Unformatted text preview: iii. If we project P to the yz plane, we get a parallelogram in the yz plane. What is the area of that parallelogram? 7. Suppose R is the parallelipiped spanned by u = h a; ; i ; v = h b 1 ; b 2 ; i ; w = h c 1 ; c 2 ; c 3 i . i. Give a formula for the surface area A in terms of a; b 1 ; b 2 ; c 1 ; c 2 ; c 3 . ii. Give a formula for the volume V in terms of a; b 1 ; b 2 ; c 1 ; c 2 ; c 3 . iii. If we wanted to nd the parallelipiped with maximal volume among all such parallelipipeds with the same surface area A , write (but don't solve) the Lagrange multipliers system in the variables a; b 1 ; b 2 ; c 1 ; c 2 ; c 3 that we'd need to solve. iv. By examining the equations in the Lagrange multipliers system, show that u ; v ; w must be mutu- ally orthogonal to maximise volume. ( Hint. Consider @ @b 1 V = @ @b 1 A .) 8. TRUE or FALSE. For each of the following statements, indicate whether it is true (T) or false (F). Support your answers. i. The graph of a solution of the di erential equation dy dt = cos y is a sinusoidal wave. ii. The graph of a solution of the di erential equation dy dt = sin t is a sinusoidal wave. iii. The curve r ( t ) = h 1 4 t 3 2 ; 4 t 3 ; cos( t 2) i is tangent to the surface x 3 + y 3 + z 3 xyz = 0 at the point (0 ; 1 ; 1)....
View Full Document
- Fall '07
- Calculus, Trigraph, Rectangle, Prism, Parallelepiped