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Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems,

(a) Graph the curve with equation

$ y(y^2 - 1)( y -2) = x(x -1)(x -2) $

At how many points does this curve have horizontal tangents? Estimate the $ x- $ coordinates of these points.

(b) Find equations of the tangent lines at the points (0, 1) and (0, 2).

(c) Find the exact r-coordinates of the points in part (a).

(d) Create even more fanciful curves by modifying the equation in part (a).

(a) $$x \approx 0.42265$$

(b) $$y=-x+1 \text { and } y=\frac{1}{3} x+2$$

(c) $$x=1 \pm \frac{1}{3} \sqrt{3}$$

(d)

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Missouri State University

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University of Nottingham

so for this probably expanded sign should be easy to take. Impressive derivative. Then computers exploding off the point. Where's a horizontal tension? The tension lines horizontal, which means that the relative zero So that's something which is have the sob numerator. Zero. So three X Square minus six x plus cheery ho zero and we probably just fact arises. Actually, that's that would probably have to use a former so ex evos one over six six prosper minus square of sixties thirty six miners twenty four twelve. So that's one prosper on minus. Um, so that's one over three square root of three. So those are the point where you have horizontal tension line and for from the point zero one actually, I'm just answering posse People's Party. They asked you to ask me which I will leave you to you. Ah, for the to use a graphing calculator and the party are also leave it to you for part B. Ah, we just have to figure out a point and a slope you're planning an exit was zero. Wife wants one. We have slow because Shiyu over negative two, which is never if one. So the equation should be why minus one equals negative X. And for zero two, the slope, It's reply it. Actually, it was. You know why I was two, Saul. Numerous ways too. Denominator is, ah, eight times for his thirty two minors. Twenty five twenty four. So is eight minus four is four plus two is six. So Slope is one of our history. So the equation should be why minus two horse won over three X and and those are the equation of tension alive.