SOLUTION KEYS TO THE HOMEWORK / SOLUTIONS
TO SIMILAR PROBLEMS
Click here for
solutions to all homework problems up to and including the last HW handed back.
THE SOLUTIONS BELOW ARE FOR THE 2010
VERSION OF THE COURSE, WHERE I USED A DIFFERENT TEXTBOOK AND ARE THUS OF SOME
BUT NOT PERFECT USE.
Click here for
a special set of problems and solutions to finding bounds for regions of
integration
NOTE: starting with HW#12,
I'm TeXing up the solutions to similar problems and putting all of these in one,
big document:
click here.
You can also use the
Cramster site (login:
mathephs AT gmail.com, password is 11235813) as an aid; do not just copy!
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[Solution key]
HW #1: Due Monday, February 8: Section 1.1: #4, #7:
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[Solution key] HW #2: Due
Friday, February 12: Section 1.2: #1, #7, #19: A common mistake in
#21 was people getting to the point where 8s = s and then claiming the lines
cannot intersect, forgetting that this has the solution s = 0.
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Several
people have also asked questions about lines in three dimensional space. I
have written a short note with an alternative explanation, showing how our
definition in three dimensions is a natural generalization of the definition
in the plane. Essentially the idea is that we regard the slope of a line in
a plane not as a number but as a vector. Thus a line with slope of 5 is the
same as a line going in the direction (1,5); a line with a slope of -2 is
the same as one going in the direction (1,-2).
Click here for more details.
- [Solution key] HW #3:
Due Monday, February 15: Section 1.3: #2c, #4, #6, #15a.
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[Solution
key] HW #4: Due Wednesday, February 17: Section 2.1: #1 (just find
the level sets, no need to graph the function), #24. (click
here for a postscript version)
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[Solution
key] HW #5: Due Monday, February 22: Section 2.2: #4.
Additional Problem #1: Let f(x) = x2 + 8x + 16 and g(x) = x2+2x-8.
Compute the limits as x goes to 0, 3 and
∞ of f(x)+g(x), f(x)g(x) and f(x)hwsolns\Math105_hw17solns.PDF/g(x).
Additional Problem #2: Compute the derivative of cos(sin(3x2 + 2x
ln x)). Note that if you can do this derivative correctly, you should be fine
for the course.
- [Solution key] HW #6:
Due Wednesday, February 24: Section 2.2: #5, #8ab, #17.
Note that for these problems you may assume that the exponential, sine and
cosine functions are continuous. Section 2.3: #1ad (hint: if you know
δf/δx by symmetry you know δf/δy).
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[Solution key] [Notes
on solutions]
HW #7: Due Friday, February 26: Section 2.3: #2b, #4ab, #5, #7c
(instead of giving the matrix of partial derivatives, just give the partial
derivatives with respect to x and y of the two coordinate functions f1(x,y,z)
= x + ez + y and f2(x,y,z) = yx2).
Click here for another solution key
to these problems.
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[Solution key] [Solutions
to similar problems]
HW #8:
Due Wednesday, March 3: Section 2.3: 12a, #13a. Section 2.4: #1,
#15.
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[Solution key] [Solutions
to similar problems] HW #9: Due Friday, March 5: Section 2.5: #2g,
#4 (by verify it means use the chain rule as well as substitute for u(x,y) and
v(x,y) and then take the derivative considering it as a function of x and y),
#7, #12 (there are two ways to do this problem).
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[Solution key] [Solutions
to similar problems] HW #10:
Due Monday, March 8: Section 2.5: #4 (now do it using the chain rule),
#5a (do not do #10), #13a or #13b (but not both), #20.
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[Solution key] [Solutions
to similar problems] HW #11: Due Friday, March 12: Section 2.6:
#2ab, #4a, #6a. DO ONE OF #16 (the famous Captain Ralph problem) OR #18.
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[Solution key] [Solutions
to similar problems] HW #12: Due Monday, March 15:
Do ONE OF Page 176: #23 (homogenous functions) OR #47 (ideal gas
law from Chemistry / Physics). Also do: Section 3.1: #1, #8a,
#11.
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[Solution key] [Solutions
to similar problems] HW #13: Due Wednesday, March 17: Section 3.2: #2, #3.
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[Solution key] [Solutions
to similar problems]] HW #14: Due Friday, March 19: Section 3.3: #7 (just find the
critical points), #13 (just find the critical point), #22 (hint: minimize the
square of the distance).
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[Solution key] [Solutions
to similar problems]] HW #15: Section 3.4: #2, #10. Review Exercises
from Chapter 2: Page 176: #21. Review Exercises from Chapter 1:
Page 91: #28.
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[Solution key] [Solutions
to similar problems]] HW #16: no problems assigned - enjoy spring break!
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[Solution key] [Solutions
to similar problems]] HW #17: From my notes
on the Method of Least Squares: Exercise 3.3, Exercise 3.9. Review
Exercises from Chapter 2: Page 177: #33b. Review Exercises from Chapter
1: Page 93: #42.
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[Solution key] [Solutions
to similar problems]] HW #18: Section 5.1: #1ac. Section
5.2: #1b. Review Exercises from Chapter 2: Page 174: #7e.
Review Exercises from Chapter 1: Page 90: Do either 18a or 18b.
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[Solution key] [Solutions
to similar problems]] HW #19:
Due Monday, April 19: Section 5.3: #2bd, #8, and do exactly one of #9
and #15.
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[Solution key] [Solutions
to similar problems]] HW #20:
Due Wednesday, April 21: Assume the probability that X equals x
is 2 exp(-2x) if x >= 0 and 0 otherwise, and the probability that Y equals y
is 3 exp(-3x) if y >= 0 and 0 otherwise. Show both of these densities are, in
fact, probability distributions (this means showing they are non-negative and
integrate to 1), and calculate the probability that X >= Y. Note exp(u) = eu.
Also do: Page 257, #15 (hint: minimize distance squared instead of distance;
this makes the algebra simpler).
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[Solution key] [Solutions
to similar problems]] HW #21:
Due Friday, April 23: Section 5.4: #1bc, #4 (hint: see
equation (6) on page 353). Also do: Page 364, #14.
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[Solution key] [Solutions
to similar problems]] HW #22:
Due Wednesday, April 28: Page 192, #20abf. Page 366: #14. Page 367: #25.
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[Solution key] [Solutions
to similar problems]] HW #23:
Due
Friday, April 30: Consider the surface (x/a)^2 + (y/b)^2 <= 1. Find a change
of variables to map this to a nice region, and then use that to find the area
of the ellipse.
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[Solution key] [Solutions
to similar problems]] HW #24: Due Friday, April 30: Section 6.2:
#1, #13.
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[Solution key] [Solutions
to similar problems]] HW #25: Due Wednesday, May 5: Section 6.2:
#21. Hint: to integrate ρ^2 / sqrt(2 + ρ^2) we can use a table
of integrals, Mathematica's Integrator, or write it as (ρ/sqrt(2+ρ^2)) * ρ and
integrate by parts (and then use tables!), or write it as (2+ρ^2)/sqrt(2+ρ^2)
- 2/sqrt(2+ρ^2).
From multivariable
calculus (Cain and Herod): Page 10-3: #5, #6, #7. Page 10.6: #8.
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[Solution key] [Solutions
to similar problems]] HW #26: NO HW DUE, though I strongly urge you to
start the problems due Monday.
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[Solution key] [Solutions
to similar problems]] HW #27: Due Monday, May 10:
From multivariable
calculus (Cain and Herod): Page 10-7: #13. Page 10-8: #14, #15, #16. Page 10-8: #17.
Page 10-10: #18, #19.
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[Solution key] [Solutions
to similar problems]] HW #28: Due Friday, May 14: Section 4.2: #1 (see
formula at the bottom of the page for help). Section 4.4: #1, #14. Section
7.1: #3b. Section 7.2: #1c. Section 8.1: #3a.
NOTE: starting with HW#12,
I'm TeXing up the solutions to similar problems and putting all of these in one,
big document:
click here.