review1 - Math 53 - Midterm 1 Review GSI: Santiago Canez 1....

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Unformatted text preview: Math 53 - Midterm 1 Review GSI: Santiago Canez 1. Eliminate the parameter to find a cartesian equation of the curve given by the parametric equations x = 2 sinh t and y = 3 cosh t. 2. Find parametric equations for the curve of intersection of the surfaces y = z 2 − x2 and z = 3 − x2 . Are there any points on the curve with zero velocity? 3. Sketch the curve given by the polar equation r = 1 + 2 cos θ and find the area of the region inside the outer loop but outside the inner loop. 4. Suppose that |a| = 4, |b| = 2, and a · b = 5. Compute the area of the parallelogram spanned by a and b. 5. Find the equation of the plane containing the points (1, 0, −2), (1, 1, 1), and (−2, 3, 1). 6. State the relations between rectangular and cylindrical coordinates, and between rectangular and spherical coordinates. 7. Identify each of the following surfaces: (a) θ = π/4, (b) φ = π/4, (c) x2 + 2x − y 2 + 4y + z 2 = 0. √ √ 8. Show that the curve x = 6 cos t − 1, y = 5, z = 6 sin t lies on the surface x2 + 2x − y 2 + 4y + z 2 = 0. 9. Let f (x, y) = 1 − x2 + y 2 . Show that f satisfies the partial differential equation yux − xuy = 0. Draw a contour map of f and describe its graph. 10. Find the tangent plane to the graph of f (x, y) = e−x (y − sin(x − y)) at (0, 0). Use this to approximate the value of f (.01, −.01). 11. Compute the following limit, or show it doesn’t exist. lim (x,y)→(0,0) 2 x+y x2 + y 2 Date: February 20, 2006 1 ...
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