# HG changeset patch # User amit # Date 1288088049 -19800 # Node ID 9ced58c5c3b6e86e3956eccbf898fd5e3b277666 # Parent f0c93ea97e4ccd2b7051ae513ec4b95d76d6efe2 Added long answer type problems in all scripts diff -r f0c93ea97e4c -r 9ced58c5c3b6 statistics/questions.rst --- a/statistics/questions.rst Thu Oct 21 18:22:07 2010 +0530 +++ b/statistics/questions.rst Tue Oct 26 15:44:09 2010 +0530 @@ -8,15 +8,13 @@ sum 2. Calcutate the mean of the given list? - student_marks=[74,78,56,87,91,82] mean(student_marks) -3. Given a two dimensional list,:: +3. Given a two dimensional list, two_dimensional_list=[[3,5,8,2,1],[4,3,6,2,1]] - how do we calculate the mean of each row? @@ -27,7 +25,6 @@ std 5. Calcutate the median of the given list? - student_marks=[74,78,56,87,91,82] median(age_list) @@ -50,5 +47,10 @@ .. A minimum of 2 questions here (along with answers) -1. Question 1 -2. Question 2 +1. Get the weighted mean of [74,64,86,76,83] where the weights are + [2,2,1,4,4] respectively. + + Hint: Readup on function average using average? + +2. Calculate the weighted standard deviation for the sequence [74,64,86,76,83], such that the weights are [2,2,1,4,4] . + diff -r f0c93ea97e4c -r 9ced58c5c3b6 statistics/script.rst --- a/statistics/script.rst Thu Oct 21 18:22:07 2010 +0530 +++ b/statistics/script.rst Tue Oct 26 15:44:09 2010 +0530 @@ -105,7 +105,7 @@ * Roll Number 015163 * Name JOSEPH RAJ S * Marks of 5 subjects: ** English 083 ** Hindi 042 ** Maths 47 ** -Science AA (Absent) ** Social 72 +Science 35 ** Social 72 * Total marks 244 * @@ -165,8 +165,9 @@ This tutorial was created as a part of FOSSEE project, NME ICT, MHRD India Hope you have enjoyed and found it useful. + Thankyou - + .. Author : Amit Sethi Internal Reviewer 1 : Internal Reviewer 2 : diff -r f0c93ea97e4c -r 9ced58c5c3b6 symbolics/questions.rst --- a/symbolics/questions.rst Thu Oct 21 18:22:07 2010 +0530 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,61 +0,0 @@ -Objective Questions -------------------- - -.. A mininum of 8 questions here (along with answers) - -1. How do you define a name 'y' as a symbol? - - - Answer: var('y') - -2. List out some constants pre-defined in sage? - - Answer: pi, e ,euler_gamma - -3. List the functions for differentiation and integration in sage? - - Answer: diff and integral - -4. Get the value of pi upto precision 5 digits using sage? - - Answer: n(pi,5) - -5. Find third order differential of function. - - f(x)=sin(x^2)+exp(x^3) - - Answer: diff(f(x),x,3) - -6. What is the function to find factors of an expression? - - Answer: factor - -7. What is syntax for simplifying a function f? - - Answer f.simplify_full() - -8. Find the solution for x between pi/2 to pi for the given equation? - - sin(x)==cos(x^3)+exp(x^4) - find_root(sin(x)==cos(x^3)+exp(x^4),pi/2,pi) - -9. Create a simple two dimensional matrix with two symbolic variables? - - var('a,b') - A=matrix([[a,1],[2,b]]) - -Larger Questions ----------------- - -.. A minimum of 2 questions here (along with answers) - -1.Find the points of intersection of the circles - - x^2 + y^2 - 4x = 1 - x^2 + y^2 - 2y = 9 - -2. Integrate the function - -x^2*cos(x) - -between 1 to 3. diff -r f0c93ea97e4c -r 9ced58c5c3b6 symbolics/quickref.tex --- a/symbolics/quickref.tex Thu Oct 21 18:22:07 2010 +0530 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,8 +0,0 @@ -Creating a linear array:\\ -{\ex \lstinline| x = linspace(0, 2*pi, 50)|} - -Plotting two variables:\\ -{\ex \lstinline| plot(x, sin(x))|} - -Plotting two lists of equal length x, y:\\ -{\ex \lstinline| plot(x, y)|} diff -r f0c93ea97e4c -r 9ced58c5c3b6 symbolics/script.rst --- a/symbolics/script.rst Thu Oct 21 18:22:07 2010 +0530 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,277 +0,0 @@ -Symbolics with Sage -------------------- - -Hello friends and welcome to the tutorial on symbolics with sage. - -{{{ Show welcome slide }}} - - -.. #[Madhu: What is this line doing here. I don't see much use of it] - -During the course of the tutorial we will learn - -{{{ Show outline slide }}} - -* Defining symbolic expressions in sage. -* Using built-in costants and functions. -* Performing Integration, differentiation using sage. -* Defining matrices. -* Defining Symbolic functions. -* Simplifying and solving symbolic expressions and functions. - -We can use Sage for symbolic maths. - -On the sage notebook type:: - - sin(y) - -It raises a name error saying that y is not defined. But in sage we -can declare y as a symbol using var function. - - -:: - var('y') - -Now if you type:: - - sin(y) - -sage simply returns the expression. - - -Thus sage treats sin(y) as a symbolic expression . We can use -this to do symbolic maths using sage's built-in constants and -expressions.. - - -So let us try :: - - var('x,alpha,y,beta') - x^2/alpha^2+y^2/beta^2 - -taking another example - - var('theta') - sin^2(theta)+cos^2(theta) - - -Similarly, we can define many algebraic and trigonometric expressions -using sage . - - -Sage also provides a few built-in constants which are commonly used in -mathematics . - -example : pi,e,infinity , Function n gives the numerical values of all these - constants. - -{{{ Type n(pi) - n(e) - n(oo) - On the sage notebook }}} - - - -If you look into the documentation of function "n" by doing - -.. #[Madhu: "documentation of the function "n"?] - -:: - n( - -You will see what all arguments it takes and what it returns. It will be very -helpful if you look at the documentation of all functions introduced through -this script. - - - -Also we can define the no. of digits we wish to use in the numerical -value . For this we have to pass an argument digits. Type - -.. #[Madhu: "no of digits"? Also "We wish to obtain" than "we wish to - use"?] -:: - - n(pi, digits = 10) - -Apart from the constants sage also has a lot of builtin functions like -sin,cos,log,factorial,gamma,exp,arcsin etc ... -lets try some of them out on the sage notebook. - - -:: - - sin(pi/2) - - arctan(oo) - - log(e,e) - - -Given that we have defined variables like x,y etc .. , We can define -an arbitrary function with desired name in the following way.:: - - var('x') - function('f',x) - - -Here f is the name of the function and x is the independent variable . -Now we can define f(x) to be :: - - f(x) = x/2 + sin(x) - -Evaluating this function f for the value x=pi returns pi/2.:: - - f(pi) - -We can also define functions that are not continuous but defined -piecewise. Let us define a function which is a parabola between 0 -to 1 and a constant from 1 to 2 . Type the following as given on the -screen - -:: - - - var('x') - h(x)=x^2 g(x)=1 - f=Piecewise( - -{{{ Show the documentation of Piecewise }}} - -:: - f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x) f - - - - -We can also define functions which are series - - -We first define a function f(n) in the way discussed above.:: - - var('n') - function('f', n) - - -To sum the function for a range of discrete values of n, we use the -sage function sum. - -For a convergent series , f(n)=1/n^2 we can say :: - - var('n') - function('f', n) - - f(n) = 1/n^2 - - sum(f(n), n, 1, oo) - - -Lets us now try another series :: - - - f(n) = (-1)^(n-1)*1/(2*n - 1) - sum(f(n), n, 1, oo) - - -This series converges to pi/4. - - -Moving on let us see how to perform simple calculus operations using Sage - -For example lets try an expression first :: - - diff(x**2+sin(x),x) - 2x+cos(x) - -The diff function differentiates an expression or a function. Its -first argument is expression or function and second argument is the -independent variable. - -We have already tried an expression now lets try a function :: - - f=exp(x^2)+arcsin(x) - diff(f(x),x) - -To get a higher order differential we need to add an extra third argument -for order :: - - diff( diff(f(x),x,3) - -in this case it is 3. - - -Just like differentiation of expression you can also integrate them :: - - x = var('x') - s = integral(1/(1 + (tan(x))**2),x) - s - - - -Many a times we need to find factors of an expression ,we can use the "factor" function - -:: - factor( - y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2) - f = factor(y) - -One can simplify complicated expression :: - - f.simplify_full() - -This simplifies the expression fully . We can also do simplification -of just the algebraic part and the trigonometric part :: - - f.simplify_exp() - f.simplify_trig() - - - -One can also find roots of an equation by using find_root function:: - - phi = var('phi') - find_root(cos(phi)==sin(phi),0,pi/2) - -Lets substitute this solution into the equation and see we were -correct :: - - var('phi') - f(phi)=cos(phi)-sin(phi) - root=find_root(f(phi)==0,0,pi/2) - f.substitute(phi=root) - -as we can see when we substitute the value the answer is almost = 0 showing -the solution we got was correct. - - - - -Lets us now try some matrix algebra symbolically :: - - - - var('a,b,c,d') - A=matrix([[a,1,0],[0,b,0],[0,c,d]]) - A - -Now lets do some of the matrix operations on this matrix - - -:: - A.det() - A.inverse() - - - -{{{ Part of the notebook with summary }}} - -So in this tutorial we learnt how to - - -* We learnt about defining symbolic expression and functions. -* Using built-in constants and functions. -* Using to see the documentation of a function. -* Simple calculus operations . -* Substituting values in expression using substitute function. -* Creating symbolic matrices and performing operation on them . - diff -r f0c93ea97e4c -r 9ced58c5c3b6 symbolics/slides.tex --- a/symbolics/slides.tex Thu Oct 21 18:22:07 2010 +0530 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,67 +0,0 @@ -% Created 2010-10-21 Thu 00:06 -\documentclass[presentation]{beamer} -\usetheme{Warsaw}\useoutertheme{infolines}\usecolortheme{default}\setbeamercovered{transparent} -\usepackage[latin1]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} -\usepackage{longtable} -\usepackage{float} -\usepackage{wrapfig} -\usepackage{soul} -\usepackage{amssymb} -\usepackage{hyperref} - - -\title{Plotting Data } -\author{FOSSEE} -\date{2010-09-14 Tue} - -\begin{document} - -\maketitle - - - - - - -\begin{frame} -\frametitle{Tutorial Plan} -\label{sec-1} -\begin{itemize} - -\item Defining symbolic expressions in sage.\\ -\label{sec-1.1}% -\item Using built-in costants and functions.\\ -\label{sec-1.2}% -\item Performing Integration, differentiation using sage.\\ -\label{sec-1.3}% -\item Defining matrices.\\ -\label{sec-1.4}% -\item Defining Symbolic functions.\\ -\label{sec-1.5}% -\item Simplifying and solving symbolic expressions and functions.\\ -\label{sec-1.6}% -\end{itemize} % ends low level -\end{frame} -\begin{frame} -\frametitle{Summary} -\label{sec-2} -\begin{itemize} - -\item We learnt about defining symbolic expression and functions.\\ -\label{sec-2.1}% -\item Using built-in constants and functions.\\ -\label{sec-2.2}% -\item Using to see the documentation of a function.\\ -\label{sec-2.3}% -\item Simple calculus operations .\\ -\label{sec-2.4}% -\item Substituting values in expression using substitute function.\\ -\label{sec-2.5}% -\item Creating symbolic matrices and performing operation on them .\\ -\label{sec-2.6}% -\end{itemize} % ends low level -\end{frame} - -\end{document}