Let r(t) = (t^2,sin t - tcost,cost + tsint), t > 0. Find the unit tangent and un
ID: 2859283 • Letter: L
Question
Let r(t) = (t^2,sin t - tcost,cost + tsint), t > 0. Find the unit tangent and unit normal vectors T(t) and N(t). Find the curvature. 10. Locate and classify all critical points of f(x, y) = y^3 + 3x^2y - 6x^2 - 6y^2 + 2. 11. Let P(x,y,z) = xe^x+2z - z^2sin y. a) Compute P(x, y, z) and P(4,0, -2). b) Find the rate of change of P at the point (4,0, -2) in the direction given by the vector v = -2i + 2j + k. 12. Find an equation for the tangent plane to the surface z = 2xy- 3x^2y^3 +y^2 at the point (2,-1, 9). Write your answer in the form Ax + By + Cz + D = 0. 13. Using the method of Lagrange Multipliers, find the absolute minimum values of f(x,y) = x^2y on the circle x^2 +y^2 = 3.Explanation / Answer
9)r(t)=<t2,sint-tcost,cost +tsint>
differentiate with respect to t
r'(t)=<2t,cost+(-cost+tsint),-sint +(sint+tcost)>
r'(t)=<2t,tsint,tcost>
|r '(t)|=[(2t)2+(tsint)2+(tcost)2]
|r '(t)|=[4(t)2+(t)2(sin2t+cos2t)]
|r '(t)|=[4(t)2+(t)2]
|r '(t)|=5 t
unit tangent vector =r'(t)/|r'(t)|
T(t)=<2t,tsint,tcost>/(5 t )
T(t)=<(2/5),(1/5)sint,(1/5)cost>
differentiate with respect to t
T'(t)=<0,(1/5)cost,-(1/5)sint>
|T'(t)|=[02+((1/5)cost)2+(-(1/5)sint)2>
|T'(t)|=(1/5)
unit normal N(t)=T'(t)/|T'(t)|
N(t)=<0,cost,-sint>
curvature k =|T'(t)|/|r'(t)|
k=(1/5)/(5 t )
k=1/(5 t )
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