SUMMARY
The maximum error in approximating cos(x) using its Taylor polynomial of order 2 on the interval [-0.25, 0.25] is determined using the Remainder Estimation Theorem. The remainder R3(x) is bounded by |R3(x)| ≤ M/3! |x|^3, where M represents the maximum absolute value of the third derivative of cos(x) over the specified interval. The third derivative of cos(x) is -sin(x), and its maximum absolute value on [-0.25, 0.25] is sin(0.25). Thus, M is sin(0.25), leading to the conclusion that the maximum error can be calculated accurately using this value.
PREREQUISITES
- Understanding of Taylor series and polynomial approximations
- Familiarity with the Remainder Estimation Theorem
- Knowledge of derivatives, specifically the third derivative of trigonometric functions
- Basic calculus concepts, including limits and continuity
NEXT STEPS
- Calculate the third derivative of cos(x) and evaluate its maximum on the interval [-0.25, 0.25]
- Explore the implications of the Remainder Estimation Theorem in other contexts
- Learn about higher-order Taylor polynomials and their error estimation
- Investigate the convergence of Taylor series for different functions
USEFUL FOR
Students studying calculus, particularly those focusing on Taylor series and polynomial approximations, as well as educators teaching these concepts in mathematics courses.