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Unformatted text preview: Math 53, Review for the Final Exam GSI: Ivan Mati´c 1. Find the area enclosed by the curve given parametrically by x = 1 2 sin 2 t , y = sin t , ≤ t ≤ π . 2. Check that the Divergence Theorem is true by showing that both sides are equal for the vector field F ( x, y, z ) = ( x, y, z ) over the sphere x 2 + y 2 + z 2 = a 2 . 3. Let f : R 2 → R be a differentiable function. Then at ( x , y ) , ∇ f is normal to the curve f ( x, y ) = f ( x , y ) . If this statement is true, prove it; if it is false, give a counterexample. 4. (a) Under what conditions is a differentiable vector field F = P i + Q j + R k conservative? (b) Is the planar vector field (− y,x ) x 2 + y 2 conservative? Explain your answer! 5. Evaluate the volume of the region bounded by √ x + √ y + √ z = 1 and the coordinate planes. 6. Find the area under the cycloid, x = t − sin t , y = 1 − cos t , and above the xaxis, for ≤ t ≤ 2 π . 7. If a string wound around the unit circle in the xyplane is unwound, its end traces an involute of the circle. (a) Find the parametric equations for the involute when the end of the string starts at (1 , 0) . (b) Find the slope of the involute at (1 , 0) . (c) Find the length of the involute after the string has been unwound one full revolution. 8. Find the volume of the ice cream cone cut from the unit ball in R 3 by the cone φ = π/ 3 (in spherical coordinates). 9. Let f : R 2 → R be a function with plenty of continuous derivatives. Suppose the partial derivatives, f x , f y at (1 , 2) satisfy f x (1 , 2) = a and f y (1 , 2) = b . (a) What is the directional derivative of f at (1 , 2) in the direction of the vector ( 1 , − 1 ) ? (b) Prove that the directional derivative of f at (1 , 2) is maximized in the direction of the gradient....
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 Fall '08
 Lanzara
 Physics, Derivative, Cos, dt, sin t cos, Ivan Mati´ c

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