FINAL EXAM MATH261 SPRING 2014
NAME:
CSU ID:
Problem Points
1
30
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
Recall:
Z 2π
0
cos2 (t)dt = π.
GOOD LUCK !!!
2
30
3
25
4
20
5
20
6
30
7
25
8
20
Total
200
Score
1. Read all sections before starting. Consider the coordinate points P1 = (1, 1, 2), P2 =
(2, −1, −1), and P3 = (1, −1, 3).
(a) (10 pts) Find the area of the triangle with vertices P1 , P2 , and P3 .
(b) (10 pts) In the form Ax + By + Cz = D, give an equation of the plane through
P1 , P2 , and P3 .
(c) (10 pts) Find the distance from P1 to the line passing through the points P2 and
P3 .
2. The equation r(t) = sin(πt)i + 3πtj + cos(πt)k, t ≥ 0, describes a particle moving along
a curve.
(a) (10 pts) Find the velocity v(t) and the acceleration a(t) for any time t.
(b) (10 pts) Find the unit tangent vector T(t) and the tangential and normal components of the acceleration: aT (t), and aN (t), for any time t.
(c) (10 pts) Find the distance travelled along the curve in the time interval 0 ≤ t ≤ 3.
3. The height(altitude) z of a mountain above the point (x, y) in the plane is given by
z = x(2 − sin y).
(a) (10 pts) Starting at the point (1, π/2), in what direction should one proceed to
go down the mountain the fastest? Your answer should be a UNIT vector of the
form ai + bj.
(b) (10 pts) Suppose a particle follows a path on the mountain such that the x and y
coordinates of the point are given√by x(t) = t, y(t) = t2 . How fast is the altitude
z of the particle changing at t = π ?
(c) (5 pts) Find the equation of the plane tangent to the mountain at the point
(1, π/2, 1).
4. Given f (x, y) = ey (y 2 − x2 ) complete the following:
(a) (10 pts) Find the critical points. (There are two. The points should be given in
the form (a, b).)
(b) (10 pts) Classify each critical point as a local maximum, local minimum or saddle
point .
5. (a) (10 pts) Find the linearization L(x, y) for f (x, y) = π1 sin(πx) + xy 2 at the point
(1, 2). The final form of the linearization must be L(x, y) = A + Bx + Cy.
(b) (10 pts) Find an upper bound on the magnitude of the error in the approximation
f (x, y) ≈ L(x, y) for the rectangle |x − 1| ≤ .1, |y − 2| ≤ .2
6. Consider the volume of a tetrahedron T bounded by the planes x + 2y + z = 2, x = 2y,
x = 0, and z = 0.
(a) (12 pts) Graph the tetrahedron, labeling each of its four vertices. Then, sketch
its projections onto the xy and xz planes, labeling appropriately. (Note that you
have THREE sketches to do.)
(b) (18 pts) Write the two triple integrals in rectangular coordinates, with integration
orders dzdydx and dydxdz, that compute the volume of the tetrahedron. Do Not
Evaluate any of the integrals.
7. Let F = x2 i + (2xy + x)j + zk. Let C be the circle x2 + y 2 = 1 in the plane z = 0
oriented counterclockwise and S the disk x2 + y 2 ≤ 1 in the xy-plane, oriented with
the normal vector n = k.
(a) (10 pts) Compute the work(circulation) of F around C by DIRECTLY setting up
and computing an appropriate line integral.
(b) (10 pts) Set up the surface integral
S (∇ × F) · ndσ completely as a double
integral in the variables x, y. Do not do the double integration.
RR
(c) (5 pts) According to Stokes’ theorem, what is the value of the integral in part b)?
Explain your answer.
8. (20 pts) Let F = x3 i + y 3 j + z 3 k. Using the Divergence Theorem, write down a triple
integral in spherical coordinates that computes the flux of F outward across the sphere
S of radius 1 centered at the origin. Evaluate the triple integral.