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Consider the surface, s, described by the equation z = x sin(x + y). (a) does (2, −2, 0) belong to this surface? if so, what is the equation of the tangent plane, t, to the surface at this point? (b) suppose r(t) = ha(t), b(t), c(t)i is a vector valued function whose associated space curve, c, is on the surface s. that is, we suppose c(t) = a(t) sin(a(t) + b(t)) for all t ∈ r. show that if r(0) = h2, −2, 0i, then r 0 (0) is parallel to t. (c) we now suppose r 0 (0) 6= 0. is the tangent line to c at r(0) on the plane t? can you infer from this a general fact about space curves on surfaces, tangent lines, and tangent planes?
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