Difference between revisions of "Scilab/C4/Integration/English"

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(Created page with ''''Title of script''': Numerical Methods for Integration '''Author: Shamika''' '''Keywords: Integration, Numerical Methods, integral''' {| style="border-spacing:0;" ! <center…')
 
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'''Keywords: Integration, Numerical Methods, integral'''
 
'''Keywords: Integration, Numerical Methods, integral'''
 +
  
  
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|-
 
|-
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 7,8- Composite Trapezoidal Rule-I
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 7,8- Composite Trapezoidal Rule-I
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| '''Composite Trapezoidal Rule''' is
+
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| '''Let us study Composite Trapezoidal Rule. This rule''' is
  
 
* The extension of '''trapezoidal rule'''
 
* The extension of '''trapezoidal rule'''
* We divide the interval '''a comma b '''into n equal intervals  
+
* We divide the interval '''a comma b '''into '''n''' equal intervals  
 
* Then,
 
* Then,
 
* '''h equal to b minus a divided by n''' is the common length of the intervals  
 
* '''h equal to b minus a divided by n''' is the common length of the intervals  
* Then '''composite trapezoidal rule '''is given by <br/> <nowiki>[</nowiki>'''The integral of the function F of x in the interval a to b is approximately equal to h multiplied by the sum of the values of the function at x zero to x n'''<br/>
+
* Then '''composite trapezoidal rule '''is given by  
 
+
* '''The integral of the function F of x in the interval a to b is approximately equal to h multiplied by the sum of the values of the function at x zero to x n'''
  
  
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| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let us solve an example using '''composite trapezoidal rule''':
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let us solve an example using '''composite trapezoidal rule''':
* Assume the number of intervals n is equal to 10.
+
* Assume the number of intervals '''n''' is equal to ten.
  
  
  
 
|-
 
|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show the code for Trap_composite.sci on Scilab Editor
+
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab editor
 +
 
 +
 
 +
Highlight
 +
 
 +
'''<nowiki>function [I1] </nowiki><nowiki>= Trap_composite(f, a, b, n)</nowiki>'''
 +
 
 +
'''x = linspace(a, b, n+1)'''
 +
 
 +
 
 +
'''I1 = (h/2)*(2*sum(f(x)) - f(x(1)) - f(x(n+1)))'''
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let us look at the code for C'''omposite Trapezoidal Rule '''on''' Scilab Editor'''
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let us look at the code for C'''omposite Trapezoidal Rule '''on''' Scilab Editor'''
 
* We first define the function with parameters''' f , a , b , n. f refers to the function we have to solve, a is the lower limit of the integral, b is the upper limit of the integral and n is the number of intervals. '''
 
* We first define the function with parameters''' f , a , b , n. f refers to the function we have to solve, a is the lower limit of the integral, b is the upper limit of the integral and n is the number of intervals. '''
* '''linspace''' function is used to create 10 equal intervals between 0 and 1
+
* '''linspace''' function is used to create ten equal intervals between zero and one
* '''We find the value of the integral and store it in I1'''
+
* '''We find the value of the integral and store it in I one'''
  
  
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| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Define the example function by typing:
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Define the example function by typing:
 
* '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one by open paranthesis two asterisk x plus one close paranthesis close quote close paranthesis'''
 
* '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one by open paranthesis two asterisk x plus one close paranthesis close quote close paranthesis'''
* Press enter
+
* Press '''enter'''
 
* Type  
 
* Type  
 
* '''Trap underscore composite open paranthesis f comma zero comma one comma ten close paranthesis'''
 
* '''Trap underscore composite open paranthesis f comma zero comma one comma ten close paranthesis'''
* Press enter
+
* Press '''enter'''
 
* The answer is displayed on the console
 
* The answer is displayed on the console
  
Line 108: Line 119:
 
|-
 
|-
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 10, 11- Composite Simpson's Rule
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 10, 11- Composite Simpson's Rule
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| In '''Composite simpson's rule''', we  
+
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Next we shall study '''Composite simpson's rule. In this rule''' we  
  
* decompose the interval''' '''<nowiki>[a comma b]</nowiki> into '''''n is greater than 1 '''''subintervals of equal length  
+
* decompose the interval''' a comma b''' into '''''n is greater than 1 '''''subintervals of equal length  
 
* Apply '''Simpson's rule''' to each interval
 
* Apply '''Simpson's rule''' to each interval
* We get the value of the integral to be<br/> <nowiki>[</nowiki>'''h by 3 multiplied by the sum of f zero, 4 into f one , 2 into f two to f n''']
+
* We get the value of the integral to be
 +
* '''h by three multiplied by the sum of f zero, four into f one , two into f two to f n'''
  
  
  
 
|-
 
|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 12- Example
+
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 12- Example  
  
  
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| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let us solve an example using '''Composite Simpson's rule'''
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let us solve an example using '''Composite Simpson's rule'''
 
* We are given a '''function one by one plus x cube d x in the interval one to two'''
 
* We are given a '''function one by one plus x cube d x in the interval one to two'''
* Let the number of intervals be 20
+
* Let the number of intervals be '''twenty'''
  
  
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|-
 
|-
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab Editor and show the code for Simp_composite.sci
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab Editor and show the code for Simp_composite.sci
 +
 +
 +
Highlight
 +
 +
'''function I = Simp_composite(f, a, b, n)'''
 +
 +
 +
'''for i = 1:(n/2)-1 '''
 +
 +
'''x1(i) = x(2*i) '''
 +
 +
'''end '''
 +
 +
'''for j = 2:n/2 '''
 +
 +
'''x2(i) = x(2*i-1) '''
 +
 +
'''end'''
 +
 +
 +
'''I = (h/3)*(f(x(1)) + 2*sum(f(x1)) + 4*sum(f(x2)) + f(x(n)))'''
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let us look at the code for '''Composite simpson's rule'''
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let us look at the code for '''Composite simpson's rule'''
 
* '''We first define the function with parameters f , a , b , n. '''<br/> '''f refers to the function we have to solve, a is the lower limit of the integral, b is the upper limit of the integral and n is the number of intervals.'''
 
* '''We first define the function with parameters f , a , b , n. '''<br/> '''f refers to the function we have to solve, a is the lower limit of the integral, b is the upper limit of the integral and n is the number of intervals.'''
 
* We find two sets of points
 
* We find two sets of points
* We find the value of the function with one set and multiply it with 2
+
* We find the value of the function with one set and multiply it with '''two'''
* With the other set we find the value and multiply it with 4
+
* With the other set we find the value and multiply it with '''four'''
* We sum these values and multiply it with h by 3 and store the final value in I
+
* We sum these values and multiply it with '''h by three and store the final value in I'''
 
* Let us execute the code
 
* Let us execute the code
  
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'''<nowiki>deff ('[y]=f(x)','y=sin*x+sin*(2*x)')</nowiki>'''
+
'''<nowiki>deff ('[y]=f(x)','y=1/(1+x^3)')</nowiki>'''
  
  
'''Simp_composite( f, 0, %pi, 20)'''
+
'''Simp_composite( f, 1, 2 20)'''
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let me clear the screen first.
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let me clear the screen first.
 
* Define the function given in the example by typing
 
* Define the function given in the example by typing
* '''<nowiki>[d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to sine asterisk x plus sine asterisk open paranthesis two asterisk x close paranthesis close quote close paranthesis]</nowiki>'''
+
* '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one divided by open paranthesis one plus x cube close paranthesis close quote close paranthesis'''
* Press enter
+
* Press '''enter'''
 
* '''Type Simp underscore composite open paranthesis f comma one comma two comma twenty close paranthesis'''
 
* '''Type Simp underscore composite open paranthesis f comma one comma two comma twenty close paranthesis'''
* Press enter
+
* Press '''enter'''
 
* The answer is displayed on the console
 
* The answer is displayed on the console
  
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| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us now look at '''Composite Midpoint Rule. It'''
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us now look at '''Composite Midpoint Rule. It'''
 
* Integrates polynomials of degree one or less
 
* Integrates polynomials of degree one or less
* Divides the interval <nowiki>[ a comma b ]into n subintervals of equal width</nowiki>
+
* Divides the interval '''a comma b into n subintervals''' of equal width
* Finds the '''midpoint '''of each interval indicated by x i  
+
* Finds the '''midpoint '''of each interval indicated by '''x i '''
* We find the sum of the values of the integral at each midpoint <br/>
+
* We find the sum of the values of the integral at each midpoint  
 
+
  
  
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| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us solve this problem using '''Composite Midpoint Rule'''
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us solve this problem using '''Composite Midpoint Rule'''
 
* '''We are given a function one minus x square d x in the interval zero to one point five'''
 
* '''We are given a function one minus x square d x in the interval zero to one point five'''
* We assume n is equal to 20
+
* We assume '''n''' is equal to''' twenty'''
  
  
Line 200: Line 232:
  
 
Show the file mid_composite.sci
 
Show the file mid_composite.sci
 +
 +
 +
Highlight
 +
 +
 +
'''function I = mid_composite(f, a, b, n)'''
 +
 +
 +
'''x = linspace(a + h/2, b - h/2, n) '''
 +
 +
 +
'''I = h*sum(f(x))'''
  
  
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| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let me clear the screen
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Let me clear the screen
 
* We define the function given in the example by typing  
 
* We define the function given in the example by typing  
* '''<nowiki>[d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one minus x square close quote close paranthesis]</nowiki>'''
+
* '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one minus x square close quote close paranthesis'''
* Press enter
+
* Press '''enter'''
* Then type <br/> '''<nowiki>[mid underscore composite open paranthesis f comma zero comma one point five comma twenty close paranthesis]</nowiki>'''
+
* Then type <br/> '''mid underscore composite open paranthesis f comma zero comma one point five comma twenty close paranthesis'''
* Press enter
+
* Press '''enter'''
 
* The answer is displayed on the console
 
* The answer is displayed on the console
  
Line 294: Line 338:
  
 
|-
 
|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 19'''
+
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 12'''
  
 
'''Title: Acknowledgement'''  
 
'''Title: Acknowledgement'''  
Line 304: Line 348:
 
* More information on this Mission is available at  
 
* More information on this Mission is available at  
  
* [http://spoken-tutorial.org/NMEICT-Intro http://spoken-][http://spoken-tutorial.org/NMEICT-Intro tutorial.org/NMEICT-][http://spoken-tutorial.org/NMEICT-Intro Intro]  
+
* [http://spoken-tutorial.org/NMEICT-Intro http://spoken-tutorial.org/NMEICT-][http://spoken-tutorial.org/NMEICT-Intro Intro]  
  
  
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Spoken Tutorial Project is a part of the Talk to a Teacher project  
 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * 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  
 
* It is supported by the National Mission on Education through ICT, MHRD, Government of India  
 
 
* More information on this Mission is available at  
 
* More information on this Mission is available at  
 +
* spoken hyphen tutorial dot org slash NMEICT hyphen Intro <br/>
  
* spoken hyphen tutorial dot org slash NMEICT hyphen Intro
+
 
 +
 
 +
 
 +
|-
 +
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"|
 +
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * This is Ashwini Patil signing off. Thank you for joining.
  
  
  
 
|}
 
|}

Revision as of 04:46, 18 December 2013

Title of script: Numerical Methods for Integration

Author: Shamika

Keywords: Integration, Numerical Methods, integral


Visual Cue
Narration
Slide 1 Dear Friends,

Welcome to the Spoken Tutorial on “ Composite Numerical Integration

Slide 2,3 -Learning Objective Slide At the end of this tutorial, you will learn how to:
  • Develop Scilab code for different Composite Numerical Integration algorithms
  • Divide the integral into equal intervals
  • Apply the algorithm to each interval
  • Calculate the composite value of the integral


Slide 4-System Requirement slide * To record this tutorial, I am using Ubuntu 12.04 as the operating system with Scilab 5.3.3 version


Slide 5- Prerequisites slide * Before practising this tutorial, a learner should have basic knowledge of Scilab and Integration using Numerical Methods
  • For Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.


Slide 6- Numerical Integration Numerical Integration is the:
  • Study of how the numerical value of an integral can be found
  • It is used when exact mathematical integration is not available
  • It approximates a definite integral from values of the
    integrand


Slide 7,8- Composite Trapezoidal Rule-I Let us study Composite Trapezoidal Rule. This rule is
  • The extension of trapezoidal rule
  • We divide the interval a comma b into n equal intervals
  • Then,
  • h equal to b minus a divided by n is the common length of the intervals
  • Then composite trapezoidal rule is given by
  • The integral of the function F of x in the interval a to b is approximately equal to h multiplied by the sum of the values of the function at x zero to x n


Slide 9- Example


* Let us solve an example using composite trapezoidal rule:
  • Assume the number of intervals n is equal to ten.


Switch to Scilab editor


Highlight

function [I1] = Trap_composite(f, a, b, n)

x = linspace(a, b, n+1)


I1 = (h/2)*(2*sum(f(x)) - f(x(1)) - f(x(n+1)))

* Let us look at the code for Composite Trapezoidal Rule on Scilab Editor
  • We first define the function with parameters f , a , b , n. f refers to the function we have to solve, a is the lower limit of the integral, b is the upper limit of the integral and n is the number of intervals.
  • linspace function is used to create ten equal intervals between zero and one
  • We find the value of the integral and store it in I one


Click on Execute on Scilab editor and choose Save and Execute the code * Click on Execute on Scilab editor and choose Save and Execute the code


Switch to Scilab Console


deff ('[y]=f(x)','y=1/(2*x+1)')


Trap_composite(f, 0, 1, 10)

* Define the example function by typing:
  • d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one by open paranthesis two asterisk x plus one close paranthesis close quote close paranthesis
  • Press enter
  • Type
  • Trap underscore composite open paranthesis f comma zero comma one comma ten close paranthesis
  • Press enter
  • The answer is displayed on the console


Slide 10, 11- Composite Simpson's Rule Next we shall study Composite simpson's rule. In this rule we
  • decompose the interval a comma b into n is greater than 1 subintervals of equal length
  • Apply Simpson's rule to each interval
  • We get the value of the integral to be
  • h by three multiplied by the sum of f zero, four into f one , two into f two to f n


Slide 12- Example


* Let us solve an example using Composite Simpson's rule
  • We are given a function one by one plus x cube d x in the interval one to two
  • Let the number of intervals be twenty


Switch to Scilab Editor and show the code for Simp_composite.sci


Highlight

function I = Simp_composite(f, a, b, n)


for i = 1:(n/2)-1

x1(i) = x(2*i)

end

for j = 2:n/2

x2(i) = x(2*i-1)

end


I = (h/3)*(f(x(1)) + 2*sum(f(x1)) + 4*sum(f(x2)) + f(x(n)))

* Let us look at the code for Composite simpson's rule
  • We first define the function with parameters f , a , b , n.
    f refers to the function we have to solve, a is the lower limit of the integral, b is the upper limit of the integral and n is the number of intervals.
  • We find two sets of points
  • We find the value of the function with one set and multiply it with two
  • With the other set we find the value and multiply it with four
  • We sum these values and multiply it with h by three and store the final value in I
  • Let us execute the code


Click on Execute and choose

Save and execute the file

Simp_composite.sci

* Save and execute the file
  • Simp underscore composite dot s c i


Switch to Scilab Console

Type


clc


deff ('[y]=f(x)','y=1/(1+x^3)')


Simp_composite( f, 1, 2 20)

* Let me clear the screen first.
  • Define the function given in the example by typing
  • d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one divided by open paranthesis one plus x cube close paranthesis close quote close paranthesis
  • Press enter
  • Type Simp underscore composite open paranthesis f comma one comma two comma twenty close paranthesis
  • Press enter
  • The answer is displayed on the console


Slide 13, 14- Composite Midpoint Rule Let us now look at Composite Midpoint Rule. It
  • Integrates polynomials of degree one or less
  • Divides the interval a comma b into n subintervals of equal width
  • Finds the midpoint of each interval indicated by x i
  • We find the sum of the values of the integral at each midpoint


Slide 15- Example


Let us solve this problem using Composite Midpoint Rule
  • We are given a function one minus x square d x in the interval zero to one point five
  • We assume n is equal to twenty


Switch to Scilab Editor


Show the file mid_composite.sci


Highlight


function I = mid_composite(f, a, b, n)


x = linspace(a + h/2, b - h/2, n)


I = h*sum(f(x))


* Let us look at the code for Composite Midpoint rule
  • We first define the function with parameters f , a , b , n.
    f refers to the function we have to solve, a is the lower limit of the integral, b is the upper limit of the integral and n is the number of intervals.
  • We find the midpoint of each interval
  • Find the value of integral at each midpoint and then find the sum and store it in I.
  • Let us now solve the example


Click on Execute and choose

Save and execute the file mid_composite.sci

* Save and execute the file mid underscore composite dot s c i


On the Scilab Console type:


clc


deff ('[y]=f(x)','y=1-x^2')


Type mid_composite(f, 0, 1.5, 20)

* Let me clear the screen
  • We define the function given in the example by typing
  • d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one minus x square close quote close paranthesis
  • Press enter
  • Then type
    mid underscore composite open paranthesis f comma zero comma one point five comma twenty close paranthesis
  • Press enter
  • The answer is displayed on the console


Slide 16- Summary Let us summarize this tutorial. In this tutorial we have learnt to:
  • Develop Scilab code for numerical integration
  • Find the value of an integral


Show Slide 17

Title: About the Spoken Tutorial Project

  • It summarises the Spoken Tutorial project
  • If you do not have good bandwidth, you can download and watch it


* Watch the video available at the following link
  • It summarises the Spoken Tutorial project
  • If you do not have good bandwidth, you can download and watch it


Show Slide 18

Title: Spoken Tutorial Workshops

The Spoken Tutorial Project Team

  • Conducts workshops using spoken tutorials
  • Gives certificates for those who pass an online test
  • For more details, please write to contact@spoken-tutorial.org


The Spoken Tutorial Project Team
  • Conducts workshops using spoken tutorials
  • Gives certificates for those who pass an online test
  • For more details, please write to contact at spoken hyphen tutorial dot org


Show Slide 12

Title: Acknowledgement

  • 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


* 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
  • spoken hyphen tutorial dot org slash NMEICT hyphen Intro



* This is Ashwini Patil signing off. Thank you for joining.


Contributors and Content Editors

Lavitha Pereira, Nancyvarkey