MATH 2D Midterm: MATH2D Midterm 2018 Spring Practice Solutions
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MATH 2D Full Course Notes
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Midterm practice problems from class: for (cid:126)r(t) = (cid:104)3 cos t, 2 sin t(cid:105) nd (a) the unit tangent vector and the tangent line at t = 0, (cid:126)r = (cid:104) 3 sin t, 2 cos t(cid:105), so the (forward) unit tangent vector (at any time t) is (cid:104) 3 sin t, 2 cos t(cid:105) (cid:126)r(t) (cid:12)(cid:12)(cid:12) (cid:126)r(t) (cid:12)(cid:12)(cid:12) = and in particular. T(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)r (cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12) . From our expression for t in (a) above we nd d dt (cid:104)(cid:0)9 sin2 t + 4 cos2 t(cid:1) 1/2 (cid:104) 3 sin t, 2 cos t(cid:105)(cid:105) (cid:19) T (0) = (2) 1/2(cid:104) 3, 0(cid:105) = (cid:28) (cid:29) = (cid:104) 1, 0(cid:105) and, since we can see from our work in (a) that (cid:10) 3. (cid:126)r = (cid:12)(cid:12)(cid:12) (cid:126)r (cid:12)(cid:12)(cid:12) dt d (cid:12)(cid:12)(cid:12) (cid:126)r (cid:12)(cid:12)(cid:12)2. (cid:126)r(t) = (cid:104) 3 sin t, 2 cos t(cid:105), so.