* 2013WT1
** Remarks: Many problems have a small box for the final answer.
** (1) 
*** (a) [12.4,12.5]
**** (i) convert parametric line equation to symmetric equations
**** (ii) find angle between a line and a plane
*** (b) 
**** (i) find the equation of the tangent plane to surface f(x,y,z)=0 at a point
**** (ii) find  partial z / partial x  at same point, same surface as in (i) 
**** (iii) approximate z at nearby point, using (ii)
*** (c) [14.5] chain rule involving some trig functions
*** (d) [14.5] chain rule, involving parameters a,b (which don't matter)
*** (e) [14.6] given fastest increase direction of f(x,y) at a point, find direction that is tangent to the level curve
*** (f) [14.6] find points on f(x,y,z)=0 with given tangent direction
*** (g) [15.7] find mass of a rectangular box with given mass density function
** (2) [14.6] hill given by z=f(x,y)
*** (a) direction of steepest ascent at a given point
*** (b) slope of the hill there
*** (c) rate of altitude change while riding a given speed in the direction
** (3) [14.8] 
*** (a) Find the minimum of f(x,y,z) subject to g(x,y,z)=0 via Lagrange multipliers
*** (b) Give geometric interpretation of (a)
** (4) [14.7] 
*** (a) Find minimum of h(x,y) on x^2+y^2 <= 1
*** (b) Explain why this gives another way of solving question (3) [ this is because of a special choice of f,g,h: here f=h when restricted to g=0, since f=h+g; note that h=h(x,y) is independent of z, unlike f and g ]
** (5) [15.4] double integral in polar coordinates
** (6) [15.3] tricky! evalute e^(y^2) in a double integral: have to change from dy dx to dx dy
** (7) [15.7] and [15.4, or equivalently 15.8] tricky! compute the volume of the solid inside both x^2 + y^2 = a x  and x^2+y^2+z^2 = a^2 : if you integrate in the order  dz dx dy , you get 2 (a^2-x^2-y^2)^{1/2}
** (8) [15.7] sketch the region bounded by y=0, z=1-x^2 (a parabolic cylinder), and y=z, and set up an integral over this region as an iterated integral
** (9) [15.9] find the volume of solid inside rho = 8 sin(phi) , and describe what this solid looks like.