The hyperboloid of two sheets from problem I. The curves on the surface are the MATH 241 -- Multivariable Calculus
Exam 1 Sample Questions
The hyperboloid of two sheets from problem I. The curves on the surface are the
slices by planes 
![implicitplot3d(`+`(`*`(`^`(x, 2)), `-`(`*`(`/`(1, 4), `*`(`^`(y, 2)))), `-`(`*`(`^`(z, 2))), `-`(1)), x = -3 .. 3, y = -3 .. 3, z = -3 .. 3, style = patchcontour, grid = [40, 40, 40], axes = normal); ...](videos/Ex1RevSol_3.gif){kind=small}
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The curve from III A:
![plot([`+`(`*`(4, `*`(cos(`*`(Pi, `*`(t)))))), `+`(`*`(3, `*`(sin(`*`(Pi, `*`(t)))))), t = 0 .. 1], scaling = constrained); 1](videos/Ex1RevSol_5.gif){kind=small}
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The cardioid from IV, together with the tangent line from part C:
![Card := plot([`*`(`+`(1, cos(t)), `*`(sin(t))), `+`(`-`(`*`(`+`(1, cos(t)), `*`(cos(t))))), t = 0 .. `+`(`*`(2, `*`(Pi)))]); -1](videos/Ex1RevSol_7.gif){kind=small}
![Tang := plot([`+`(`*`(`+`(1, sqrt(2)), `/`(1, 2)), `*`(`/`(1, 2), `*`(u, `*`(sqrt(2))))), `+`(`-`(`*`(`+`(1, sqrt(2)), `/`(1, 2))), `*`(u, `*`(`+`(1, `*`(`/`(1, 2), `*`(sqrt(2))))))), u = -1 .. 1]); -...](videos/Ex1RevSol_8.gif){kind=small}
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