Part 6/8:
First-degree approximation ( T_1(x) = 1 )
Second-degree approximation ( T_2(x) = 1 - \frac{x^2}{2} )
Fourth-degree approximation ( T_4(x) = 1 - \frac{x^2}{2} + \frac{x^4}{4!} )
As we could observe, higher-degree approximations bring the polynomial curve closer to that of the cosine function, showcasing the power of Taylor polynomials in capturing function behavior accurately.
Visualizing Taylor Polynomial Approximations
Graphing the approximated functions alongside the original cosine curve reveals just how effective Taylor polynomials can be. As we increase the degree of the polynomial, the approximation not only offers a better fit near the point of tangency (( a = 0 )) but also extends its accuracy over a wider range of ( x ).