# w08 - line is horizontal or vertical b Verify that the...

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Math 152 Workshop 8 Spring 2011 1. If A is a positive number, the parametric curve ± x = t 3 - t y = A 1+ t 2 looks like: For one value of A , the “self-intersection” of the curve is perpendicular. The picture looks like: Find that value of A . 2. The parametric curve ± x = 3 sin t + 1 2 cos t y = - sin t + 3 2 cos t is a tilted ellipse whose graph is shown to the right. What are the dimensions and location of the box containing the ellipse? Note The sides of the box are vertical and horizontal and also are tangent to the ellipse. 3. Investigate the family of curves deﬁned by the parametric equations x = t 2 , y = t 3 - ct , where c is a positive constant. a) Graph the curves for c = 1 4 , c = 1 and c = 4. What features do all the curves have in common? (You may need to adjust the graphing window as you change c .) How does the shape change as c increases? Find the x - and y -coordinates of all points where the tangent

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Unformatted text preview: line is horizontal or vertical. b) Verify that the point ( c, 0) is on the curve for any c > 0. How many tangent lines does the curve have at the point ( c, 0)? What are their slopes? Check your answer numerically (for c = 1 4 , c = 1 and c = 4) by drawing the tangent lines on your graphing calculator. c) Consider the curve corresponding to c = 1 3 . Part of this curve is a loop. Find the length of that loop. 4. Find the Cartesian coordinates of all points of intersection of the curves with polar equations r 2 = 4 sin θ and r = 1-sin θ . Sketch these curves on the same coordinate axes. Warning (0 , θ 1 ) and (0 , θ 2 ) represent the same point always ; so do ( r, θ ) and (-r, θ ± π )....
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w08 - line is horizontal or vertical b Verify that the...

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