c. Find the angle between ??? ?????? ????.
2. Find the equation of the line that passes through (2, ?1, 5) that is perpendicular to the plane 3?? ? 2?? = 7.
3. Find the equation of the plane that passes through ??1(1, 2, ?1), ??2(2, 3, 1), and ??3(3, ?1, 2).
4. Solve the initial value problem for ??(??) by integrating and using the initial conditions to find the constants of integration.
????(??) = ?12??2 , ?2??? ????????? ??(1) = ?6, 8
3 ? ; ???(1) = ?5, 2?
5. If 2( ) ,r t t t= , find the curvature of ( )r t at 3t = .
6. If 2( , ) 3 yg x y yx e= + , cosx r ?= , and siny r ?= , draw a tree diagram for ??(??, ??),
write out a formula for g
r
?
? , and find
g
r
?
? in terms of r and ? .
7. Find the directional derivative of 2ln),( xyxyyxf += at the point (1,2) in the
direction from )0, 4
3 (?P to ( )1,0Q .
8. Find the absolute extrema of the given function on the indicated closed and bounded set R:
??(??, ??) = ???? ? 2??
where R is the triangular region with vertices (0,0), (0,4), and (4,0).
9. Find the area in the first quadrant bounded by ?? = 1 and ?? = sin (2??), with ??
4 ? ?? ?
??
2 .
10. Set up a double integral to find the volume of the tetrahedron bounded by the coordinate planes and the plane ?? = 4 ? 4?? ? 2??. Be sure to sketch both a 3-D drawing, a 2-D drawing of the footprint, and a representative rectangle in your region.
11. Convert ? ? ? (?? 2 + ??2 + ?? 2) ?18???2???2
???2+??2 ?9???2
0
3
0 ???? ???? ???? into a triple integral in
spherical coordinates. Draw a 3-D picture and show how you get the boundaries for ??, ??, ?????? ??. Look where x and y go from and dont just assume ?? ???????? ???????? 0 ???? 2??. You do not have to evaluate the integral.
12. Set up the integral to find the volume in the first octant of the solid bounded above by
the sphere 2 2 2 2x y z+ + = and below by the plane z y= . Use rectangular coordinates.
Do not solve the triple integral(s). Draw the 3-D and 2-D drawings with the
representative rectangle.
13. Set up a triple integral in cylindrical coordinates to find the volume of the solid
bounded above by the paraboloid 2 2 2 ( , ) 8F x y x y= ? ? and below by the paraboloid
2 2
1 ( , )F x y x y= + . Do not solve the triple integral(s). Draw the 3-D and 2-D drawings
with the representative rectangle.
Urgent
Q8, Q9, Q10, Q11, Q12, Q13
6 question Only.
