Exam-3-Additional-Review-Problems

Exam-3-Additional-Review-Problems - CALCULUS 4 QUIZ #3...

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Page 1 of 7 6x 9y 16 z 5 = 3 8 x3 + () 9 16 y1 1z 2 + 0 = n x xx 0 n y yy 0 + n z zz 0 + 0 = n i 3 8 j 9 16 + k 1 + = n i z x 3 1 , j z y 3 1 , + k 1 + = z y 3 1 , 3 2 82 = 9 16 = y z x 2 8z = x 2 y z 0 = Differentiate implicitely with respect to "y". z x 3 1 , 3 1 42 = 3 8 = x z xy 4z = 2x y x z 0 = Differentiate implicitely with respect to "x". (a.) Find an equation of the tangent plane and parametric equations for the normal line to the surface: x 2 y 2 7 = at the point 3 1 , 2 , . (1.) Determine the following about the tangent planes & normals to surfaces CALCULUS 4 QUIZ #3 REVIEW Part 1 / SPRING 09
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xt () 3 3 8 t + = yt 1 9 16 t = zt 2 t = z y x Surface, Tangent Plane, & Normal Line (b.) Find a point on the surface : z3 x 2 y 2 = at which the tangent plane is parallel to the plane: 6x 4y + z 5 = . n i 6 j 4 + k 1 + = Page 2 of 7
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z x x 0 y 0 , () 6x 0 = z y x 0 y 0 , 2 y 0 = 0 6 = 2 y 0 4 = x 0 y 0 , z 0 , 12 , 1 , = y z x Parallel Plane to Tangent Plane Determines Point Page 3 of 7
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(c.) Find an equation of the tangent plane to the ellipsoid x 2 y 2 + 3z 2 + 12 = at the point 22 , 1 , () ; find parametric equations of the normal line to the ellipsoid at that point; find the acute angle that the tangent plane at the point , 1 , makes with the xy-plane. wx y , z , x 2 y 2 + 4z 2 + 12 = Δ w i 2x j 2y + k 8z + = Δ w , 1 , i 4 j 4 + k 8 + = 4x 2 4y 2 + 8z 1 + 0 = 4x 4y + 8z + 24 0 = xy + 2z + 6 0 = xt 2t + = yt + = zt 12 t + = The xy-plane is z0 = . k i 1 j 1 + k 2 + 2 = k i 1 j 1 + k 2 + cos θ = cos θ 2 6 = 2 3 = θ cos 1 2 3 = 0.615 = θ 35.3degrees = Page 4 of 7
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z x y Tangent plane @ (2,2,1) & Normal Line (d.) Find 2 unit vectors that are normal to the surface sin x z () 4 cos y z 4 = at the point ππ , 1 , . wx y , z , sin x z 4 cos y z = Δ w i z cos x z j 4z sin y z + k x cos x z 4y sin x z + + = Δ w i 1 cos π j 41 sin π + k π cos π 4 π sin π + + = Page 5 of 7
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Page 6 of 7 1 2 3 , 2 3 , 1 2 3 , 2 3 , λ 3 2 = 3 2 , 3 2 2 1 () 2 + 1 2 λ 2 = z 1 λ = y 1 λ = x 3 2 1 λ = Δ w i 2x λ j 2y λ + k 2 z λ + = PQ i 3 j 2 + k 2 + = Δ w + k 2 z + = wx y , z , x 2 y 2 + z 2 = (e.) Find all points on the surface x 2 y 2 + z 2 1 = at which the normal line is parallel to the line thru P1 2 , 1 , and Q4 0 , 1 , .
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Exam-3-Additional-Review-Problems - CALCULUS 4 QUIZ #3...

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