MA 126 (Calculus-II) Midterm test #3
Show your work. Fri, Dec 2
1. (10 pts) Find the distance from the point P(3,โ1,0) to the plane 2x+ 4yโ4z= 14.
2. (15 pts) Two planes are given: 2xโyโ4z= 5 and x= 2z+ 1.
(a) Find parametric equations and symmetric equations for these line of intersection of
the planes.
(b) Determine the angle between these planes.
3. (10 pts) Find the volume of the parallelepiped with adjacent edges P Q,P R, and P S,
where P(2,โ1,1), Q(6,0,4), R(3,2,0), S(0,3,5).
4. (15 pts) Write its first four nonzero terms of the Maclaurin series for the function
y=3
โ1โx2.
5. (10 pts) (a) Find a vector perpendicular to the plane through the points P(1,0,โ2),
Q(0,3,5) and R(โ1,2,2).
(b) Find the area of the triangle P QR.
6. (15 pts) Find the Taylor series for the function y= cos xcentered at a=ฯ. Write a
general formula for the series and also write down its first three terms.
7. (10 pts) Show that the equation
2x2+ 2y2+ 2z2+ 12xโ4y+ 8z+ 10 = 0
represents a sphere. Find its center and radius.
8. (15 pts) Recall Maclaurin series for exand ln(1 + x). Then use multiplication to find
first three nonzero terms for the Maclaurin series of the function y=eโxln(1 + 2x).
[Bonus] Find the distance between two skew lines
x+ 5
0=yโ4
2=z+ 1
โ3
and xโ3
2=y
6=zโ2
0
