Math1011_Week09_Prelim2A

Math1011_Week09_Prelim2A - Math011 Spring 2002 Prelim II-A1...

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Math011 Spring 2002 Prelim II-A1 A point sweeps out a curve given parametrically by x = sin² θ , y = sin θ , 0 θ 5 π /2 Sketch this curve, label the initial and final positions of the point and precisely describe the motion of the point throughout this period of time. Eliminate the parameter to give a Cartesian equation in x and y whose graph contains the parametric curve.
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Math011 Spring 2002 Prelim A1 A point sweeps out a curve given parametrically by x = sin² θ , y = sin θ , 0 θ 5 π /2 Sketch this curve, label the initial and final positions of the point and precisely describe the motion of the point throughout this period of time. If possible, eliminate the parameter first, which will indicate the shape of the curve. 2 22 Replace sin with y in the x = sin equation. x = sin ( ) so Cartesian equation is x = y y θθ θ = 2 t x y 0 0 0 π /2 1 1 π 0 0 3 π /2 1 -1 2 π 0 0 5 π /2 1 1 Initial Point (0,0) Final Point (1,1) The function starts at (0,0) and moves clockwise up to (1,1), then it retraces its steps counterclockwise back to (0,0) and continues counterclockwise to (1,-1). Lastly, it reverses itself going clockwise all the way to (1,1) Eliminate the parameter to give a Cartesian equation in x and y whose graph contains the parametric curve. 2 As above, x = y (Doc #011p.17.02)
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Math011 Fall 2002 Prelim A2 A curve is given parametrically by the following equations: 33 ( ) 4cos ( ) 4sin 0 2 xt t yt π ==≤ dy dx dt dt Find formulas for and dy 4 dx Find at t= dy dx Find the (x, y) coordinates for each point on the curve where does not exist. 2 2 dy dx Find
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Math011 Fall 2002 Prelim A2 A curve is given parametrically by the following equations: 33 ( ) 4cos ( ) 4sin 0 2 xt t yt π ==≤ dy dx dt dt Find formulas for and 32 2 Remember the Chain Rule ( ) '( ) 4(3cos ) (cos ) 12cos sin ( ) '( ) 4(3sin ) (sin ) 12sin cos d dt x t y t =⇒ = = −⋅ = = 2 dy 4 dx Find at t= 2 2 / cos sin cos / sin 44 tan () t a n 1 dy tt dx ππ == = = =− dy dx Find the (x, y) coordinates for each point on the curve where does not exist. sin cos 3 22 2 3 2 -tan - does not exist when the denominator is 0, cos 0 when and :4 c o s ( ) 0 , y = 4 s i n ( ) 4 c o s ( ) 0 , y = 4 s i n ( ) 4 does not exist at (0, 4) a ttt tx = = = = = nd (0, -4) 2 2 dx Find 2 2 23 2 4 '/ (t a n) / sec 1 / dx (4cos ) sin sin = = (Doc #011p.17.03)
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Math011 Spring 2004 Prelim A3 Functions f(t) and g(t) are graphed below. The lines L and M are tangent to the graphs of f(t) and g(t) at t = 1. The line L has slope 1.6, and the line M has slope -0.96. Sketch the parametric curve described by the equations x = f(t), y = g(t). Find the equation of the line tangent to this curve at t = 1.
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Math011 Spring 2004 Prelim A3 Functions f(t) and g(t) are graphed below. The lines L and M are tangent to the graphs of f(t) and g(t) at t = 1. The line L has slope 1.6, and the line M has slope -0.96. Sketch the parametric curve described by the equations x = f(t), y = g(t).
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This note was uploaded on 11/22/2009 for the course CHEM 2070 at Cornell.

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Math1011_Week09_Prelim2A - Math011 Spring 2002 Prelim II-A1...

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