BRIEF QUIZ ANSWERS
In this document you will find brief answers to quiz questions.
If the quiz question is a homework problem, and the answer
is in the book, then the brief answer that you
will see below will be only ``Answer is in the book.''
QUIZ
Week 1
Week 2
- The angle is Pi/3.
- Yes, they are on a line.
- (Solution 1)
Let P be a plane containing vectors a
and b. The vectors a+b and a-b
lie in P, so (a+b) x (a-b) is either zero
or is normal to P. In either case,
(a+b) x (a-b) is
perpendicular to all vectors in P, in particular
to both a and b.
(Solution 2) (a+b) x (a-b) = -2(a x b),
which is perpendicular to both a and b.
- Yes, points A, B, C and D lie on a plane. In fact, B, C and D
lie on a line.
Week 3
- angle = arccos(4/square_root(66))
- (e-1)*i + ((e^(-1) - 1)/2)*j. (Answer is a VECTOR!)
- length = 5/3
- curvature = 2, radius of curvature = 1/2.
Week 4
- Many answers are possible. One example: z = x^2 + 2*y^2.
- Many answers are possible. Examples:
(square_root(2), 3*Pi/4, -1) or
(-square_root(2), -Pi/4, -1) are both cylindric coordinates
for the point whose rectangular coordinates are (-1,1,-1). Note:
(square_root(2), -Pi/4, -1) is not correct, since these are the
cylindric coordinates for (1, -1, -1).
- Many answers are possible. One example:
tan(phi) = 1.
- Sketch.
Week 5
- The tangent plane is z = 2 - 2(x-1) - (y-2).
- The critical points are (0,0), (1,0), (-1,0).
- The minimum value is -1. (It occurs at (x,y) = (1,1/2).)
- A continuous function on a closed,
bounded set R attains a minimum value and a maximum value
on R.
Week 6
Week 7
No quiz.
Week 8
- -2
- 5(x-1) + 4(y-2) + 3(z-3) = 0.
- There are two points: if s = square_root(6), then the points
are (1/s, 2/s, 3/s) and (-1/s, -2/s, -3/s).
- The max value is 2. (It occurs at (0,1) and (0,-1).)
Week 9
- (-1,-2) is a minimum.
- (1,1) is a minimum.
- Many answers are possible. Example: F(x,y) = (x-1)^2 - (y-1)^2.
- A max occurs at (Pi/2,Pi/2). A min occurs at (-Pi/2,-Pi/2). The
point (Pi/2,-Pi/2) is a saddle point.
Week 10
Week 11
Test 2.
Week 12
No quiz.
Week 13
- volume = 24*Pi.
- area = (2/3)(2^(3/2) - 1)
- area = 1/9.
- div(F) = x^2+y^2+z^2, curl(F) = (-2yz,-2xz-2xy).
Week 14
- (5^(3/2) - 1)/12.
- F = grad(f) for f(x,y,z) = xy+xz+yz.
The value of the integral is f(1,2,3) - f(0,0,0) = 11.
- curl(F) = (-y, -z, -x), which is not the zero vector.
Week 15
Last modified on Sep 6, 2000.