Sections 10.7-10.9, 11.1, 11.3-11.6

To Study for Test 2:
1) Try #2-9 on Old Test 2, and all of Old Test 3 except #1, 3 and #8, 10, 11.
2) Know everything on this test review.
3) Study your homework problems and class notes from these sections.

Formulas given on Test 2:

For projectiles:  

r(t) = ( i + ( 

For F(x, y) = 0 

For F(x, y, z) = 0 

The equation of a tangent plane to the surface z = f(x, y) at P(, , z) is 

    z - z = (, )(x - ) + (, )(y - ) 

The linear approximation for z = f(x, y) at (a, b) is 

The increment of z = f(x, y) is 

The differential of z = f(x, y) is 

     dz = (a, b)dx + (a, b)dy 

The equation of the tangent plane to F(x, y, z) = 0 at P(, , z) is 

The symmetric equations of the normal line to F(x, y, z) at P(, , z) is  

For Section 10.7 (Old Book 13.1/13.2) you need to know: 

1) How to sketch simple space curves of the form r(t) = < f(t), g(t), h(t) > by setting up a table: 

t  |  x = f(t) |  y = g(t) |  z = h(t) for 3 - 4 values of "t" and plot the points (x, y, z).

2) That for c = constant,  < c, cos(mt), sin(mt) > is a circle on the x = c plane.

3) That < t, cos(mt), sin(mt) > is a helix along the x -axis. 

4) That the equation of a line segment from Po to P₁ is

      r(t) = (1 - t)ro + r₁,   0 < t < 1
    or    r(t) = ro + t*(r₁ - ro),   0 < t < 1

5) How to find r'(t) given r(t). 

6) That parametric equations for a tangent line to a curve at is: 

where r'() = <a, b, c>. 

For Section 10.8 (Old Book 13.3) you need to know: 

1) How to use the given arc length formula. 

2) How to use the given formulas for Unit Tangent Vector, Normal Vector, Binormal Vector, and Curvature. 

3) That T is perpendicular to N, and B is perpendicular to both T and N 

4) That the equation for a tangent plane at is: 

where N() = <a, b, c>. 

For Section 10.9 (Old Book 13.4) you need to know: 

v(t)| = speed 

   v(t) = r'(t) 

   a(t) = v'(t) = r''(t) 

   v(t) =   

   r(t) =  

2) How to use the given formulas for projectiles. 

For Section 11.1 (Old Book 14.1) you need to know: 

1) The equation for a plane is: 

         z  = ax + by + c 

2) The equation for a sphere is: 

   And a hemi-sphere is: 

3)How to find the domain of f(x, y). 

4) How to graph level curves for z = f(x, y) by graphing several values of z = k in 2-D. 

5) How to describe level surfaces for w = f(x, y, z) by replacing w = k for several values of k in 3-D. 

For Section 11.3 (Old Book 14.3) you need to know: 

1) How to find partial derivatives, and for z = f(x, y). 

For Section 11.4 (Old Book 14.4) you need to know: 

1) How to find the equation of a tangent plane to the surface z = f(x, y) at P(, , z) using 

    z - z = (, )(x - ) + (, )(y - ) 

2) How to find the linear approximation for z = f(x, y) at (a, b) using 

3) How to find the increment of z = f(x, y) using 

4) How to find the differential of z = f(x, y) using 

     dz = (a, b)dx + (a, b)dy

For Section 11.5 (Old Book 14.5) you need to know: 

1) How to use the chain rule.  For z = f(x, y, w, ....) and x(s, t, .....), y(s, t, .....), and w(s, t, .....) 

    and so on ......... 

2) Know how to use the formula for dy/dx for F(x, y) = 0 and dz/dx and dz/dy for F(x, y, z) = 0

For Section 11.6 (Old Book 14.6) you need to know: 

1) The gradient of f(x, y) is 

2) The directional derivative of f(x, y) in the direction of the unit vector u is 

3) The maximum value of the directional derivative,  and it occurs when u has the same direction as  

4) How to find the equation of the tangent plane to F(x, y, z) = 0 at P(, , z) using 

5) How to find the symmetric equations of the normal line to F(x, y, z) at P(, , z) using