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Math 2130: Final Exam (May 18, 2015)
Your Name: Your Netid:
Your Section Number and/or Time:
This exam has 14 pages with 10 problems adding up to 200 points. You have 150 minutes. If you need extra space, you can use the other side of each page or the last mostly blank page. No calculators or books are allowed.
Please show all of your work (including the intermediate steps). Writing clearly and legibly will improve your chances of receiving the maximum credit that your solution deserves. Good luck!
Problem # Max pts Earned 1 20 2 20 3 25 4 15 5 25 6 20 7 20 8 15 9 15 10 25 Total 200
Academic Integrity is expected of all students of Cornell University at all times, whether in presence or absence of members of the faculty.
Understanding this, I declare I shall not give, use, or receive unauthorized aid in this examination.
Signature of the Student:
Unless specifically requested, answers on this exam NEED NOT be simplified or completely numerical. Arithmetic expressions are fine. Ordinarily, answers like
ln
3 x^2 (x^4 − 1) − 4 x^3 (x^3 + 1) (x^4 − 1)^2
4 x^3
(x^4 + 1).^5
are completely acceptable.
Problem 1) (20 Points)
1a) Find a parametrization of the line segment (or just the equation of the line) between the points (1, 2 , 3) and (4, 6 , 7) in R^3.
1b) Let C be the line segment described in part a), oriented as starting from (4, 6 , 7) and ending at (1, 2 , 3). Use your parametrization in part a) to calculate the value of the line integral ∫
C
y dx + z dy − 2 x dz.
Problem 3) (25 Points) Let
x = u^2 − v^2 y = 2 uv.
Suppose z = f (x, y) is a differentiable function.
3a) Express ∂z ∂u
in terms of ∂z ∂x
∂z ∂y
, u, and v.
3b) Show that
1 4(u^2 + v^2 )
[(
∂z ∂u
∂z ∂v
) 2 ]
∂z ∂x
∂z ∂y
Problem 5) (25 Points) Set up as an integral or sum of integrals the double integral of the function ex (^2) y 2 over the trapezoid in R^2 with vertices (0, 0), (3, 0), (5, 2), and (1, 1). Include all limits of integration but do not attempt to compute the value of the double integral.
Problem 6) (20 Points) Consider the (solid) region in R^3 bounded by the four planes z = 0, x = 1, y = 2, and x + 2y + 3z = 6.
Let f (x, y, z) be a differentiable function within this region. Set up the triple integral of f (x, y, z) over this region. Include all limits of integration but do not attempt to compute the value of the integral.
Problem 8) (15 Points) Using the method of separation of variables, find a solution of the form u(x, y) = X(x)Y (y) to the partial differential equation
ux = uy + 2u
which also satisfies u(x, 0) = e^5 x.
Problem 9) (15 Points) The method of separation of variables applied to a certain partial differential equation (with boundary conditions) defined for 0 ≤ x ≤ 1 produces the series solution
u(x, t) =
a 0 2
- Σ∞ n=1 cos (nπx) [an cos (4nπt) + bn sin (4nπt)]
with orthogonality relation
∫ (^1)
0
cos (nπx) cos (mπx) dx = 0
when n and m are unequal non-negative whole numbers. (Take this as given - you don’t have to verify it.)
It is desired to fit the initial conditions u(x, 0) = f (x) and ut(x, 0) = g(x). Use the given orthogonality relations to obtain formulas for the coefficients a 0 as well as for the ak and bk when k = 1, 2 ,... in the above solution. Of course your answer will be in terms of the functions f (x) and g(x).
10b) Using the full collection of solutions generated by the separation of vari- ables procedure in part 10a), write down a general series solution satisfying the above equation with these boundary conditions.
10c) Write down the orthogonality relations you would expect to hold for the functions X(x) you found in part 10a).
More space - Be sure and indicate clearly on the original problem page if you are continuing some problem on this page.
