calculus

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!

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.

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.

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 :)

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!

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.

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.

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