Cal3samplemid2

# Cal3samplemid2 - y 2 and the plane z = 1 y(b Find a...

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CALCULUS III SAMPLE MIDTERM 2 Problem 1. (20 points) (a) Find the indicated power using De Moivre’s Theorem: (2 3 + 2 i ) 5 ; (1 - i ) 8 (b) Find the indicated roots and sketch the roots in the complex plane: The ﬁfth roots of 32; The cube roots of 1 + i Problem 2. (20 points) (a) Solve the initial-value problem 2 y 00 + 5 y 0 + 3 y = 0 , y (0) = 3 , y 0 (0) = - 4 (b) Solve the boundary-value problem y 00 + 4 y 0 + 13 y = 0 , y (0) = 2 , y ( π/ 2) = 1 Problem 3. (20 points) (a) Find a vector function that represents the curve of intersection of two surfaces: the cone z = p x 2
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Unformatted text preview: + y 2 and the plane z = 1 + y . (b) Find a parametric equation for the tangent line to the curve at the speciﬁed point. x = e-t cos t, y = e-t sin t, z = e-t ; (1 , , 1) Problem 4. (20 points) Find the tangential and normal components of the acceleration vector: ~ r ( t ) = t ~ i + cos 2 t ~ j + sin 2 t ~ k at time t = π/ 8. Problem 5. (20 points) If ~ r ( t ) 6 = ~ 0, show that d dt | ~ r ( t ) | = 1 | ~ r ( t ) | ~ r ( t ) · ~ r ( t ). 1...
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