1. We learned that when a force Fi is applied to triangle A, the x-component of the force that

ends up actually being applied to the pin P is

f ( , ) = F sin ( ) sin ( ) cos ( ) .

Assuming F is constant and using critical point analysis, with 2 nd derivative and/or

boundary tests as needed, find the points in the domain {( , ) | 0 < /2 }

that minimize f .

2. Note that C2 in the problem means C sub 2

Let f ( x, y ) = ( 1 x ^4 y^ 4 ) ^1/4 and let S be the surface given by the graph z = f ( x, y ) .

a) (1 pt) What is the domain of f ? What happens to the values of f as the point ( x, y )

gets near the edge of this domain?

b) (1 pt) Make a sketch of the surface S in 3-dimensional space, and also a (separate)

sketch of the contour plot of f .

c) (1 pt) Let C 2 denote the curve in xy-plane given by 2

cos ( t ) , 2 sin ( 2t . Let C <(23/4)cos(t),1/2sin(2t)>

denote the curve on the surface S which whose projection on to the xy-plane is C 2 .

Find the parametric equations r = r ( t ) for C.

d) (1 pt) Add to the sketches you gave in in part (b):

i) C in the graph of S and

ii) C 2 on the contour plot of f .

e) (2 pts) Let z ( t ) denote the z-component of parametric equations r = r ( t ) of C you

found in part (c). Find the points where z ( t ) has its local maxima and minima, and

add these in to the sketch in part (b).

f) (1 pt) Set up the function h ( t ) which gives the square of the distance from the origin

to a variable point on the curve C2 , and then find the local maxima and minima of

h ( t ) .