 Welcome to the session of Expansion of Functions by using Taylor Series Part 2. This is Swati Nikam, Assistant Professor, Department of Humanities and Sciences, Valchur Institute of Technology, Sulapur. At the end of this session, students can create the Expansion of Functions about any point with the help of Taylor Series. Friends, in earlier video, I have explained you about Taylor Series. Let us see one of its expression. Assuming that f of x plus h can be expanded in ascending powers of h, it is expressed as f of x plus h is equal to f of x plus h times f dash of x plus h square by 2 factorial f double dash of x plus h cube upon 3 factorial f third derivative at x plus and so on plus h raise to n upon n factorial f nth derivative at x plus and so on. Let us call this representation number one as Taylor Series. Now, let us have some examples so that we will get use or implementation of this Taylor Series. Example number one, by using Taylor Series, find square root of 9.12 correct to 5 places of decimals. We know that Taylor Series is given by the representation I have already explained. Same I have represented over here and call it as representation number one. Here f of x plus h is equal to square root of x plus h which is equal to under root of 9.12 which is equal to under root of 9 plus 0.12. That we can rewrite it as f of x is equal to square root of x and h is equal to 0.12. Now, in order to use Taylor Series, let us find out some successive derivatives of f of x. So, f of x is square root of x which can be written as x raise to half. First order derivative f dash of x is equal to 1 by 2 into x raise to minus 1 by 2. Second derivative f double dash of x is equal to minus 1 by 4 into x raise to minus 3 by 2. Third order derivative f triple dash of x is equal to 3 by 8 into x raise to minus 5 by 2 and so on. Now, we have to put all these values in equation number one. So, f of x plus h is equal to square root of 9 plus 0.12 is equal to f of x is x raise to half plus h times f dash of x is half into x raise to minus half plus h square by 2 factorial into f double dash of x is minus 1 by 4 into x raise to minus 3 by 2 plus h cube upon 3 factorial into third order derivative 3 by 8 into x raise to minus 5 by 2 plus and so on. Now, let us substitute x is equal to 9 so that after putting value of x is equal to 9, we have f of 9 is equal to square root of 9 is equal to 3 f dash of 9 is equal to half into 9 raise to minus 1 by 2 which is 1 by 2 into 1 by 3 f double dash of 9 is equal to minus 1 by 4 into 9 raise to minus 3 by 2 which is equal to minus 1 by 4 into 1 by 27 f triple dash at 9 is equal to 3 by 8 into 9 raise to minus 5 by 2 which is equal to 3 by 8 into 1 upon 243. So, here in all these representation we have replaced x by 9 and got these values. So, that under root of 9.12 can be calculated with the help of Taylor series as f of x 9 is equal to 3 plus h that is 0.12 into f dash of 9 is 1 by 2 into 1 by 3 plus h square 0.12 whole square upon 2 factorial f double dash of 9 that is minus 1 by 4 into 1 by 27 plus h cube that is 0.12 whole cube upon 3 factorial into f triple dash at 9 that is 3 by 8 into 1 upon 243 plus and so on which is again equal to 3 plus. Now, let us do these calculations 0.12 divided by 3 into 6 minus 0.12 square divided by 2 factorial is 2 into 4 into 27 next plus 0.12 whole cube multiplied by 3 whole divided by 3 factorial is 6 into 8 multiplied by 243 plus and so on which is again equal to 3 as it is plus 0.02 minus 0.01 double 4 divided by 216 plus 0.005184 divided by double 1 double 64 plus and so on which is equal to 3 plus 0.02 minus 0.00006 plus 0.00000004 and so on. So, see the further terms are tending towards 0 and they becomes very negligible. Hence, we can neglect the higher order terms and hence we can get our solution as 3.0199. You can verify this answer by using before we proceed further. Friends, write the conversion of minutes to degree and conversion of degrees to radiance. I hope you have written your answer. So, we know that 16 minutes is equal to 1 degree and 1 degree into pi by 180 is equal to 0.01745 radiance. Now, let us solve one more example, example number 2. Using Taylor's theorem, find approximate value of sine 30 degrees and 30 minutes. Now, we know that Taylor's series is f of x plus h is equal to f of x plus h times f dash of x plus h square by 2 factorial f double dash of x plus h cube upon 3 factorial f triple dash of x plus and so on. Let us call this as equation number 1. Friends, always remember that we have three different forms of Taylor's series. According to your example, you have to choose the correct one and before you write your solution, you have to state that Taylor's series. So, we are going to use this representation of Taylor's series. Before we proceed, we first convert minutes to degrees and then degrees to radiance. So, it is like this. We know that 16 minutes is 1 degree and therefore 30 minutes is equal to half degree and 1 degree into pi by 180 is equal to 0.01745 radiance. Therefore, half degrees into pi by 180 is equal to 0.008727 radiance so that sine of 30 degrees 30 minutes becomes sine of 30 degrees plus half degree which is again shifted to radiance and as sine of pi by 6 plus 0.0087. Now, here f of x given function is supposed to be sine x and h is equal to 0.0087. In order to use Taylor's series for the function f of x, let us calculate its successive derivatives of few orders. f of x is sine x, so f dash of x is cos x, f double dash of x is minus sine x, f triple dash of x is minus cos x and so on. Now, putting all these values in equation number 1 that is Taylor's series, we get representation for sine of 30 degrees 30 minutes as sine x which is f of x plus h times f dash of x is cos x plus h square by 2 factorial into f double dash of x is minus sine x plus h cube upon 3 factorial into f triple dash of x is minus cos x plus and so on. Now, putting x is equal to pi by 6 and h is equal to 0.0087 in this representation of Taylor's series we get therefore, sine of 30 degrees 30 minutes is equal to sine of pi by 6 plus 0.0087 which is h into cos of pi by 6 plus h square that is 0.0087 whole square upon 2 factorial into minus sine of pi by 6 plus h cube is 0.0087 whole cube divided by 3 factorial into minus cos of pi by 6. So, in this slide I have already calculated the values of function f and their successive representation at pi by 6. So, f of x gives f of pi by 6 as sine of pi by 6 which is 1 by 2 f dash of pi by 6 is cos of pi by 6 that is root 3 by 2 f double dash of pi by 6 is minus sine of pi by 6 which is minus 1 by 2 f triple dash of pi by 6 is minus cos of pi by 6 which is minus under root of 3 by 2. Now, let us put all these values in the Taylor series representation here so that we get first term sine of pi by 6 as 1 by 2 plus 0.0087 into under root of 3 by 2 plus 0.0087 whole square upon 2 into minus half plus 0.0087 whole cube upon 6 into minus under root of 3 by 2 plus and so on and therefore, after calculating this we get sine of 30 degrees 30 minutes is equal to 0.5 plus 0.00753 minus 0.00001 minus 0.0000006 plus and so on. So, see if you observe the nature of terms again they becomes negligible and hence we can neglect the higher order terms and therefore, sine of 30 degrees 30 minutes is equal to 0.5075. So, again you can verify your answer with the correct one by calculating this value on your calculator preferred a textbook called Higher Engineering Mathematics by P. S. Grewal.