Chapter 10
Write the equation of the plane containing the point (3,0,-4) and orthogonal to the line given by r(t) = (12-t, 1+8t, 4+6t).
-(x-3) + 8(y-0) + 6(z+4) = 0
-x + 8y + 6z = -27
Evaluate: lim(x,y)->(0,0)[(x-4y)/(6y+7x)]. Explain.
DNE because along x-axis, the limit is 1/7.
Along y-axis is -2/3.
Evaluate ∬D 4xy-y3 dA, where D is the region bounded by y=√x and y=x3.
55/156
Evaluate ∫C xy-4z ds , where C is the line segment from (1,1,0) to (2,3,-2).
43/2
(-2,3,-3)
Write the equation of the line through the point (-7,2,4) and parallel to the line given by x=5-8t, y=6+t, z=-12t in vector form, parametric form, and symmetric form.
r(t) = (-7,2,4) + t(−8,1,−12) = (−7−8t,2+t,4−12t)
x=-7-8t, y=2+t, z=4-12t
(-7-x)/8 = y-2 = (4-z)/12
Use chain rule to determine dz/dx.
z=x2y4-2y
y=sin(x2)
2xsin4(x2) + 2x(4x2sin3(x2)-2)cos(x2)
Evaluate ∬D 2xy dA, where D is the portion of the region between the circles of radius 2 and radius 5 centered at the origin that lies in the first quadrant.
609/4
Evaluate ∫C √(1+y) dy , where C is the portion of y=e2x from x=0 to x=2.
Find the normal vector n to the plane given by 2x-y+z=4.
n = (2,-1,1)
True or False: If a and b are vectors with angle T between them, then a x b = |a| |b| sin(T). Explain.
False. The left side of the equation is a vector. The right side is a scalar number.
Find the linear approximation to z=4x2-ye2x+y at (-2,4).
12-24(x+2)-5(y-4) = -24x-5y-16
Evaluate ∭E 2x dV, where E is the region under the plane 2x+3y+z=6 that lies in the first octant.
9
Use Green's Theorem to evaluate ∮C xy dx + x2y3 dy , where C is the triangle with vertices (0,0), (1,0), (1,2) with positive orientation.
2/3
Find the cross product of axb where a=(3,2,4) and b=(2,0,6).
(12,-10,-4)
Is the line through the points (2,0,9) and (-4,1,-5) parallel, orthogonal, or neither to the line given by r(t)=(5,1-9t,-8-4t)? Why?
Not parallel because the two vectors are not scalar multiples of each other.
Not orthogonal because the dot product of the two vectors is -47, which is not 0.
Find the equation of the tangent plane to z=ln(2x+y) at (-1,3).
z = 2(x+1)+(y-3) = 2x+y-1
Evaluate ∭E y dV , where E is the region that lies below the plane z=x+2 above the xy-plane and between the cylinders x2+y2=1 and x2+y2=4.
0
Evaluate ∫C F.dr where F(x,y)=y2i+(3x-6y)j and C is the line segment from (3,7) to (0,12).
-1079/2
Find the tangent plane and normal line to x2+y2+z2=30 at the point (1,-2,5).
2(x-1)-4(y+2)+10(z-5)=0
r(t)=(1,-2,5)+t(2,-4,10)=(1+2t,-2-4t,5+10t)
Find the arc length function for r(t)=[2t,3sin(2t),3cos(2t)]
2√10 t
Find and determine all the critical points of
f(x,y)=(3x+4x3)(y2+2y)
(0,-2) and (0,0) are saddle points
Evaluate ∭E x2+y2 dV where E is the region portion of x2+y2+z2=4 with y>0.
(128/15) pi
Use Stokes Theorem to evaluate ∬curlF.dS where F=(x2z2,y2z2,xyz). S is the part of the paraboloid that lies inside x2+y2=4 oriented up.
0
Determine if the following vector field is conservative and find a potential function for the vector field if it is conservative.
F=(2x3y4+x)i+(2x4y3+y)j
f(x,y) = (1/2)x4y4+(1/2)x2+(1/2)y2+c