Contents

Preliminary Quiz

For Penn State student, access quiz here

import ipywidgets as widgets

Queation 1

image

Consider the feasible region given by the following inequalities. whose boundary lines are graphed above.

\[\begin{split} x+2y \geq 6 \\ x+y \geq 4 \\ x \geq 0 \\ y \geq 0 \\ \end{split}\]

Which ONE of the following labels best indicate the feasible region described above?

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Answer: III

Question 2

Let

\[ f(x) = x_1^2+x_3^2+x_3^4, p =(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}) , g(t)=f(x+tp) \]

where \(t\) is a scalar variable. Find \(g^{'}(0)\) at \(x=(1,1,1)\)

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Answer:

\[ 3\sqrt{3} \]

Question 3

Determine the global maximum and global minimum of the function

\[ f(x) = x^3-3x^2+5 \]

on the interval \([-2,3]\)

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Answer:

Global maximum is 5; global minimum is -15.

Question 4

Find the global minimizer and the global minimum value of the function

\[ f(x,y)=(x-2)^2+y^2+2(y-4)^2+2x^2+2xy+4y-16x+1. \]

That is, determine the point \((x_0,y_0)\) which yields the global minimum value,

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Answer: \(x_0 = 3\), \(y_0 = 1\)

\(f_{min} = 1\)

Problem 5

Which of the following matrices has an inverse?

\[\begin{split} \begin{bmatrix} 3&4 \\ 6&8 \\ \end{bmatrix} \end{split}\]
\[\begin{split} \begin{bmatrix} 0&-4 \\ 0&10 \\ \end{bmatrix} \end{split}\]
\[\begin{split} \begin{bmatrix} 4&-10 \\ 2&5 \\ \end{bmatrix} \end{split}\]
\[\begin{split} \begin{bmatrix} 1&4 \\ 0&3 \\ \end{bmatrix} \end{split}\]
\[\begin{split} \begin{bmatrix} 0&0 \\ 5&7 \\ \end{bmatrix} \end{split}\]
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Answer:

\[\begin{split} \begin{bmatrix} 1&4 \\ 0&3 \\ \end{bmatrix} \end{split}\]

Question 6

Let

\[\begin{split} A=\left( \begin{matrix} 1&-1&0\\ -1&2&-1\\ 0&-1&1 \end{matrix} \right) \end{split}\]

Compute the eigenvalues of \(A\)

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Answer:

1,0,3

Question 7

Let

\[\begin{split} a= \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix} \in R^4 \end{split}\]

and \(A=aa^T\)

Find the eigenvalues and corresponding eigenvectors of \(A\)

Write out solution.

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\[\begin{split} \lambda =4 , x = \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix} \end{split}\]
\[\begin{split} \lambda = 0, x = \begin{bmatrix}1\\-1\\0\\0 \end{bmatrix} \begin{bmatrix}1\\0\\-1\\0\end{bmatrix} \begin{bmatrix}1\\0\\0\\-1\end{bmatrix} \end{split}\]

Question 8

How would you code \(ax^2\) in python?

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\(3*x**2\)

Question 9

When writing in python, what will be the output after the following statements?

m = 92
n = 35
print(m > n)

True
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True

Question 10

NumPy is a library for the Python programming language which provides support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. For example,

x = np.array([1, 2, 3])

will set x to a vector array with components (1, 2, 3). To utilize the above array function as listed, what do you need to do prior to using it?

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import numpy as np