The curve C in the x-y plane is given parametrically by ( x ( t ) , y ( t ) ) , where d x d t = t sin ( 2 t 3 ) and d y d t = cos ( t 2 4 ) for 1 ≤ t ≤ 6 . a. Find the slope of the line tangent to the curve C at the point where t = 3 . b. For 1 ≤ t ≤ 6 , what is the value of t at which the line tangent to the curve C is vertical? c. Find the length of the curve C for 1 ≤ t ≤ 6 .

The equation respenting the left table is

Determine whether the following statements are true and give an explanation or counterexample. bold a. when using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution. bold b. if a region is revolved about the y-axis, then the shell method must be used. bold c. if a region is revolved about the x-axis, then in principle it is possible to use the disk/washer method and integrate with respect to x or the shell method and integrate with respect to y.

Abby is preparing fruit punch for her party. she starts with a bowl filled with liquid that has 1/2 fruit juice and 1/2 sprite, but she decides it's too fruity. so she pours out 1/4 of the liquid, and then fills the bowl up again with sprite. what fraction of the punch is now sprite?