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Unformatted text preview: Math 53, Review for the Final Exam GSI: Ivan Matic 1. Find the area enclosed by the curve given parametrically by x = 1 2 sin 2 t , y = sin t , t . 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 x-axis, for t 2 . 7. If a string wound around the unit circle in the xy-plane 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 = / 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