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Calculus help - worded question. - May 25th 2010, 06:38 AM

I'm not sure of the answer, nor how to get it - I'm completely stumped. Can someone give me a hand with this?

Quote:
A particle is moving along a curve with the equation
y = x^2 + 1
Consider the family of straight lines passing through the origin
y = px
where p is a real constant.

i. Find a set of values of p for which the equation x^2 + 1 = px, when it has two distinct solutions.
ii. Explain the geometric significance of the result.
Bleh. This is one of my exam questions.


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Re: Calculus help - worded question. - May 25th 2010, 07:00 AM

Recall the quadratic formula for solving ax^2 + bx + c = 0:

x = [ -b +/- sqrt(b^2 - 4ac)] / 2a

The determinant, d, of the quadratic equation is b^2 - 4ac. If d < 0, then there are no real solution. If d = 0, there is one real solution. If d > 0, there are two real solutions.

Given the equation x^2 + 1 = px, you want to rearrange into standard quadratic form, then solve for the values of p that give you a positive determinant. Try it yourself now, and if you still get stuck come back and read the rest.

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|
|
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--

Rearranging we have: x^2 - px + 1 = 0. That makes our determinant d = p^2 - 4. Since we want a positive determinant, let d > 0. Now we have:

p^2 - 4 > 0
p^2 > 4
|p| > 2

So if p > 2 or if p < -2, then there are two solutions to x^2 + 1 = px. That means that if p belongs to this set, then the particle will pass through the line y = px twice as it travels along y = x^2 + 1. If p = 2 or p = -2, the particle will only touch the line once (meaning that the line is tangent to the particle's curve). If -2 < p < p, the particle will never touch the line.

Good luck on your exam.


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Re: Calculus help - worded question. - May 25th 2010, 07:09 AM

Lol. Why the hell do I overthink these things? Thanks man. Really appreciate it. (:

I guess its the way its worded. We're really only taught a bunch of equations and formulae and are expected to know how to apply it to something like this.

Thanks. (:


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Re: Calculus help - worded question. - May 25th 2010, 07:42 AM

Quote:
Originally Posted by Composure View Post
Lol. Why the hell do I overthink these things? Thanks man. Really appreciate it. (:

I guess its the way its worded. We're really only taught a bunch of equations and formulae and are expected to know how to apply it to something like this.

Thanks. (:
Welcome.

I admit I hesitated a little once I'd worked out my answer. It's the right answer from what I understand of the question, but it doesn't seem like something I'd expect from a calculus class (my experience with high-school calculus being lots and lots of derivatives and little else).

Usually the way I attack word problems is to make one list of all the information I'm given in the question, then a second list of everything I need to figure out for the solution, then figure out how to get from one to the other. Using your question as an example, we had two curves - one fixed and one with a parameter - and we needed to find all values of the parameter for which the curves would intersect twice. Finding intersections of curves just requires setting the two equations equal and solving; and since the resultant equation was quadratic, finding solutions just needed the quadratic equation.


The atoms that make up you and me were born in the hearts of suns many times greater than ours, and in time our atoms will once again reside amongst the stars. Life is but an idle dalliance of the cosmos, frail, and soon forgotten. We have been set adrift in an ocean whose tides we are only beginning to comprehend and with that maturity has come the realization that we are, at least for now, alone. In that loneliness, it falls to us to shine as brightly as the stars from which we came.
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Re: Calculus help - worded question. - May 25th 2010, 07:53 AM

Quote:
Originally Posted by Xujhan View Post
Welcome.

I admit I hesitated a little once I'd worked out my answer. It's the right answer from what I understand of the question, but it doesn't seem like something I'd expect from a calculus class (my experience with high-school calculus being lots and lots of derivatives and little else).

Usually the way I attack word problems is to make one list of all the information I'm given in the question, then a second list of everything I need to figure out for the solution, then figure out how to get from one to the other. Using your question as an example, we had two curves - one fixed and one with a parameter - and we needed to find all values of the parameter for which the curves would intersect twice. Finding intersections of curves just requires setting the two equations equal and solving; and since the resultant equation was quadratic, finding solutions just needed the quadratic equation.
Well, yes. That's the matter with this exam... It's all worded questions, mostly solved simply with quadratic formulae and derivative functions, but its not what I expected from this class either. My whole class is having trouble with it, including two people who scored the top 0.75% of the state in a higher math class than the one I'm in now - with a question like that, its hard to break down and figure out which formulae to use, even more so when its using formulae we learned last year.

You'd expect questions which require the product rule and the division rule (f'(x) = (u(x)*v'(x) - v(x)*u'(x)) / v(x)) or something similar... not the quotient rule. S:


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