SOLUTION KEYS TO THE HOMEWORK / SOLUTIONS
TO SIMILAR PROBLEMS
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.