From 07bf3e2c3142b4a973c11753ec08397bd2b63016 Mon Sep 17 00:00:00 2001 From: Amit Sethi Date: Tue, 26 Oct 2010 15:44:09 +0530 Subject: Added long answer type problems in all scripts --- statistics/script.rst | 5 +- symbolics/quickref.tex | 8 -- symbolics/script.rst | 277 ------------------------------------------------- symbolics/slides.tex | 67 ------------ 4 files changed, 3 insertions(+), 354 deletions(-) delete mode 100644 symbolics/quickref.tex delete mode 100644 symbolics/script.rst delete mode 100644 symbolics/slides.tex diff --git a/statistics/script.rst b/statistics/script.rst index 5398e21..5409abf 100644 --- a/statistics/script.rst +++ b/statistics/script.rst @@ -105,7 +105,7 @@ The following are the fields in any given line. * 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 @@ we have learnt 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 --git a/symbolics/quickref.tex b/symbolics/quickref.tex deleted file mode 100644 index b26d168..0000000 --- a/symbolics/quickref.tex +++ /dev/null @@ -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 --git a/symbolics/script.rst b/symbolics/script.rst deleted file mode 100644 index b9014bf..0000000 --- a/symbolics/script.rst +++ /dev/null @@ -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 --git a/symbolics/slides.tex b/symbolics/slides.tex deleted file mode 100644 index 4fc3634..0000000 --- a/symbolics/slides.tex +++ /dev/null @@ -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} -- cgit From bca2280d4b9f140fb627bedf5b0cec996274bb9e Mon Sep 17 00:00:00 2001 From: Amit Sethi Date: Tue, 26 Oct 2010 16:04:50 +0530 Subject: Reviewed manipulating strings --- manipulating-strings/script.rst | 15 +++++++++++---- progress.org | 2 +- 2 files changed, 12 insertions(+), 5 deletions(-) diff --git a/manipulating-strings/script.rst b/manipulating-strings/script.rst index 7873575..50d5e32 100644 --- a/manipulating-strings/script.rst +++ b/manipulating-strings/script.rst @@ -17,7 +17,7 @@ .. #. basic datatypes .. Author : Puneeth - Internal Reviewer : + Internal Reviewer : Amit External Reviewer : Checklist OK? : [2010-10-05] @@ -36,6 +36,7 @@ upper to lower case and vice-versa and joining a list of strings. .. #[punch: reversed returns an iterator. should we still teach it?] + We have an ``ipython`` shell open, in which we are going to work, through out this session. @@ -59,7 +60,7 @@ valid name of a day of the week or not. ``s`` could be in any of the forms --- sat, saturday, Sat, Saturday, -SAT, SATURDAY. We shall now be solving the problem only for the forms, +SAT, SATURDAY. For now, shall now be solving the problem only for the forms, sat and saturday. We shall solve it for the other forms, at the end of the tutorial. @@ -69,6 +70,7 @@ So, we need to check if the first three characters of the given string exists in the variable ``week``. As, with any of the string data-types, strings can be sliced into +.. #[Amit: Sequence data type???] sub-strings. To get the first three characters of s, we say, :: @@ -82,7 +84,7 @@ As we already know, the last element of the string can be accessed using ``s[-1]``. Following is an exercise that you must do. - +.. #[Amit: I don't know I am not sure about the sentence formation.] %%1%% Obtain the sub-string excluding the first and last characters from the string s. @@ -127,7 +129,7 @@ So, we obtain the reverse of s, by simply saying, :: s[::-1] - +.. #[amit: I think using reversed in not required after this] Now, to check if the string is ``s`` is palindromic, we say :: @@ -157,6 +159,9 @@ Let's try it out. Note that these methods, do not change the original string, but return a new string. +.. #[amit: I wish we could include this right when s.upper() is used so +.. that it is clear] + Following is an exercise that you must do. %%2%% Check if ``s`` is a valid name of a day of the week. Change the @@ -171,6 +176,8 @@ Please, pause the video here. Do the exercise and then continue. s.lower()[:3] in week +.. #[amit: May be a sentence or two about what our original problem was and +.. how this helps in solving it. One can loose the flow.] We just convert any input string to lower case and then check if it is present in the list ``week``. diff --git a/progress.org b/progress.org index 77573e4..806a080 100644 --- a/progress.org +++ b/progress.org @@ -40,7 +40,7 @@ | 6.5 LO: | Assessment | 3 | Anoop | | | |---------+----------------------------------------+-------+----------+---------------------------------------+-----------| | 7.1 LO: | manipulating lists | 3 | Madhu | | | -| 7.2 LO: | manipulating strings | 2 | Punch | Pending | | +| 7.2 LO: | manipulating strings | 2 | Punch | Amit | | | 7.3 LO: | getting started with tuples | 2 | Nishanth | | | | 7.4 LO: | dictionaries | 2 | Anoop | Pending | | | 7.5 LO: | sets | 2 | Nishanth | | | -- cgit From c8175a8b193cf0b093540c3b8bf0272c15c16d3a Mon Sep 17 00:00:00 2001 From: Amit Sethi Date: Tue, 26 Oct 2010 16:08:02 +0530 Subject: changed the name of symbolics to getting started with symbolics --- .../getting_started_with_lists.rst | 137 ---------- getting-started-with-symbolics/quickref.tex | 8 + getting-started-with-symbolics/script.rst | 277 +++++++++++++++++++++ getting-started-with-symbolics/slides.tex | 67 +++++ 4 files changed, 352 insertions(+), 137 deletions(-) delete mode 100644 getting-started-with-lists/getting_started_with_lists.rst create mode 100644 getting-started-with-symbolics/quickref.tex create mode 100644 getting-started-with-symbolics/script.rst create mode 100644 getting-started-with-symbolics/slides.tex diff --git a/getting-started-with-lists/getting_started_with_lists.rst b/getting-started-with-lists/getting_started_with_lists.rst deleted file mode 100644 index a82de02..0000000 --- a/getting-started-with-lists/getting_started_with_lists.rst +++ /dev/null @@ -1,137 +0,0 @@ -Hello friends and welcome to the tutorial on getting started with -lists. - - {{{ Show the slide containing title }}} - - {{{ Show the slide containing the outline slide }}} - -In this tutorial we will be getting acquainted with a python data -structure called lists. We will learn :: - - * How to create lists - * Structure of lists - * Access list elements - * Append elements to lists - * Deleting elements from lists - -List is a compound data type, it can contain data of other data -types. List is also a sequence data type, all the elements are in -order and there order has a meaning. - -We will first create an empty list with no elements. On your IPython -shell type :: - - empty = [] - type(empty) - - -This is an empty list without any elements. - -* Filled lists - -Lets now define a list, nonempty and fill it with some random elements. - -nonempty = ['spam', 'eggs', 100, 1.234] - -Thus the simplest way of creating a list is typing out a sequence -of comma-separated values (items) between square brackets. -All the list items need not have the same data type. - - - -As we can see lists can contain different kinds of data. In the -previous example 'spam' and 'eggs' are strings and 100 and 1.234 -integer and float. Thus we can put elements of heterogenous types in -lists. Thus list themselves can be one of the element types possible -in lists. Thus lists can also contain other lists. Example :: - - list_in_list=[[4,2,3,4],'and', 1, 2, 3, 4] - -We access list elements using the number of index. The -index begins from 0. So for list nonempty, nonempty[0] gives the -first element, nonempty[1] the second element and so on and -nonempty[3] the last element. :: - - nonempty[0] - nonempty[1] - nonempty[3] - -We can also access the elememts from the end using negative indices :: - - nonempty[-1] - nonempty[-2] - nonempty[-4] - --1 gives the last element which is the 4th element , -2 second to last and -4 gives the fourth -from last element which is first element. - -We can append elements to the end of a list using append command. :: - - nonempty.append('onemore') - nonempty - nonempty.append(6) - nonempty - -As we can see non empty appends 'onemore' and 6 at the end. - - - -Using len function we can check the number of elements in the list -nonempty. In this case it being 6 :: - - len(nonempty) - - - -Just like we can append elements to a list we can also remove them. -There are two ways of doing it. One is by using index. :: - - del(nonempty[1]) - - - -deletes the element at index 1, i.e the second element of the -list, 'eggs'. The other way is removing element by content. Lets say -one wishes to delete 100 from nonempty list the syntax of the command -should be :: - - a.remove(100) - -but what if their were two 100's. To check that lets do a small -experiment. :: - - a.append('spam') - a - a.remove('spam') - a - -If we check a now we will see that the first occurence 'spam' is removed -thus remove removes the first occurence of the element in the sequence -and leaves others untouched. - - -{{{Slide for Summary }}} - - -In this tutorial we came across a sequence data type called lists. :: - - * We learned how to create lists. - * How to access lists. - * Append elements to list. - * Delete Element from list. - * And Checking list length. - - - -{{{ Sponsored by Fossee Slide }}} - -This tutorial was created as a part of FOSSEE project. - -I hope you found this tutorial useful. - -Thank You - - - * Author : Amit Sethi - * First Reviewer : - * Second Reviewer : Nishanth diff --git a/getting-started-with-symbolics/quickref.tex b/getting-started-with-symbolics/quickref.tex new file mode 100644 index 0000000..b26d168 --- /dev/null +++ b/getting-started-with-symbolics/quickref.tex @@ -0,0 +1,8 @@ +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 --git a/getting-started-with-symbolics/script.rst b/getting-started-with-symbolics/script.rst new file mode 100644 index 0000000..b9014bf --- /dev/null +++ b/getting-started-with-symbolics/script.rst @@ -0,0 +1,277 @@ +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 --git a/getting-started-with-symbolics/slides.tex b/getting-started-with-symbolics/slides.tex new file mode 100644 index 0000000..4fc3634 --- /dev/null +++ b/getting-started-with-symbolics/slides.tex @@ -0,0 +1,67 @@ +% 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} -- cgit