MATHS ASSIGNMENT.docx
- 1. MATHS ASSIGNMENT NAME : V.CHARAN CHANDRA SEC : M TOPIC : Explain The Application of Taylor’s and Maclaurin to find value in the calculation.
- 2. INTRODUCTION TAYLOR’S SERIES : The Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function’s derivatives at a single point.
- 3. MACLAURIN SERIES : A Maclaurin series is a function that has expansion series that gives the sum of derivatives of that function. An Maclaurin series is a Taylor’s series of function ‘0’.
- 4. LITERATURE SURVEY The Taylor series is named for mathematician Brook Taylor, who first published the power series formula in 1715. CONSTRUCTION OF TAYLOR’S SERIES: Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the Taylor series as-is and generalize the form of the coefficients, or one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series (such as those above) to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the
- 5. Taylor series by repeatedly applying integration by parts. The use of computer algebra systems to calculate Taylor series is common, since it eliminates tedious substitution and manipulation. The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor’s Theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent.
- 6. METHODOLOGY Problem for Solving Taylor’s series : Q. Determine the Taylor series at x=0 for f(x) = ex Sol. F(x) = ex Differentiate the given equation, F’(x) = e^x F’’(x) =e^x F’’’(x) = e^x At x=0, we get F’(0) = e^0 =1 F’’(0) = e^0=1 F’’’(0) = e^0 = 1 When Taylor series at x= 0, then the Maclaurin series is
- 7. PROBLEMS ON MACLAURIN SERIES : Expanding e^x:
- 8. CONCLUSION Maclaurin Series displays the expansion series for the given function. How to Use the Maclaurin Series in Calculator? The procedure to use the Maclaurin series calculator is as follows: Step 1: Enter two functions in the respective input field Step 2: Now click the button “=” to get the result Step 3: Finally, the expansion series for the given function will be displayed in the Display bar. What is Meant by Maclaurin’s Series? In Mathematics, the Maclaurin series is defined as the expanded series of the given function. In this series, the approximate value of the function can be calculated as the sum of the derivatives of the function. This method is sometimes called Taylor’s
- 9. series if the function is expanded around zero, rather than some other values. --------------------------------------------------------------------- Taylor Series displays the Taylor series for the given function and the limit. How to Use the Taylor Series in Calculator? The procedure to use the Taylor series calculator is as follows: Step 1: Enter the function and the limit in the respective input field Step 2: Now click the button “=” to get the series Step 3: Finally, the Taylor series for the given function will be displayed in the new window What is Meant by the Taylor Series? In Mathematics, the Taylor series is defined as the expression for the given function as an infinite series, in which the terms are expressed for the value of the function’s derivative at a single point.
- 10. The Taylor series is used to draw the conclusion of what a function looks like. There is a special case of a Taylor series called the Maclaurin’s Series. This series helps to reduce many mathematical proofs and are used in power flow analysis. THANK YOU