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diff --git a/using_sage_for_calculus/script.rst b/using_sage_for_calculus/script.rst new file mode 100644 index 0000000..f465248 --- /dev/null +++ b/using_sage_for_calculus/script.rst @@ -0,0 +1,327 @@ +.. Objectives +.. ---------- + +.. By the end of this tutorial you will -- + +.. 1. Get an idea of the range of things for which Sage can be used. +.. #. Know some of the functions for Calculus +.. #. Get some insight into Graphs in Sage. + + +.. Prerequisites +.. ------------- + +.. Getting Started -- Sage + +Script +------ + +.. L1 + +{{{ Show the title slide }}} + +.. R1 + +Hello Friends and Welcome to the tutorial on 'Using Sage for Calculus'. + +.. L2 + +{{{ show the 'objectives' slide }}} + +.. R2 + +At the end of this tutorial, you will be able to, + + 1. Learn the range of things for which Sage can be used. + #. Perform integrations & other Calculus in Sage. + #. Perform matrix algebra in sage. + +.. L3 + +{{{ show the 'pre-requisite' slide }}} + +.. R3 + +Before beginning this tutorial,we would suggest you to complete the +tutorial on "Getting started with Sage". + +Let us begin with Calculus. We shall be looking at limits, +differentiation, integration, and Taylor polynomial. + +.. L4 + +{{{ open sage notebook }}} + +.. R4 + +We have our Sage notebook running. In case, you don't have it running, +start is using the command, ``sage --notebook``. + +.. R5 + +To begin with, let us find the limit of the function x*sin(1/x), at x=0. +To do this we say + +.. L5 +:: + + lim(x*sin(1/x), x=0) + +.. R6 + +As expected, we get the limit to be 0. + +It is also possible to limit a point from one direction. For +example, let us find the limit of 1/x at x=0, when approaching from +the positive side. + +.. L6 +:: + + lim(1/x, x=0, dir='right') + +.. R7 + +We get the limit from positive side. +To find the limit from the negative side, we say, + +.. L7 +:: + + lim(1/x, x=0, dir='left') + +.. L8 + +{{{ Show the 'differential expression' slide }}} + +.. R8 + +Let us now see how to perform differentiation, using Sage. We shall +find the differential of the expression ``exp(sin(x^2))/x`` w.r.t ``x``. +For this, we shall first define the expression, and then use the ``diff`` +function to obtain the differential of the expression. So, switch to the sage +notebook and type + +.. L9 +:: + + var('x') + f = exp(sin(x^2))/x + diff(f, x) + +.. R9 + +And we get the expected differential of the expression. + +.. L10 + +{{{ Show the slide 'Partial Differentiation' }}} + +.. R10 + +We can also obtain the partial differentiation of an expression with one of the +vriables. Let us differentiate the expression +``exp(sin(y - x^2))/x`` w.r.t x and y. Switch to sage notebook and type + +.. L11 +:: + + var('x y') + f = exp(sin(y - x^2))/x + diff(f, x) + diff(f, y) + +.. R11 + +Thus we get our partial differential solution. + +.. L12 + +{{{ Show the 'integration' slide }}} + +.. R12 + +Now, let us look at integration. We shall use the expression obtained +from the differentiation that we calculated before, ``diff(f, y)`` +which gave us the expression ---``cos(-x^2 + y)*e^(sin(-x^2 + y))/x``. +The ``integrate`` command is used to obtain the integral of an +expression or function. So, switch to sage notebook and type. + +.. L13 +{{{ Switch to sage }}} +:: + + integrate(cos(-x^2 + y)*e^(sin(-x^2 + y))/x, y) + +.. R13 + +As we can see, we get back the correct expression. The minus sign being +inside or outside the ``sin`` function doesn't change much. + +Now, let us find the value of the integral between the limits 0 and +pi/2. + +.. L14 +:: + + integral(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y, 0, pi/2) + +.. R14 + +Hence we get our solution for the definite integration. +Let us now see how to obtain the Taylor expansion of an expression +using sage. We will obtain the Taylor expansion of ``(x + 1)^n`` up to +degree 4 about 0. + +.. L15 +:: + + var('x n') + taylor((x+1)^n, x, 0, 4) + +.. R15 + +We easily got the Taylor expansion,using the function ``taylor()``. +This brings us to the end of the features of Sage for Calculus, that +we will be looking at. + +.. L16 + +{{{ Show the 'More on Calculus' slide }}} + +.. R16 + +For more on calculus you may look at the Calculus quick-ref from the Sage +documentation at the given link. + +.. L17 + +{{{ show the 'Equation' slide }}} + +.. R17 + +Next let us move on to Matrix Algebra. +Let us begin with solving the equation ``Ax = v``, where A is the +matrix ``matrix([[1,2],[3,4]])`` and v is the vector +``vector([1,2])``. + +.. R18 + +To solve the equation, ``Ax = v`` we simply say + +.. L18 + +{{{ Switch back to sage notebook page }}} +:: + + A = matrix([[1,2], + [3,4]]) + v = vector([1,2]) + x = A.solve_right(v) + x + +.. R19 + +To solve the equation, ``xA = v`` we simply say. +The left and right here, denote the position of ``A``, relative to x. + +.. L19 +:: + + x = A.solve_left(v) + x + +.. L20 + +{{{ show the 'Summary' slide }}} + +.. R20 + +This brings us to the end of this tutorial. In this tutorial we have learned to + +1. Use functions like lim(), integrate(), integral(), solve() +#. Use sage for performing matrix algebra, integrations & other calculus +operations using the above mentioned functions. + +.. L21 + +{{{ Show the 'Evaluation' slide }}} + +.. R21 + +Here are some self assessment questions for you to solve. + + 1. How do you find the limit of the function x/sin(x) as x tends to 0 from the + negative side. + + #. Solve the system of linear equations + x-2y+3z = 7 + 2x+3y-z = 5 + x+2y+4z = 9 + +Try the xercises and switch to next slide for solutions. + +.. L22 + +{{{ Show the 'Solutions' slide }}} + +.. R22 + + 1. To find the limit of the function x/sin(x) as x tends to 0 from negative +side, use the lim function as: lim(x/sin(x), x=0, dir'left') + + #. A = Matrix([1, -2, 3], [2, 3, -1], [1, 2, 4]]) + b = vector([7, 5, 9]) + x = A.solve_right(b) + x + +.. L23 + +{{{ Show the 'FOSSEE' slide }}} + +.. R23 + +FOSSEE is Free and Open-source Software for Science and Engineering Education. +The goal of this project is to enable all to use open source software tools. +For more details, please visit the given link. + +.. L24 + +{{{ Show the 'About the Spoken Tutorial Project' slide }}} + +.. R24 + +Watch the video available at the following link. It summarizes the Spoken +Tutorial project. If you do not have good bandwidth, you can download and +watch it. + +.. L25 + +{{{ Show the 'Spoken Tutorial Workshops' slide }}} + +.. R25 + +The Spoken Tutorial Project Team conducts workshops using spoken tutorials, +gives certificates to those who pass an online test. + +For more details, please write to contact@spoken-tutorial.org + +.. L26 + +{{{ Show the 'Acknowledgements' slide }}} + +.. R26 + +Spoken Tutorial Project is a part of the "Talk to a Teacher" project. +It is supported by the National Mission on Education through ICT, MHRD, +Government of India. More information on this mission is available at the +given link. + +.. L27 + +{{{Show the 'Thank you' slide }}} + +.. R27 + +Hope you have enjoyed this tutorial and found it useful. +Thank you! |