Calculus 3 Sample Final Exam - Math 241 Spring, 2011

1.	Sketch several (two or three) level curves of the function f(x,y) = x^2y. Answer: family of curves are x^2y = Constant
2.	Let z = f(x,y), where x = t^2, y = t^3 . Find d^2z/dt^2 in terms of t and partial derivatives of f with respect to x and y. Answer: d^2z/dt^2 = 2zx + 6zy + 4t^2 zxx + 9t^4 zyy + 12t^3 zxy
3.	Find ∂z/∂x if z = z(x,y) is defined implicitly by xy^2 + 3z = e^z . Answer: ∂z/∂x = y / (xy^2 + 3z - 3)
4.	Find the absolute minimum and maximum of f(x,y) = x^2 - 4x + 5y^2 on the circle x^2 + y^2 = 1. Answers: min = -3, max = 6
5.	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. Answer: m = π/2 Click here to continue with this exam.

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