Question

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 \( z \). \[

Answer 1

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.