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Wednesday, February 16th, 2011
3:30 am
[eidothea]
2 related rate problems I can't solve (cross-posted to math-help)
I have 2 related rate problems I can't solve. Any suggestions, hints, help are appreciated!

First problem: A highway patrol airplane flies 3 miles above a level, straight road at a constant rate of 120 mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 miles the line-of-sight distance is decreasing at a rate of 160 mph. Find the car's speed along the highway.

What I've done so far:

Here's how I've labeled the given information
Y=3 (vertical distance from the road to the plane)
dx/dt=120 mph (rate of change of the plane)
s=5 (distance from plane to car)
ds/dt=-160mph (rate of change from plane to car)

I need to find the rate of change (speed) of the car.

What I can't figure out: What's an equation I can use to relate these rates? Once I have that I should be able to differentiate with respect to time and plug in my known values to find the speed of the car. I just can't figure out how to relate these variables!

Second problem: A man 6 feet tall walks at a rate of 5 ft/sec toward a streetlight that is 16 feet above the ground. A ball is dropped from the same height from a point 30 feet away from the light. How fast is the ball's shadow moving along the ground 1/2 second later? (Assume the ball falls a distance s=16t^2 in t seconds.)


What I've done so far:

I've labeled the given information as follows:
y=50 (height of pole)
x(t) is the rate of change of the shadow, which is what I need to solve for.

Then I plugged t=1/2 into the equation s=16t^2 to get s(1/2)=4 feet. Then I subtracted 4 feet from 50 feet to get 46 feet, the height of the ball after 1/2 second.

I'm assuming that the related rate equation has something to do with similar triangles....like 50 is to ? as 46 is to ? but that's as far as I got.

Thanks in advance for the help!

Current Mood: hopeful
Tuesday, February 3rd, 2009
6:04 pm
[otakuchibijosh]
Systems of ODEs
I'm a graduate in chemical engineering taking a graduate ODE course. And, I'm having a little trouble with a couple homework problems. We're doing systems of first order ODEs which I've done previously in an undergrad ODE course a number of years ago, and slightly in a linear algebra course.

The first one I'm having trouble with is:

y' = [2 1; -1 1]y

I found the eigenvalues to be 1.5 +/- 0.866i
I don't deal with complex numbers very often, so I'm a little rusty. I know the two eigenvectors are complex conjugates, but I'm having a difficult time finding them. A little help?

The second one is x" + 3x' +2x = 0
The professor mentioned looking in the book to see how a higher order ODE can be converted to a series of first order ODEs, but the book only has like half a page on it, and I don't quite understand it.

Thanks.
Thursday, December 18th, 2008
6:05 pm
[ihatepeoplealot]
Calculus texts
Hey calculus,
I'm heading in to Calculus II next semester. We're going to use Stewart's "Calculus: Early Transcendentals" 6e, which is the same book I've used in high school and previously in Calculus I.
Do you all have any opinions about the book?

Would you recommend some companion text I can read to compliment my understanding? I like this text, but at the same time some concepts allude me. (Riemann series, washer method/cylindrical shell, etc)
Sometimes I think to myself, "I'd get this more if they included some proof" as I'm terrible at simply memorizing formulas and rules. But I realize that many of the concepts required to "fully" understand some things gone over in Calculus are too complicated for me to grasp.

I'd appreciate any commentary or musing about the above.
1:23 am
[neuroticloving]
I need some help :(
What is the derivative of R = pq with respect to p:  when p = f(q) and q = h(p) in other words p is a function of q and vice versa?  (This should be a product.)

Thank you soo much :)
Sunday, September 28th, 2008
3:22 pm
[heart_over_head]
How do you find the power series for 1 / ((1 - x^2)^2)?

THanks a lot.
Wednesday, April 16th, 2008
2:20 pm
[miami_shadows]
Finding Maclaurin series
Can anyone verify whether or not the correct Maclaurin series for the following function:


is the following?



Or am I completely off?
Thanks for any help!
Saturday, March 1st, 2008
6:58 pm
[soccer_gal]
General Formula
Hello,

I've been trying to find the general formula for the following problem but I can't seem to find anything in common.

Find a general formula for f^k (x) (the kth derivative of f) if f = 1 / (x^2-1)

f=1/[(x^2)-(1)]

So I found f ' and I got:
2x/ (x^2-1)^2

f '' :
2 (3x^2-1) / (x^2-1)^3

so I'm having difficulty with finding the general formula.
I know the denominator changes you'll have (x^2-1) always, but since the power is always increasing so...you'll have something like...
(x^2-1) ^k+1.
Friday, February 29th, 2008
11:05 pm
[calculus4all]
Calculus Help Blog
Hello! How do you do?
I've decided to begin a new blog called Conversational Calculus. I know how a lot of people can struggle with calculus and I think that I have gotten pretty good at explaining it. So I am working through the concepts preseanted to most students of calculus in the approximate order that they come and posting entires that explain the ideas in a thorough but casual manner, along with my subjective opinions on how important different ideas are and so on.
I'm going to try my best to post regularly. Just click on my name to go to my blog.
Friday, January 4th, 2008
6:48 pm
[lurel]
heeeelp~!
My calculus teacher assigned my class an extra credit project, but I don't know what to do... ;_; Any suggestions? something artsy that I can do by myself would be great... Just FYI, I'm horrible at calculus (and math in general)... so something easy would be nice....

Please help!!! I need the extra credit!!!

Thank you very much!
~Lurel
Tuesday, December 18th, 2007
11:30 pm
[rantipole6]
Rate of Change Problems and Speeding Tickets
I tutor high school kids and I've been compiling a list of ways that math can be useful in everyday life. One of the items on my list is "You can use calculus to fight a speeding ticket in court." I've heard professors making this claim and it makes sense given that you can use integrals in rate of change problems to determine the speed of an object using measurements from another speeding object. But truth to tell, I've never heard any actual cases of someone using this in court. So here are a few questions I have that maybe y'all can help me with:

1. Have their been any actual traffic court cases where calculus was used to dispute a machine reading on car speed and if so, what was the outcome?

2. If you wanted to calculate a reading on your car's speed when you were caught by the police, how would you go about determining the speed of the cop car when their machine reading was taken?
Thursday, December 13th, 2007
9:37 pm
[_teaspoon_]
I have no idea how to approach this problem!

If f(x)={ e^x, x<ln2
             2, x≥ln2
then lim f(x) as x→ln2 =?

Also: lim (x→infinity) of (5x²+7x-3)/(2+3x-11x²) = -5/11, right?

Sorry about the awful formatting and thanks for any help!
Wednesday, December 12th, 2007
8:32 pm
[gumlets]
Thanks so much to the people who helped me on my last FRQ: The FRQs are the only thing keeping my grade above a D (I failed two quizes, because I wasn't prepared Dx) I have another one to do, so if you could please help me again, I'd really appreciate it. I figured out most of it, I just need to be sure that its correct.

Thank you so so so much in advance!
Sunday, December 9th, 2007
3:11 pm
[pottergirl26]
I really need help!!!
Hi and I am new here and I'm in serious need of rescuing.

I have a not so great teacher this term for my multivariable class and now I'm completely lost. We're supposed to be on arithmetic and geometric means but I have no idea how to do anything since my teacher makes us go to the board all the time to figure out things on the spot. We've been given a take home test for which we're allowed to get any and all the help we want. I'm terrified of failing because I only have a vague idea of how to solve these problems.

1. There are many quadrilaterals ABCD with AB=10, BC=8, CD=11, DA=15. Show that if ABCD is the quadrilateral with the greatest area, then the points A, B, C, and D are concyclic. What is its area?

2. Find the largest value of constant k so that x^3+1_>_kx (that's greater or equal to)
a) solve using calculus
b)solve not using calculus

3. Find the maximum value of 4xy^2(10-x-y) over positive numbers x and y.
a)solve using calculus
b)solve not using calculus

4. The region R shown to the right is bounded above by a circular arc of radius 5 centered at the origin and below by a horizontal line segment. FInd the minimum and maximum value of y/(1+x^2^+y^2) over R.

a) solve using calculus
b) solve not using calculus

I will be eternally grateful to whoever can help since my grade is on the line. Please ~bows~
Thursday, November 29th, 2007
11:53 pm
[hammerforge]
Help!
I need a second pair of eyes. I have done something for my Calculus 3 class that is either clever or utter crap, and I am unsure of which. I am on the chapter for line integrals and have the following problem: A 160 lb man carries a 25 lb can of paint up a helical staircase that encircles a silo with a radius of 20 ft. Over the course of his climb, 9 lbs of paint leaks out? If the silo is 90 ft in height, and the man makes exactly 3 complete revolutions, how much work is done against gravity? From a physics perspective, this is quite simple. Gravity is a conservative field and so the only actual distance that matters is his height above the earth. And at 90', his height is not enough for the gravitational field to vary from its constant of 32'/s^2. Lacking any variable for time, we can express paint loss (and hence mass) as a function of height. !
  • height = 0g = gravity
  • M = Mass man
  • P = Mass paint
  • Delta P can be expressed as height/10. As integration of 1/10 will give us this value, delta P is mearly -1/10.
  • Work for a 160 lb man to carry a can of paint 90 ' vertically (532800 ft lbs/s^2),
  • Work for a 160 lb man to carry a 16 lb can of paint 90 ' vertically (506880 ftlbs/s^2),
This gives us the function of g(M+P-(1/10)32 dz to be integrated which will express all the values as a linier function of height. If this were my phsyics homework I would be done. Sadly this is not Physics but Calculus 3. I must express this as a Line Integral. For which I need to obtain i and j values that will go away when this problem is complete. 

Expressing the helix as a function of Height
  • 90' = 6z(pi) (for 3 full revolutions)
  • 2z(pi>/30) will give us a theta function
  • Radius = 20'
  • cos( 2z(pi>/30))i+sin(2z(pi>/30))j

Now that we have dummy values for i and j (but hopefully valid dummy values) we can attempt to express this as a line integral.

  • Integrate: F * T dz 0
  • Integrate <0i+0j+gk>*<20cos(2pi(z)/30)i+20sin(2pi(z)/30)j+(M+P-(1/10))k>
  • Integrate: g(M+P-(1/10))dz 0<z<90
  • Integrate 32(160+25-(1/10)dz 0<z<90

This gives a value of 532512 ftlbs/s^2
506880 ftlbs/s^2 < 532512 ftlbs/s^2 < 532800 ft lbs/s^2

While this is within the range that I would expect it to be, I cannot help but feel that this equation is a little bit too simple for a Calculus 3 line integral any opinions?

Monday, November 26th, 2007
9:32 pm
[gumlets]
I'm having trouble with this AP Calculus AB FRQ. I don't think my answer is right; does anyone know how to solve this / did I get it right? Thanks so much!

3 large images behind the cut.Collapse )
Monday, November 19th, 2007
12:04 am
[mudifier]
Unit Circle?
So, I have a calculus question on hand about double integrals and volume.  It's a practice problem without any solution, so I was wondering if people could give input.

"Compute the integral over the first quadrant of the unit circle centered at the origin of the function f(x,y) = 2xy"

How do you even set up this equation for the circle?

I'm completely lost.
Friday, September 28th, 2007
2:41 pm
[pouncewolf]
Question
This may be an irrelevant question, because I think it deals more with physics, but I am trying to figure out a formula where it shows how much force/pounds of pressure a car going at 60km/hr can produce.

Also how much applied force does it take to injure/break a bone?

Does anyone know or know where to point me in the right direction?


It would be much Appreciated!!

Current Mood: frustrated
3:01 am
[hammerforge]
Determining arc length of a vector.
So, I am in Calculus 3 right now and we are covering vector calculus. This week we are covering arc lengths.
While I understand the formula for arc length, I have a specific problem that I do not understand the answer which the book computed. It is not so much a question of the application of the formula as something the book did in its solutions manual without any obvious reason.
The problem is formated as:

r(t) = 21/2t i + etj + e-t k

r'(t) = 21/2 + et -e-t


|r'(t)| = ( (21/2)2 + (et)2 + (-e-t)2)(1/2)
This is where my answer diverges from the books...According to my understanding of math, 21/221/2=21/2. According to the book, the '2' simply disapears.

After arbitrarilly dropping the '2', the book continues on its merry way. with:

|r'(t)| = (e2t + e-2t)1/2

|r'(t)| = ((et + e-t)2)1/2

|r'(t)| = (et + e-t)

Integrate |r'(t)| ... and so on and so forth...

What I fail to understand is how and why the '2' is arbitrarilly dropped. In my understanding, the 2 should stay in and be integrated to a value of 21/2t . Resultantly my answer and the books answer differ by the value of 21/2.

Can anyone explain what happens to the 2?
Monday, September 17th, 2007
11:59 pm
[viccro]
I've got another question; I'm just starting vector cal, but I left my multivariable notes at home, so until I return to get them I'm struggling to remember how to do easy things like gradients.

Captain Ralph is in trouble near teh sunny side of Mercury. The temperature of the ship's hull when he is at location(x,y,z) will be given by T(x,y,z) = e^(-x^2-2y^2-3z^2)
He is currently at 1,1,1.

a) in what direction should he proceed in order to decrease the temp most rapidly? [For this one I took the gradient at 1,1,1 and said that moving in a direction of -grad(T) would satisfy...so in the direction(2e^-6, 4e^-6, 6e^-6)]

b) if the ship travels at e^8 m/s, how fast will the temperature decrease if he proceeds in that direction? [I feel that this should be a really simple division or something, but it's escaping me. all units in T(x,y,z) are in meters, though.]

c) unfortunately, the metal of the hull will crack if cooled at a rate greater than sqrt(14)e^2 deg/sec. Describe the set of possible directions in which he may proceed to bring the temp down at no more than that rate. [I have absolutely no clue how to do this. Help!!]

Thanks so much...you guys always explain things really well!

Current Mood: confused
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