Problem Set 5 Solutions
Question 1
(a). Rewrite f (x) = 2x = e(log 2)x , so the nth derivative of f (x) is
f (n) (x) = (log 2)n e(log 2)x = (log 2)n 2x .
(b). There are many ways to do this problem, including solving a linear system directly, or by using
Homework set Module 1
Fundamentals of Machine Learning and Data Analytics
IEOR290 Sec. 4 Spring 2016
Due Tuesday, February 2nd at noon (post pdf on bCourses)
Work individually on this problem set
1. Problem 1.3 in the text, (graded).
2. Work problems 1.4,
Homework set Module 2
Fundamentals of Machine Learning and Data Analytics
IEOR290 Sec. 4 Spring 2016
Due Thursday, February 11th at 11:59pm (post pdf on bCourses)
Work individually on this problem set
To be submitted and graded:
1. Problem 1.2 from the te
Math 128A, Spring 2016
Question 1
Problem Set 02
Show that oating point arithmetic sums
n
1
1
1
1
= 1 + 2 + 2 + + 2
k2
2
3
n
sn =
k=1
with accuracy O(n) from left to right, while summing from right to left gives accuracy O(log n).
Question 2
Suppose a and
MATH 128A Quiz 1
September 4, 2015
Name:
And now these three remain: faith, hope and love. But the greatest of these is love.
Problem 1. Circle True or False.
a. True or False If f (x) is continuous and f (1) = 1, f (1) = 1, then f (x) must have
exactly 1
Math 126 Homework 11 (Due Monday Nov 23)
1. In this question, you will construct a test function that is positive on the interval (1, 1)
and vanishes outside of this interval. Recall a test function must be innitely dierentiable and compactly supported.
D
3. A)
function [t,w]=idec(f,dfdy,a,b,ya,p,maxiter,tau)
% a,b: interval endpoints with a < b
% ya: vector y(a) of initial conditions
% f: function handle f(t, y) to integrate (y is a vector)
% p: number of euler substeps / correction passes
% tau: user-spe
2 Consider the iteration
xn+1 =
x3n + 3axn
.
3x2n + a
(a) What is it intended to compute? (b) Given a = 2 and x0 = 1,
compute x1 and x2 . (c) Define and determine the order of convergence of
this iteration.
4 (a) Derive a numerical integration formula
Z 1
Math 128A, Spring 2016
Programming Project 02
For the following two problems, write and debug MATLAB codes and make sure they run
with the test autograders from the course web page. Test them thoroughly on test cases of
your own design. When you are convi
UC Berkeley Math 128A, Spring 2016
Problem Set 09 SOLUTIONS
Question 1 The position (x(t), y(t) of a satellite orbiting around the earth and moon is described by the
second-order system of ordinary differential equations
x00 = x + 2y 0 b
y 00 = y 2x0 b
Math 128A, Spring 2016
Problem Set 08
Question 1 (a) For arbitrary real s find the exact solution of the initial value problem
y 0 (t) =
1
y(t) + y(t)3
2
with y(0) = s > 0.
(b) Show that the solution blows up when t = log(1 + 1/s2 ).
Question 2 (a) Find t
Name; 3 0 WW
Name
1a
1b
2a
2b
20
3a
3b
4b
Total
Page 1 of 10 Name: GSI:
(1a) Suppose a3" is a sequence of real numbers deﬁned by x0 = 1 and
1
xn+1 = 2910,1 — 5.23% =
Assume can —> a: for some as as n ——) 00. Show that
l
ixn+1 _ 55' S ‘lxn "' 37'2-
2
Cam
Math 128A, Fall 2016.
Homework 1, due Sep 7th.
Prob 1. Show that if k k is a vector norm and A is a non-singular matrix, then x 7 kAxk is a(nother)
vector norm. What happens if A is singular?
Prob 2. Prove that A 7 maxi,j |aij | a norm on the space of mat
Problem Set 6 Solutions
Problem 1
(a). https:/en.wikipedia.org/wiki/Sophomores_dream
(b). Write
1
xx dx = SN + RN ,
0
where for any integer N 1,
N
n
n
SN =
,
nn .
RN =
n=1
n=N +1
We need to choose N 1 such that RN < 1012 , which will guarantee that the ni