172k views
4 votes
Use the Taylor series of degree 5 about \( a=0 \) for \( f(a)= \) tin \( x \) to approximate the following integent. Firs, calculate the Taylor serias of degree 5 abosit \( a=0 \) for nin \( z \). \[

User Badnaam
by
8.3k points

1 Answer

2 votes

Final Answer:

1. For
\( \tan(x) \), the fifth-degree Taylor series expansion about ( a = 0 ) is
\( x + (x^3)/(3) + (2x^5)/(15) \).

2. For
\( \sin(x) \), the fifth-degree Taylor series expansion about ( a = 0 ) is
\( x - (x^3)/(6) + (x^5)/(120) \).

3. These approximations can be used to estimate values of
\( \tan(x) \) and
\( \sin(x) \) near ( x = 0 ) by substituting the corresponding series expressions.

Explanation:

The Taylor series expansions for
\( \tan(x) \) and
\( \sin(x) \) centered at ( a = 0 ) up to the fifth degree provide polynomial approximations that capture the behavior of these functions in the vicinity of zero.

For
\( \tan(x) \), the expansion
\( x + (x^3)/(3) + (2x^5)/(15) \) is derived from the derivatives of
\( \tan(x) \) evaluated at ( a = 0 ). The series is designed to closely mimic the behavior of
\( \tan(x) \)for small values of ( x ).

Similarly, for
\( \sin(x) \), the expansion
\( x - (x^3)/(6) + (x^5)/(120) \) is obtained by evaluating the derivatives of
\( \sin(x) \) at ( a = 0 ). This polynomial approximation is particularly accurate for ( x ) near zero, capturing the sine function's oscillatory behavior.

These Taylor series are powerful tools in calculus and mathematical analysis, enabling the estimation of function values and behavior in regions around a chosen center. As the degree of the series increases, the accuracy of the approximation improves, making them valuable for various mathematical and scientific applications.

User Faisal Sayed
by
8.0k points