<transcript><audio_94>s being greater than zero and the interior of the unit circle corresponds to the left half of the z plane there is a one to one there in fact its not even too difficult to see that its</transcript><transcript><audio_95>you have a one to one mapping between s and z not just on the imaginary axis but everywhere you have got an inverse so you know you can go from s to z or you can go from z to s</transcript><caption><image_96>The image is a whiteboard with a mathematical equation written on it. The equation is written in black marker and is written on a piece of paper. The paper is white and the equation is in black ink. 

The equation is for the equation "2(x) = 2(2) + 2(1) = (-2) = +2 + 2" and it is written as "Num Den" in English. Below the equation, there is a note that reads "1 = 1/2/2k1" in Hindi.

On the right side of the image, there are two logos, one of which is the logo of the Indian Institute of Science and Technology (CDEEP) and the other is the emblem of the institute. The image appears to be taken in a classroom or a similar setting.</caption><transcript><audio_97>so we have luckily arrived at the</transcript><transcript><audio_98>s is 1 minus z inverse by 1</transcript><caption><image_99>The image is a whiteboard with black text written on it. The text reads "This transformation meets our requirements". Below the text, there is a formula written in black ink that reads "8 = -7". The formula is written in a simple, sans-serif font and is centered on the whiteboard. The background of the image is blurred, but it appears to be a desk or a table with a computer monitor and other office supplies.</caption><caption><image_100>The image is a whiteboard with black text written on it. The text reads "This transformation meets our requirements". Below the text, there is a mathematical equation written in black marker. The equation is written in a simple, sans-serif font and is centered on the whiteboard. The background of the image is plain white. On the top right corner, there are two logos, one for the company "CCEP" and the other for "CDEEP". On the bottom left corner, the company's logo is also visible.</caption><transcript><audio_101>now as a point for further reflection i put the following question to you how do you interpret this as an approximation of derivatives in terms of shifts</transcript><caption><image_102>The image is a photograph of a whiteboard with black text written on it. The text reads "This transformation meets our requirements" and is written in a cursive font. Below the text, there is a formula written in black ink that reads "8 = 1 = 2 = 1". The formula appears to be a mathematical equation, as there are two equations written on the whiteboard. The background of the image is white, and there is the logo of the company "CCEEP IT Bombay" on the top right corner. The logo is a shield with a blue background and a white border.</caption><transcript><audio_103>and the hint is to expand this in terms of a power series of course you have one minus z inverse but one by one plus z inverse can be expanded as a power series and that will give you some insight on what you mean</transcript><transcript><audio_104>on what are we trying to do in terms of approximating derivatives by shifts its not too simple but its interesting of course thats besides the main point of the discussion we have got what we wanted we are happy we have got a rational function</transcript> s being greater than zero and the interior of the unit circle corresponds to the left half of the z plane there is a one to one there in fact its not even too difficult to see that its you have a one to one mapping between s and z not just on the imaginary axis but everywhere you have got an inverse so you know you can go from s to z or you can go from z to s so we have luckily arrived at the s is 1 minus z inverse by 1 now as a point for further reflection i put the following question to you how do you interpret this as an approximation of derivatives in terms of shifts and the hint is to expand this in terms of a power series of course you have one minus z inverse but one by one plus z inverse can be expanded as a power series and that will give you some insight on what you mean on what are we trying to do in terms of approximating derivatives by shifts its not too simple but its interesting of course thats besides the main point of the discussion we have got what we wanted we are happy we have got a rational function
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