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MATH 51 FINAL EXAM (MARCH 15, 2010)
Max Murphy Jonathan Campbell Jon Lee Eric Malm 11am 11am 10am 11am 1:15pm 2:15pm 1:15pm 1:15pm Xin Zhou Ken Chan (ACE) Jose Perea Frederick Fong 11am 1:15pm 11am 11am 1:15pm 1:15pm 1:15pm
Your name (print):
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Instructions: Find your TA’s name in the table above and circle the time that your TTh section meets. During the test, you may not use notes, books, or calculators. Read each question carefully, show all your work, and circle your final answer. Each of the 16 problems is worth 10 points. You have 3 hours to do all the problems. Good luck!
Total
- Suppose the temperature at point (x, y) is f (x, y) = x^2 − 4 x+y^2 +9.
1(a). Find the hottest point(s) and the coldest point(s) on the ellipse
4 x^2 + 9y^2 = 36.
1(b). Find the hottest point(s) and the coldest point(s) on the region
4 x^2 + 9y^2 ≤ 36.
- Consider the following system of equations:
x + 3y + 7z = b 1 3 x − y + 11z = b 2 x − y + az = b 3.
For which values of a does the system have exactly one solution? [Here b 1 , b 2 , and b 3 are given constants.]
4(a). Let v 1 , v 2 , and v 3 be vectors in R^5. Prove that there is a nonzero vector x that is perpendicular to each of those vectors.
4(b). Suppose that v 1 , v 2 , and v 3 are nonzero vectors that are orthog- onal to each other. Prove that {v 1 , v 2 , v 3 } is linearly independent.
5(b). Suppose that
v 1 =
(^) v 2 =
(^) v 3 =
Find the matrix A for T with respect to the standard basis for R^3.
[Hint: You may use your answer to 5(a). However, it is easier to find A directly, without using the matrix B.]
- Find A−^1 , where A =
- Let f (x, y) be the temperature at point (x, y), and suppose that
∂f ∂x
∂f ∂y
8(a). Find the directional derivative of f at (1, 2) in the direction
v =
[
]
8(b). An insect crawls along an isotherm (i.e., a level set of f ) with speed 3. At time t = 0, the insect is at the point (1, 2) and its x- coordinate is increasing. Find its velocity at time 0.
9(a). Let f (x, y) = sin(xy) + xy^2. Find the linear approximation L(x, y) to f (x, y) at the point (π, 3).
- Suppose that A is a matrix whose row reduced echelon form is
rref(A) =
10(a). Find a basis for the nullspace N (A) of A.
10(b). Let c = A
. Find all solutions of Ax = c.
10(c). Is there a b ∈ R^3 such that Ax = b has no solutions? Explain.
10(d). Is there a b ∈ R^3 such that Ax = b has exactly one solution? Explain.
- The position of a particle at time t is u(t) = (x(t), y(t)). Let r(t) and θ(t) be the polar coordinates of the particle’s position at time t (so that x = r cos θ and y = r sin θ). Suppose that r(0) = 5 and θ(0) = π/3. Find the particle’s velocity u′(0) in terms of r′(0) and θ′(0). (Your answer should be an expression involving r′(0) and θ′(0).)
13(a). Find the eigenvalues of the matrix A =
[
]
13(b). Find an eigenvector with eigenvalue λ = 3 for the matrix
M =
- Suppose F : R^2 → R^3 is defined by F (x, y) =
x^2 y + sin y e^7 y 2 x + 3y^2
Find the Jacobian matrix (i.e, the matrix for the total derivative) DF (x, y).
- In the following sentences, A is an n×n matrix and R is its reduced row echelon form. For each sentence, circle T if it is always true, F if it is always false, and S if it is sometimes true and sometimes false (i.e, if more information is needed to determine whether it is true or false). No explanations are necessary.
T F S (1) If b is in C(A), then Ax = b has a solution. T F S (2) If N (A) = { 0 }, then Ax = b has exactly one solution. T F S (3) If R has a pivot in every column, then A is invertible. T F S (4) If a column of R has no pivot, then 0 is an eigenvalue of A. T F S (5) If A is symmetric, then A is diagonalizable. T F S (6) If 3 vectors in R^5 are linearly independent, then the matrix whose columns are these 3 vectors is an invertible matrix. T F S (7) If n vectors in Rn^ are linearly independent, then the matrix whose columns are these n vectors is invertible. T F S (8) If a differentiable function f : Rn^ → R has a minimum at x, then x is a critical point of f. T F S (9) If x is a critical point of f : Rn^ → R, then f has a local maximum or a local minimum at x. T F S (10) If Q is a positive definite quadratic form, then its associated symmetric matrix is invertible.
