11-30-2009, 06:29 PM | #1 |
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[High School AP Calc] a few problems
yo here's a few take home questions I got. All help is appreciated. :]
1. An object moves along the x-axis with initial position x(0)=2. The velocity of the object at time t >= 0 is given by v(t)=sin((pi/3)t). What is the acceleration of the object at time t=4? What is the position of the object at time t=4? What is the total distance traveled by the object over the time interval 0 <= t <= 4? I got: a(4) = -pi/6 x(4) = (4pi + 9)/(2pi) total distance = 15/(2pi) 2. Let f(x)= x^3 + px^2 + qx. Find the values of p and q so that f(-1)=-8 and f'(-1)=12. Find the value of p so that the graph of f changes concavity at x=2. Under what conditions of p and q will the graph of f be increasing everywhere? for the first part I got p = -2 and q = 5 for the second part I got p = -6 for the last part I said whenever p is greater than or equal to 0 and q is greater than 0 3. Let f(x)= 4 - x^2. For 0 <= w <= 2, let A(w) be the area of the triangle formed by the coordinate axes and the line tangent to the graph of f at the point (w, 4 - w^2). Find A(1). For what value of w is A(w) a minimum? I know what's going on here, but I'm having trouble coming up with the equation for A(w). 4. Consider the curve given by the equation y^3 - 3xy = 2. Find dy/dx. Write an equation for the line normal to the curve at the point (1,2). What is the concavity of the curve at that point? edit: whoops I messed up, the dy/dx value I got is y/(y^2 - x) and the normal line is y - 2 = (-3/2)(x - 1) don't know how to determine the concavity there. would it be from the second derivative?
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yeaorwgh. Last edited by insanefreddy926; 12-1-2009 at 04:20 PM.. Reason: huh? |
10-21-2010, 11:43 AM | #2 |
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Re: [High School AP Calc] a few problems
For part 4, differentiate it again implicitly. If you differentiate it successfully, you will be able to replace your dy/dx's with the value of the derivative you got for your normal. Simplify, and you will get a positive or negative result, which will tell you of the curve's concavity at that point.
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10-21-2010, 01:01 PM | #3 |
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Re: [High School AP Calc] a few problems
A(w) is, as the problem says, formed by the axes and the tangent line to the parabola for 0<=w<=2 (these are your bounds because they keep things bound between the top of the parabola and the x axis).
Tangent line to f(x) is defined by the derivative of f(x), or f'(x) = -2x, which tells you the slope of the tangent at a given point. General equation for tangent line here: y = f(w) + f'(w)(x-w) Or, here, y = (4-w^2) - 2w(x-w) x intercept (when y=0, plug into tangent equation above). This acts as our triangle's base. x = (4+w^2)/2w y intercept (when x=0, plug into tangent equation above). This acts as our triangle's height. y = 4+w^2 The triangle's area here for A(1) is therefore 1/2*(b)(h) or, for w=1, (1/2)*(5/2)*(5) = 6.25. Regarding the minimum triangle area, we know that A(w) = ((4+w^2)/2w)*(4+w^2))/2 = (4+w^2)^2/(4w) So we need to find A'(w)=0, which means finding the derivative of A(w) -- which equals: (4w(2(4+w^2)*2w) - 4*(4+w^2)^2)/(16w^2) = 0 via quotient rule Simplified: (w^2+4)*(3w^2-4) = 0 or 3w^4+8w^2-16=0 Which holds only when w = positive or negative (2/rad3) Since w must be positive here, the triangle's area is minimized when w = 2/rad3
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10-22-2010, 07:08 AM | #4 |
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Re: [High School AP Calc] a few problems
You were supposed to give a lead towards the answer, rather than completely solving it. Ah well.
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10-22-2010, 08:09 AM | #5 |
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Re: [High School AP Calc] a few problems
<--- does not give a ****
if the OP doesn't know how to solve it, then he's screwed anyway if it's on a test i personally feel having the full answer is the best way to learn something. You can see all the steps and how things are applied.
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10-22-2010, 11:19 AM | #6 | |
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Re: [High School AP Calc] a few problems
As long as Rubix is doing math homework for people I have some crazy Actuarial P-Exam questions I don't get...
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10-22-2010, 12:53 PM | #7 |
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Re: [High School AP Calc] a few problems
Post away and I shall enlighten
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10-22-2010, 12:58 PM | #8 |
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Re: [High School AP Calc] a few problems
I just realized the OP's post was like a year ago
lmfao
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10-22-2010, 01:00 PM | #9 | |
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Re: [High School AP Calc] a few problems
Nah I can figure them out and I have a solutions manual anyway. I'm just freaking out about my midterm on Tuesday.
EDIT: yeah I think you might be a bit too late on that one.
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