Review Before Next Class
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Recall what all of the following parametrically-defined curves look like.
\(\displaystyle \big\langle t^3+1, t \big\rangle\) for \(-1 \lt t \lt 1\)\(\displaystyle \big\langle \cos(t), \sin(t) \big\rangle\) for \(-\pi/3 \lt t \lt 3\)\(\displaystyle \big\langle \cos(t), \sin(2t) \big\rangle\) for \(0 \lt t \lt 2\pi\)\(\displaystyle \big\langle \mathrm{e}^t, \sin(t) \big\rangle\) for \(0 \lt t \lt \mathrm{e}\)
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Recall what all of the following surfaces look like:
\(\displaystyle x^2+y^2+z^2 = 49\)\(\displaystyle x^2+y^2+z = 49\)\(\displaystyle x^2+y^2-z = 49\)\(\displaystyle x^2+y^2-z = 0\)\(\displaystyle x^2-y^2-z = 0\)\(\displaystyle x^2-y^2-z^2 = 49\)\(\displaystyle x^2+y^2-z^2 = 49\)\(\displaystyle x^2+y+z = 49\)\(\displaystyle x+y+z = 49\)\(\displaystyle x+y = 49\)\(\displaystyle y = 49\)