DATS 6203 Homework 2

Machine Learning II
DATS 6203
Homework 2:
• Show ALL Work, Neatly and in Order.
• No credit for Answers Without Work.
• Submit a single pdf file includes all of your solutions.
• DO NOT submit individual files or images.
• For coding questions, submit ONE .py file and include your comments.
Note 1: Please read chapter 5 of neural network design book (listed in the syllabus) and then
answer the following questions.
E.1:
Practice finding the derivatives of these functions:
i. f(x) = sin(6x−1).
ii. f(x) = x
8 +30+
1
x
4
.
iii. f(x) = e
(
1
x
)+( 1
x
2
)
.
iv. f(x) = sin2
(6x−1).
E.2:
Finding when a function is increasing/decreasing and concave up/down. When is the function
f(x) = 2x
3 +24x
2 −54x decreasing? When is it concave up? Plot the function and find your
check your answer?.
E.3:
Finding critical points, local max/min, global max/min, and inflection points. Find all critical
points and inflection points of f(x) = 2x
3 +24x
2 −54x. Classify the critical points as local min,
local max, or neither. Find the global max and min of this function on [−3,3] and on (−∞,0).
Plot the function and find your check your answer?.
1
E.4:
i. Find the gradient vector of f(x, y) = x
2 +y
2
.
ii. What are the gradient vectors at (1,2),(2,1)and(0,0)? Plot the function in 3D space and
check your answers?
E.5:
i.Find the gradient vector of f(x, y) = 2xy+x
2 +y.
ii.What are the gradient vectors at (1,1),(0,−1)and(0,0)? Plot the function in 3D space and
check your answers?
iii. Find the gradient vector of f(x1, x2) = x
2
1 +2x1x2 +x
2
2 +x1x
2
2
.
E.6:
i. Find equation of the line: has slope 3 and y-intercept (0, -0.5)
ii. Find equation of the line: passes through (4, 8) and (6, 14).
iii. Find equation of the line: passes through (3, 2) and is perpendicular to y = 5x + 3.
iv. Find equation of the line: has b = 3 and passes through (2, 1).
v. Find equation of the line: has passes through (6, 4) and (1, -1).
E.7:
Find the eigenvalues and eigenvectors of the given matrix by hand and check results by the
computer (use Python to check your results).
i. 
2 0
0 5
ii. 
5 1
4 5
iii.
3 5
3 1
2
E.8:
i. Consider the set of all continuous functions that satisfy the condition f(0) = 0. show that this
is a vector space.
ii. Show that the set of 2x2 matrices is a vector space.
E.9:
Which of the following sets of vectors are independent? Find the dimension of the vector space
spanned by each set. (Verify your answers using Python).)
i.


1
2
3

,


1
0
1

,


1
2
1


ii. sin(t), cos(t), cos(2t)
iii. 1+t, 1−t
iv.




1
2
2
1




,




1
0
0
1




,




3
4
4
3




E.10:
Expand x =

1 2 2 T
in terms of the following basis set.
v1 =


−1
1
0

, v2 =


1
1
−2

, v3 =


1
1
0


3