the same direction?  If not, explain physically which car has the better direction.
(iii)  Calculate a_N, the normal component of acceleration, for each vehicle.  Graph a_N for each car for 0<t<2.  Based on this graph, which car has a better chance of staying on the road?  Although this is only one example, based on your results, is it better to accelerate through a curve or decelerate?
(iv)  (10 pts additional extra credit):  Prove mathematically why your result in (iii) is true in general (i.e. can you prove that it is alway better to accelerate/decelerate through a curve?).



Section 9.6

• Omit


Section 10.1 

• 3, 4,   Be sure to descibe and sketch the domain
• 19, 20, Be sure to label each graph (you should have two graphs for 19, and 3 graphs for 20).
• 37, 39  Be sure to label each curve in your contour plot.
• 47 (explain your reasoning in a sentence)
  

Section 10.2

• 10, 11  (hint, find two paths which approach (0,0) but do not have the same limit.  You may have to consider y=kx^??
• 55, 56  Convert (x,y) to polar coordinates.  Take the limit as r goes to zero and theta can be anything.  Write the value of the limit or say it "d.n.e."  (does not exist).

Section 10.3

• 1, 3, 7, 8
• 11, 14
• 51
• 53


Section 10.4

• 4, 5 (tangent plane only)
• 9, 16
• 19, 22
• 29, 30


Section 10.5

• 1 Find dg/dt both ways - one by substituting in x(t) and y(t) first and then differentiating, and the other by using the chain rule.
• 3, 6
• 10 write out the appropriate chain rule for two only.
• Wheat Production Problem (problem 28 from Stewart, S 11.5):  Wheat production in a given year, W, depends on average temperature, T, and the annual rainfall, R.  Scientists estimate that the average temperature is rising at a rate of 0.15 degrees C per year, and rainfall is decreasing at a rate of 0.1 cm/year.  They also estimate that, at current production levels,  (partial W)/(partial T) = -2 and (partial W)/(partial R) = 8.
1. What is the significance  of the signs of these partial derivatives?
2. Estimate the current rate of change of wheat production.
• 15
• 22, 23


Section 10.6

• 2, 6, 9
• 13, 14, 19
• 28, 33
• 41
• 57
  

Section 10.7

• 3 (by hand), 4 (by hand)
• 14 (use technology - print out graph or describe what you see)
• 15 
• 37, 38 (use Lagrange Multiplier - you should probably do Section 10.8 problems and then come back to these).

Section 10.8

• 3 (see example 8.1 and discussion prior to example 8.1)
• 13
• 24 (see example 8.4).

Section 11.1

• 10, 15,
• 23, 28, 34
• 53, 54
• 74

Section 11.2

• 1, 6
• 9, 16
• 21
• 26

Section 11.3

• 4, 5
• 11, 21, 24
• 31, 42

Section 11.4

• 4,7
• 10, 13

Section 11.5

• 3, 8
• 20, 30

Section 11.6

• 3, 15, 
• 21, 24,  29  Hint for 29:  determine the domain of the region for r and z first.
• 33, 36

Section 11.7

• 1,6
• 10, 13
• 19, 22
• 27, 30
•  40, 44, 55

Section 11.8   OMIT

Section 12.1

• 1, 6
• 11
• 14
• 27, 20, 33

Section 12.2

• 3, 8
• 19, 24
• 28, 31, 32, 33
• 38 (assume F=<F_1, F_2>, i.e. F is a 2-dimensional vector field).

Section 12.3

• 3, 9, 12, 15
• 26, 30

Section 12.4

• 15, 18
• 22, 25
• 31, 32, 33, 34, 35, 36
• 38, 41, 42, 43, 44

Section 12.5

• 3, 
• 5, 8, 15,
• 21

Section 12.6

• 19
• 28, 31
• 37, 41, 46

Section 12.7

• 1
• 9, 14
• 21 (can evaluate directly or use the Divergence Theorem)
  

available at http://www-math.cudenver.edu/~mkawai/2421FA06/HW/index.html (courtesy of Mike Kawai). Problems with parenthesis around them do not have complete solutions (but have hints as to how to approach them):

Section 8.1:  26, (39), 44, (45)

Section 8.2:  18a,b; 22, 28, 56

Section 8.3:  8, (13), (23), 30

Section 8.4:  6, (17), 28, (49), (55)

Section 8.5:  4, 12, (17), 24

Section 9.2:  14, 32

Section 9.3:  14, 34

Section 9.4:  6, 8

Section 10.1:  2

Section 10.2:  12, 56

Section 10.3:  4, 14

Section 10.7:  6, 12

Section 11.1:  26, 38, 52

Section 11.2:  14, 18, 26, 38

Section 11.3:  10, 20, 34, 42

Section 11.4:  16

Section 11.5:  6, 19, 31

Section 11.6:  26, 30

Section 11.7:  34, 41