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pract_final - MATH 32A/3 Practice Final Exam Dr Frederick...

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MATH 32A/3: Practice Final Exam Dr. Frederick Park Note: This Exam is Slightly Longer than the Actual Final 1. Find the equation of the tangent plane and the normal line to the surface given by xy + yz + zx = 3 at the point (1 , 1 , 1). 2. Find the points on the sphere x 2 + y 2 + z 2 = 1 where the tangent line is parallel to the plane 2 x + y - 3 z = 2. 3. Use the chain rule to find ∂w/∂t where w = x + ( y 2 /z ), y = t 3 +4 t , x = e 2 t , and z = t 2 - 4. 4. Use a tree diagram to write out the chain rule for the case where w = f ( t, u, v ), t = t ( p, q, r, s ), u = u ( p, q, r, s ), and v = v ( p, q, r, s ) are all differentiable functions. 5. The length x of a side of a triangle is increasing at a rate of 3 in/s, the length y of another side is decreasing at a rate of 2 in/s, and the contained angle θ is increasing at a rate of 0.05 radians/s. How fast is the area of the triangle changing when x = 40 in, y = 50 in, and θ = π/ 6? 6. If yz 4 + x 2 z 3 = e xyz , find ∂z/∂x and ∂z/∂y . 7. Directional Derivative. Let f = f ( x, y, z ) be a differentiable function: (a) When is the directional derivative of f a maximum?
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