Homework Due 3/7/2014 at 8:00am

Webwork section 11.4, section 11.5 and section 11.6.

webwork questions & more webwork questions

How would you use polar coordinates to solve this?


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Status:     Enrolled ('C')
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***** Data about the problem: *****

Problem ID:                   6
Value:                        1
Max attempts                  unlimited
Random seed:                  2513
Status:                       0.5
Attempted:                    yes
Last answer:
        AnSwEr0001: dne
        AnSwEr0002: 5
Number of correct attempts:   0
Number of incorrect attempts: 10
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A. lim(x,y)→(0,0)(x+12y)2x2+144y2= ___

B. lim(x,y)→(0,0)5x3+10y3x2+y2= ___

(Hint for B: use polar coordinates, that is x=rcos(θ),y=rsin(θ) )
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well,  I would substitute in the numerator and denominator the suggested coordinate transformation, and take advantage of the fact that the denominator takes an especially simple form, since x^2+y^2=r^2cos^2(θ)+r^2sin^2(θ) =r^2(cos^2(θ)+sin^2(θ))=r^2, and also of the fact that r->0 as (x,y)->(0,0).

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How do I find an integer for this problem?
Problem ID:                   6
Source file:                  Local/Dartmouth/setStewartCh15S3/problem_6.pg
Value:                        1
Max attempts                  unlimited
Random seed:                  3762
Status:                       0.75
Attempted:                    yes

Number of incorrect attempts: 12
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The gas law for a fixed mass of an ideal gas at absolute temperature T, pressure P, and volume V is PV= mRT, where R  is the gas constant. Find the partial derivatives

∂P∂V=____

∂V∂T=____


∂T∂P=____


∂P∂V∂V∂T∂T∂P================   =
=______ (an integer) 
∂P∂V∂V∂T∂T∂P=         
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Well, it looks like you correctly computed the individual partial derivatives, so all you need to do is to multiply them together, and then maybe to substitute the ideal gas law back into the product so you can cancel everything except an integer constant
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I got the answer for most parts, but don't understand why the lim along the
y-axis is 0. I am not sure how to get the lim along the line. I have
guessed multiple times, but am unsure what to do with the constant.

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Well, x=0 on the y-axis, so the value of the numerator is 0 while the value of the denominator is 5y^2; in other words the value of the function on the y-axis is 0 except at the origin, where it is undefined. Since the limit of zero is zero, we're done.

Homework Due February 28

Webwork section 11.2 and section 11.3

Java and Webwork

It seems that Java is required for images in webwork section problems 11.1 #'s13 & 15 to be properly displayed; therefore I'm not going to require or grade these problems.

Homework due February 21st

Webwork section 10.9 and section 11.1.

The boat question on the test.

A small boat is traveling at a speed of 5 mph in relative to the water. The boat is travelling in flowing water with a current velocity given by –3, –3 . In what direction should the boat head to travel due east? (Represent the direction with a unit vector.) 

10:04AM&10:16AM
Can you explain how to solve the boat question yet?


2:21PM
Ok I think I finally woke up. Is this problem really as simple as it seems? Is the answer really only the unit vector form of any vector that when added to <-2,-4> produces a vector of form <a,0> with a >0?

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Yep.  This is actually a hugely important problem,  faced every day by people navigating aircraft or boats.  The direction that the boat is facing, is called the heading--in your case it should have been the direction of the vector <3,4> (since this has length 5), which is give by the unit vector <3/5,4/5>. The track or course over ground or course bearing is the direction you are actually moving toward, in this case given by the unit vector <1,0>.  To compensate for the effect of an ocean current or wind it's often necessary to have a heading that is in a different direction than your intended track.  Here's a wiki article on the subject, I'll update this page if I find a better link. and here's a link to a navigation article.

practice exam answers

<https://sites.google.com/a/asu.edu/mat267-practice-exam/the-practice-test/TaylorMAT267_test1_fall_2011Answers.pdf?attredirects=0&d=1>

webwork question

(1 pt) 


How do I format an answer for this problem?
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Find the domain of the vector functions, r(t), listed below.
You may use "-INF" for −∞ and use "INF" for ∞ as necessary, and use "U" for a union symbol if a union of intervals is needed.
a) r(t)=⟨ln(7t),t+18‾‾‾‾‾‾√,118−t√⟩


b) r(t)=⟨t−8‾‾‾‾‾√,sin(3t),t2⟩



c) r(t)=⟨ e−8t,tt2−9√,t1/3⟩
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  AnSwEr0001: (-inf,inf)U(-inf,inf)U[0,inf)
        AnSwEr0002: (-inf,inf)
        AnSwEr0003:
Number of correct attempts:   0
Number of incorrect attempts: 18
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Your formatting is fine.  What seems to be missing here seems to be a basic lack of knowledge about the domains of some basic functions and/or maybe a lack of understanding what "the domain of a function" means.  In all cases the domain of a vector function is at most the intersection of the natural domains of the component functions.  The domain of a real valued function f(x) is the set of all values x for which f(x) is defined as a (finite!) real number.  Given a graph of the function in the xy-plane, it will be the values on the x-axis over which a vertical line intersects the graph of the function.  See this wiki page for the graph of the log function, this one for the graph of the square root.  For an example, the domain of ln(x) is (0, INF), the domain of √x is [0,INF), the domain of 1/√x  is (-INF, 18]; together the domain of r(t)=<ln(9-t),√t, 1/√(3+t)> will be the intersection of the intervals (-INF,9), [0,INF), (-3,INF), which gives [0,9).