MAT 201 - FALL 2013
Homework 1
Ryan Szypowski
Due October 10, 2013
1. Consider a nite-precision oating-point system in base b = 3. Suppose
the signicand x has 4 thrigits (yes, I just made that up. base 3 digits
are now called thrigits. or perhaps trigits?
MAT 201 - FALL 2012
Homework 2 Solutions
Ryan Szypowski
Due October 11, 2012
1. Consider nding x such that x = ex . We will tackle this both as a root nding
problem and a xed point problem. To that end, dene
f (x) = x ex
and
g(x) = ex .
(a) Using the inte
MAT 201 - FALL 2012
Homework 1
Ryan Szypowski
Due October 4, 2012
1. Let xA = 1.001 and yA = 1.000 be approximations of the unknown values xT and yT .
Assume that the approximations are rounded correctly to the shown degits. Using
interval analysis, deter
MAT 201 - FALL 2012
Homework 3
Ryan Szypowski
Due October 23, 2012
1. In this problem, well explore the fact that Newtons method is only guaranteed to
converge if we start suciently close to a root.
Let
f (x) = tan(x) x.
(a) Show that there exists a root,
MAT 201 - FALL 2012
Homework 5
Ryan Szypowski
Due November 20, 2012
1. Compute the natural cubic spline interpolant of the function
f (x) = sin
x
2
using nodes x0 = 0, x1 = 1, x2 = 2. Is the choice of a natural boundary condition a
good choice for this fu
MAT 201 - FALL 2012
Homework 4
Ryan Szypowski
Due November 8, 2012
1. In this problem, we will study polynomial interpolation of the function
2
f (x) = ex .
(a) Determine the Newton-form quadratic polynomial interpolant of f (x) through
the nodes x0 = 1,
MAT 201 - FALL 2012
Homework 5 Solutions
Ryan Szypowski
Due November 20, 2012
1. Compute the natural cubic spline interpolant of the function
f (x) = sin
x
2
using nodes x0 = 0, x1 = 1, x2 = 2. Is the choice of a natural boundary condition a
good choice f
MAT 201 - FALL 2012
Homework 3
Ryan Szypowski
Due October 23, 2012
1. In this problem, well explore the fact that Newtons method is only guaranteed to
converge if we start suciently close to a root.
Let
f (x) = tan(x) x.
(a) Show that there exists a root,
MAT 201 - FALL 2012
Homework 6 Solutions
Ryan Szypowski
Due December 4, 2012
1. Let
2
4
f (x) = x ,
and I(f ) =
f (x) dx.
0
(a) Use the trapezoid rule to compute T1 (f ), T2 (f ), T4 (f ) to approximate I(f ).
Solution: Here, we have
2
(f (0) + f (2) = 16
MAT 201 - FALL 2012
Homework 4 Solutions
Ryan Szypowski
Due November 8, 2012
1. In this problem, we will study polynomial interpolation of the function
2
f (x) = ex .
(a) Determine the Newton-form quadratic polynomial interpolant of f (x) through
the node
MAT 201 - FALL 2012
Homework 2
Ryan Szypowski
Due October 11, 2012
1. Consider nding x such that x = ex . We will tackle this both as a root nding
problem and a xed point problem. To that end, dene
f (x) = x ex
and
g(x) = ex .
(a) Using the intermediate v
MAT 201 - FALL 2012
Homework 1 Solutions
Ryan Szypowski
Due October 4, 2012
1. Let xA = 1.001 and yA = 1.000 be approximations of the unknown values xT and yT .
Assume that the approximations are rounded correctly to the shown degits. Using
interval analy
MAT 201 - FALL 2013
Project 2
Ryan Szypowski
Due December 13, 2013
Instructions
In this project, you will study the Trapezoid rule and its behaviour on periodic integrands.
The deliverable for the project should be a short written report in which you deve
MAT 201 - FALL 2013
Homework 2
Ryan Szypowski
Due October 24, 2013
1. Consider nding x such that x = ex . We will tackle this both as a
root nding problem and a xed point problem. To that end, dene
f (x) = x ex
and
g(x) = ex .
(a) Using the intermediate v
MAT 201 - FALL 2013
Homework 3
Ryan Szypowski
Due November 18, 2013
1. Using the secant method with p0 =
1
4
and p1 = 1, nd p4 for approximating a root of
f (x) = x ex .
2. Consider
f (x) = 3 + x 2x2 + x4 .
(a) Find the linear interpolant of f (x) through
MAT 201 - FALL 2013
Homework 2
Ryan Szypowski
Due October 24, 2013
1. Consider nding x such that x = ex . We will tackle this both as a root nding
problem and a xed point problem. To that end, dene
f (x) = x ex
and
g(x) = ex .
(a) Using the intermediate v
MAT 201 - FALL 2013
Homework 3 (Brief) Solutions
Ryan Szypowski
Due November 18, 2013
1. Using the secant method with p0 =
1
4
and p1 = 1, nd p4 for approximating a root of
f (x) = x ex .
Solution: We compute the following:
p2 = p1 f (p1 )
p1 p0
f (p1 ) f
MAT 201 - FALL 2013
Homework 4
Ryan Szypowski
Due December 5, 2013
1. Let
f (x) = x3 + 4,
4
0
and consider approximating I(f ) =
f (x) dx.
(a) Find T1 (f ), T2 (f ), and T4 (f ).
(b) First, perform a u-substitution to convert the above integral into an in
MAT 201 - FALL 2013
Homework 4 Solutions
Ryan Szypowski
Due December 5, 2013
1. Let
f (x) = x3 + 4,
and consider approximating I(f ) =
4
0
f (x) dx.
(a) Find T1 (f ), T2 (f ), and T4 (f ).
Solution: We have the following:
40
(f (0) + f (4) = 36
2
40
(f (0
MAT 201 - FALL 2013
Project 1
Ryan Szypowski
Due November 21, 2013
Instructions
In this project, you will study the fours rootnding schemes discussed in class.
The deliverable for the project should be a short written report in which you develop a
narrati
MAT 201 FALL 2013
Name:
Test 2
Please show all of your work. Answers without justifaction may be worth 0.
Please make your answers easy to read. This means it should be clear what you are doing
from one step to the next and your work should be legible.
1)
MAT 201 - FALL 2013
Homework 1 Solutions
Ryan Szypowski
Due October 10, 2013
1. Consider a nite-precision oating-point system in base b = 3. Suppose
the signicand x has 4 thrigits (yes, I just made that up. base 3 digits
are now called thrigits. or perhap
Chapter 1
Mathematics on Computers
1.1
1.1.1
Floating Point Numbers
Basic Denition
In order to perform mathematics on computers, we need to be able to represent the common types of numbers. Integers are represented by integer
types of numbers, while real
1. It is believed some psychtropic meds cause endocrine problems. Synthroid 0.15 mg is ordered
for a client. Available is Synthroid 150 mcg/tablet. How many tabs will you administer?
2. Dr. Order: Administer 1000 units of heparin IV every hour. Solution a