MATHS 208 General Mathematics 2 SUMMER SESSION, 2019
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MATHS 208
SUMMER SESSION, 2019
General Mathematics 2
1. (a) Consider the function f given by f (x, y) = 3x4 - 6x2 - 12xy + 2y . Find the directional2 derivative of f at the point (1, -1) in the direction of the vector (1, 1).
(b) Find the relative maxima, minima and saddle points of the function f (x, y) = x3 - 3xy + y3
.
(c) Evaluate the integral
)
(d) Evaluate the integral
) dx
2. (a) Define the sequence ●a也 { by
a也 = / 、也
Does the limit of this sequence exist? Justify your answer.
(b) Which of the following two series converges and why?
(i) / 、 ,
(c) Find the Taylor series of the function f defined by
f (x) = sin(2x)
about x = π/4. Use summation notation to express your answer.
3. (a) Let
┌ 1 2 2 2┐ ┌4┐
'1 2 5 2' , '7'
You may use the fact that the matrix [A}b] row reduces to
┌ 1 2 0 2 2 ┐
( 0 0 1 0 1 )
' 0 0 0 0 0 '
(i) Give a basis for col(A).
(ii) Give a basis for null(A).
(iii) Find the general solution to Ax = b.
┌4┐
(iv) Show that the vector c = (5) does not belong to the column space col(A).
'8'
(b) Consider the inconsistent system
┌(' ┐)' ┌ ┐x(x)2(1) = ┌('┐)' .
(i) Find the least squares solution to this system.
(ii) Find a vector b such that it is orthogonal to each column vector of
┌ 1 0┐
(0 1)
' ' .
4. (a) The Kakapo is an endangered native New Zealand bird. It is now found only in a few carefully protected reserves. This question is about one of these reserves. Let x也 be the number of chicks (baby Kakapo) in the reserve after n years and let y也 be the number of adult breeding birds after n years. Initially there are 30 adult birds and 20 chicks. With a yearly state vector vn = ╱ \y(x)也(也) , the model has the form vn = A也v0 ,
╱ \ ╱ \
(i) Find v2 , the state vector after 2 years.
(ii) Find all eigenvalues of matrix A. Then use these eigenvalues, or otherwise, to find
how many Kakapo (chicks and adult birds) survive in the long run.
(iii) The number k in the matrix A = \ represents the probability a Kakapo
chick will survive its first year and become an adult. In the scenario above, k = 0.1. Conservationists can increase the value of k by providing protection for the chicks. What must the value of k be increased to if the Kakapo numbers are to be kept from declining?
(b) Let B be the matrix
┌ ┐
(i) Find the eigenvectors and eigenvalues of B .
(ii) Write the matrix B in the form VDV.1 , where D is a diagonal matrix and V is a
matrix to be found.
5. (a) Solve the initial value problem = ty - 2t, y(0) = 1.
(b) Find the general solution to the differential equation = - - sin(t).
(c) Given the initial value problem y上 = y2 + t, y(0) = 1, use Euler’s method to estimate y(1). Use the step size h = 0.5.
Copy the following table and fill in the missing entries, then indicate which entry gives your estimate of y(1).
n |
也 |
也 |
f (t也 , y也 ) |
y也 + hf (t也 , y也 ) |
0 |
0 |
1 |
|
|
1 |
0.5 |
|
|
|
2 |
1 |
|
|
|
6. (a) Find a fundamental set of solutions of
y上上 - y = 0.
Verify that the solutions are linearly independent.
(b) Solve the initial-value problem
y上上 - 2y上 + y = 0, y(0) = a, y上 (0) = 1.
where a is a given real number.
(c) Solve the linear system
x上 = x + 2y
y上 = 2x + y
for x(t) and y(t), with initial conditions given by x(0) = 1 and y(0) = 2.
2022-02-11