
Obtain Taylor's series expansion for about the point upto the fourth degree term.
Answer
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Hint: We will look at the Taylor's series expansion for a function about a point . Then we will find the derivative of the given function. As Taylor's series expansion has terms with higher-order derivatives, we will compute them for the given function. We will then substitute the value in the derivatives obtained. We will put the obtained results in Taylor's series expansion.
Complete step-by-step solution
The Taylor's series expansion a function about a point is given by
The given function is . We have to find the Taylor's series expansion of the given function upto the fourth degree term. So, we will compute upto the fourth derivative of the given function.
The value of the function at is .
The first derivative of is . The value of the first derivative at is .
The second derivative of is . The value of the second derivative at is .
The third derivative of is . The value of the second derivative at is .
The fourth derivative of is . The value of the second derivative at is
Now, substituting all these values in the Taylor's expansion series, we get
Simplifying the above equation, we get
The above equation is the Taylor's series expansion up to the fourth degree term of the function .
Note: In this type of question, it is necessary that we are familiar with the derivatives of standard functions. It is also important that we know the values of trigonometric functions for standard angles. This will make the calculations a little bit easier. It is useful to calculate every derivative separately so that we can avoid making errors in the calculations.
Complete step-by-step solution
The Taylor's series expansion a function
The given function is
The value of the function at
The first derivative of
The second derivative of
The third derivative of
The fourth derivative of
Now, substituting all these values in the Taylor's expansion series, we get
Simplifying the above equation, we get
The above equation is the Taylor's series expansion up to the fourth degree term of the function
Note: In this type of question, it is necessary that we are familiar with the derivatives of standard functions. It is also important that we know the values of trigonometric functions for standard angles. This will make the calculations a little bit easier. It is useful to calculate every derivative separately so that we can avoid making errors in the calculations.
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