Calculus
3 Final Exam - Math
241 Fall, 2007
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1. |
Find ∂z/∂x if z = z(x,y) is defined implicitly by xy2 + 3z = ez Hint: Differentiate the expression xy2 + 3z = ez implicitly. Answer: ∂z/∂x = y2 / (xy2 + 3z -3) |
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2. |
Find the absolute minimum and maximum of f(x,y) = x2 - 4x + 5y2 on the circle x2 + y2 = 1. Hint: Try using Lagrange Multipliers to find the location on the circle for extreme values of f(x,y). Answer: fmax = +6 at (-1/2,± √3 /2) fmin = -3 at (1,0) |
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3. |
Find the mass of a circular lamina with radius 1 and density at each point equal to the square of the distance from the point to the center. Hint: dM = r2 dA = r2 r dr dθ Answer: M = π / 2 |
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4. |
Sketch several (two or three) level curves of the function f(x,y) = x2y. Answer: Family of hyperbolas. |
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5. |
Let z = f(x,y), where x = t2, y = t3. Find d2z / dt2 in terms of t and partial derivatives of f with respect to x and y. Hint: d/dt = ∂/∂x dx/dt + ∂/∂y dy/dt Answer: d2z / dt2 = 2fx = 6t fy + 4t2 fxx + 12t3 fxy + 9t4 fyy |
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6. |
In the following integral, change the order of integration to dx dz dy. Give all necessary pictures. 3 9 – x2 3z ∫ ∫ ∫ f(x,y,z) dy dz dx -3 0 0 27 9 √(9-z) Answer: ∫ ∫ ∫ f(x,y,z) dx dz dy 0 y/3 -√(9-z) |
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7. |
Find the area of the portion of the plane x + 3y + 2z = 6 that lies in the first octant. Hint: One method is to find intercepts, find vectors, and use cross product. Hint: Second method is to find surface integral. (Easier) Answer: A = 3 √14 |
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8. |
Let F = ra r, where r = <x,y,z>, r = | r |. Find all real values of a for which div F = 0 Hint: Del Operator = ∂/∂x i + ∂/∂y j + ∂/∂z k and r = √ (x2 + y2 + z2 ) Answer: a = -3 |