1.Analyze the floating point format
IEEE Standard for Binary FloatingPoint Arithmetic (IEEE 754) is the most widelyused standard for floatingpoint computation, and is followed by many CPU and FPU implementations. The standard defines formats for representing floatingpoint numbers and special values together with a set of floatingpoint operations that operate on these values. It also specifies four rounding modes and five exceptions (Michael L Overton). 2.How floating point numbers are stored in memory
An IEEE754 float (4 bytes) or double (8 bytes) has three components (there is also an analogous 96bit extendedprecision format under IEEE854): a sign bit telling whether the number is positive or negative, an exponent giving its order of magnitude, and a mantissa specifying the actual digits of the number. Using singleprecision floats as an example, here is the bit layout: seeeeeeeemmmmmmmmmmmmmmmmmmmmmmm meaning
31 0 bit # s = sign bit, e = exponent, m = mantissa In the internal representation, there is 1 bit for the sign (S), 8 bits for the exponent (E), and 23 bits for the mantissa (m). The number is stored as follows, with high memory to the right: Byte 0 Byte 1 Byte 2 Byte 3
3.The difficulty of manipulating and using floating point numbers in c calculations There are two reasons why a real number might not be exactly represented as a floatingpoint number. The most common situation is illustrated by the decimal number 0.1. Although it has a finite decimal representation, in binary it has an infinite repeating representation. Thus when β = 2, the number 0.1 lies strictly between two floatingpoint numbers and is exactly represented by neither of them (Cleve Moler). Floatingpoint representations...
...Numerical Methods Questions
1 f(x) = x3 – 2x – 5
a) Show that there is a root β of f(x) = 0 in the interval [2,3]. The root β is to be estimated using the iterative formula
,2 5 2 0 2 1 1 = =++ x x x n n
b) Calculate the values of x1, x2, x3, and x4, giving your answers to 4 sig fig.
c) Prove that, to 5 significant figures, β is 2.0946
2 Use the iterative formula
n
n n cox x x − =+ 1 1 With x1 = 0.5 to find the limit of the sequence x1, x2, x3,……. Correct to 2 decimal places.
3 Starting with x0 = 1, use the iterative formula
( ) 5lnln1 =++ xx n
To find, to 2 decimal places, the value of x1, x2, x3, and x4.
4 The equation 2x = x3 has two roots. Show that the intervals are [1,2] and [9,10].
5 f( x ) = x 3 – 2 –
x 1
, x ≠ 0.
(a) Show that the equation f( x ) = 0 has a root between 1 and 2.
An approximation for this root is found using the iteration formula
x n + 1 =
3 1
nx 1 2
+ , with 0 x = 1.5.
(b) By calculating the values of x 1, x 2, x 3 and x 4, find an approximation to this root, giving your answer to 3 decimal places.
(c) By considering the change of sign of f( x ) in a suitable interval, verify that your answer to part (b) is correct to 3 decimal places.
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical...
...DEPARTMENT OF COMPUTER SCIENCE
FACULTY OF INFORMATION TECHNOLOGY AND APPLIED SCIENCES
LEAD CITY UNIVERSITY, IBADAN
FIRST SEMESTER, 2013/2014 ACADEMIC SESSION
LECTURERINCHARGE: PROF. B. A. OLUWADE
CSC 403: NUMERICAL COMPUTATION II (with MATLAB)
TUTORIAL QUESTIONS
1
(a)
What do you understand by the Euler method ?
(b)
Let
y/ = 3y + 1
y(0) = 2
be an initial value problem. Using Euler method, present an approximation of y(5)
using a step size of 1.
2
(a)
State Simpson’s rule.
(b)
Write MATLAB code for finding the numerical approximation of a
definite integral using (a) above.
3
(a)
Write the meaning of the following MATLAB commands and illustrate
with at least one example each:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
ceil (x)
round (x)
floor (x)
fix (x)
eye
zeros
ones
rem (x, y)
fprintf
xlabel ( )
(b)
With the aid of examples, explain the meaning and order of evaluation of
arithmetic operators and expressions in MATLAB.
1
(a)
Write a MATLAB code for evaluating sin(x)2 + cos(x)2
(b)
4
Write a MATLAB code for drawing the following :
y(x) = sin(3x)
3x
for 5 ≤ x ≤ 15
5
(a)
Write a MATLAB code for finding the approximate solution of an initial
value problem using the Euler method.
(b)
The midpoint rule, otherwise referred to as the rectangle method, is an
algorithm for computing an approximation to a definite integral by finding the area of...
...Experiment #4
Weighing and Volumetric Techniques – Accuracy and Precision
Abstract
The purpose of this experiment is to become familiar with proper techniques for using the analytic balances, graduated cylinder, burette and pipette and determine which is more accurate and/or precise. In this experiment, the burette and pipette were more exact than the graduated cylinder and the analytic balance gave a very accurate and precise answer.
Results / Report
1) Weighing Copper
Copper sample #: 1
Balance #: 14
Temperature: 22.8EC
Weighing Attempt # Mass of Copper Sample (g)
1 3.1234
2 3.1233
3 3.1235
4 3.1232
Average Mass
= (3.1234g + 3.1233g + 3.1235g + 3.1232g) ÷ 4
=3.1234g
Standard Deviation
(  ) (  )
3.1234 0 0
3.1233 
3.1235
3.1232 
Σ =
Or 0.0001
2) Burette Readings
a) Graduated Cylinder b) Pipette
Initial Volume (mL) 44.72 44.91
Actual amount added (mL) Actual amount added (mL)
1st 34.81 9.91 1st 34.98 9.93
2nd 24.90 9.91 2nd 25.00 9.98
3rd 15.05 9.85 3rd 15.08 9.92
Graduated Cylinder average
( 9.91±0.01 mL + 9.91±0.01 mL + 9.85±0.01 mL ) ÷ 3
= 9.89±0.03 mL
Pipette average
(9.93±0.01 mL + 9.98±0.01 mL + 9.92±0.01 mL) ÷ 3
= 9.94±0.03...
...Erica Alonso
Chemistry Honors 1
Mr. Cunningham
1.07 Accuracy and Precision
Procedure
Access the virtual lab and complete the experiments.
Data
• Below is the table that you will complete for the virtual lab. Either type your results into this table or print the table from the virtual lab (it must be submitted to receive full credit for this assignment.)
Part I: Density of Unknown Liquid
Trial 1 Trial 2 Trial 3
Mass of Empty 10 mL graduated cylinder (grams) 26 25.6 26
Volume of liquid (milliliters) 8.6 8.7 8.5
Mass of graduated cylinder and liquid (grams) 36.5 36.5 36.7
Part II: Density of IrregularShaped Solid
Mass of solid
(grams) 38.384 41.435 41.951
Volume of water (milliliters) 51 50 52
Volume of water and solid (milliliters) 57 55 58
Part III: Density of RegularShaped Solid
Mass of solid (grams) 28.1 26.1 26.2
Length of solid (centimeters) 5.25 5 4.5
Width of solid (centimeters) 3 4 3.5
Height of solid (centimeters) 2.5 3 2
Calculations
Show all of your work for each of the following calculations and be careful to follow significant figure rules in each calculation.
Part I: Density of Unknown Liquid
1. Calculate the mass of the liquid for each trial. (Subtract the mass of the empty graduated cylinder from the mass of the graduated cylinder with liquid.)
Trial 1 10.5
36.526= 10.5
Trial 2 10.9
36.525.6= 10.9
Trial 3 10.7
36.726= 10.7
2. Calculate the density of the unknown liquid for each...
...01.07 Accuracy and Precision: Balance Lab Worksheet
Data
* Below is the table that you will complete for the virtual lab. Either type your results into this table or print the table from the virtual lab (it must be submitted to receive full credit for this assigment.)
* To print from the virtual lab.
1. Be sure the data table is viewable.
2. Rightclick (PC) or CommandClick (Mac) on the table and select print.
Part I: Density of Unknown Liquid 
 Trial 1  Trial 2  Trial 3 
Mass of Empty 10 mL graduated cylinder (grams)  25.50  25.50  25.50 
Volume of liquid (milliliters)  8.10  8.30  8.10 
Mass of graduated cylinder and liquid (grams)  35.50  36.00  35.50 
Part II: Density of IrregularShaped Solid 
Mass of solid (grams)  38.285  42.345  42.577 
Volume of water (milliliters)  51.00  50.95  52.90 
Volume of water and solid (milliliters)  55.50  55.90  56.95 
Part III: Density of RegularShaped Solid 
Mass of solid (grams)  27.00  26.50  25.50 
Length of solid (centimeters)  5.25  5.00  4.50 
Width of solid (centimeters)  3.00  4.00  3.50 
Height of solid (centimeters)  2.50  3.00  2.00 
Calculations
Show all of your work for each of the following calculations and be careful to follow significant figure rules in each calculation.
Part I: Density of Unknown Liquid
1. Calculate the mass of the liquid for each trial. (Subtract the mass of the empty...
...However, this type of theodolites are confined for locations where the support is not stable or where space for using other such instruments is limited.
Direction Theodolites In this type of theodolite, the circle is arranged to be fixed while the telescope is aimed on a number of signals. Readings on the circle are for every direction. These theodolites are very good option for triangulation purpose.

Transit This is a specialized type of theodolite has a telescope that can "flip over" ("transit the scope") to let easy backsighting and doubling of angles for error reduction. Some transit instruments are capable of reading angles directly to thirty seconds. However "transit" is considered to be a theodolite type having less precision. It also lacks such features as scale magnification and micrometers. In spite of the fact that newer precise electronic theodolites have become very popular, the transit is still used as a lightweight tool on construction sites. There are further sub types of transit like the vernier transit theodolite which can measure vertical and horizontal angles but there are other transits which cannot measure vertical angles.
Total Station
A total station is considered to be a superior surveying tool compared to the theodolite due to its digital integration and allinclusive features. A total station incorporates the functions of theodolite to determine angles and distances by an electronic distance meter. Total stations...
...Precision describes the closeness of results that have been obtained in exactly the same way while accuracy indicates the closeness of the measurement to its true value. This experiment was used to determine the accuracy and precision of different volumetric measuring devices, as well as determining the density of an unknown metal. This lab was to help understand the application to volumetric measurements.
Part 1:
First, the nexttosmallest beaker was cleaned, dried, and weighed on the scale where it’s mass was determined. The container was then tared so the scale would only read the mass of the water. The smallest beaker was used to measure out 14 mL of water. Then, that waster was poured into the preweighed beaker and put on the scale. Once the measurement was recorded the beakers were both cleaned and dried. This process was repeated two more times.
Second, a 100 mL graduated cylinder was used to measure out 14 mL of water. Then, that water was poured into the preweighed beaker and put on the scale. Once the measurement was recorded, the beaker and graduated cylinder were both cleaned and dried. This process was repeated two more times.
Third, a 25 mL buret was used to measure out 14 mL of water. Then, that water was poured into the preweighed beaker and put on the scale. Once the measurement was recorded, the beaker and buret were both cleaned and dried. This process was repeated two more times.
After recording the mass...