Assignment #9

The video explains how animators at Pixar use the splitting and averaging of surfaces to make 3d animations used within their movies. They use Pascals triangle, which is used to create smooth curves and shapes for the animations. Pascals triangle does not work for all surfaces used in animation and other equations are required. They discuss how by splitting and averaging the shapes an  infinite amount of time, the two points used will infinitely continue to get closer until they reach a specific limit and come together at the shapes original midpoint.

Assignment #8

1)

A)     ʃsin u du = -cos u + C
B)     ʃcos u du = sin u +C
C)     ʃtan u du = -ln |cos u| + C
D)     ʃcot u du = ln |sin u| + C
E)     ʃsec u du = ln |sec u + tan u| +C
F)     ʃcsc u du = -ln |csc u + cot u| + C

2)
You have to set u equal to 2x. This is because it is within the function of U^1/2. du then equals 2dx, but it cant because of (4x+1)dx

Assignment #7

1) The general solution to (x^n dx) is {x^(n+1)]+C. The C is important because it states whether or not there was a constant present before the integral was taken.

2)
A.  sin(x)dx=-cos(x) S C -S -C  :going right to left      south carolina - south -carolina
B.  cos(x)dx=sin(x) S C -S -C  :going right to left        south carolina - south -carolina
C.  sec^2(x)dx=tan(x)     - it sounds easy to remember if you say it to yourself a few times quickly
D.  csc^2(x)dx=-cot(x)     - opposite of sec^2    all of the C's get negated
E.  sec(x)tan(x)dx=sec(x)   -  its a pattern secxtanxsecx
F.  csc(x)cot(x)dx=-cot(x)  - its a pattern except it gets negated cscxcotx -cscx