### Estimating \( f(x) = \cos(x) \) at \( x = \pi \) **Instructions:** 1. Show your work and circle only the final answer. 2. Provide your answer in scientific notation rounded to 5 digits of precision. Note: - \( x = \pi \) - Function: \( f(x) = \cos(x) \) - Required precision: 5 digits in scientific notation ### Taylor Polynomial of Degree 3 and Error Estimation This educational content is focused on understanding and applying Taylor polynomials. Follow the tasks outlined below to enhance your comprehension and calculation skills: #### 1. Writing the Taylor Polynomial \( P_3(x) \) **Objective:** Write the Taylor Polynomial \( P_3(x) \) of degree 3 at \( x = \pi \) for the function \( f(x) = \cos x \). **Instructions:** - Show your work clearly. - Circle the polynomial \( P_3(x) \). - Use exact expressions; do not use decimals. #### 2. Evaluating the Exact Error **Objective:** Using the Taylor polynomial \( P_3(x) \) you derived in Question 1, evaluate the exact error \( |f(0) - P_4(0)| \), where \( P_4(x) \) is the Taylor polynomial of degree 4. **Instructions:** - Use the findings from Question 1 as a base for your calculations. - Ensure precision in your calculations to find the exact error. This exercise enhances your ability to work with Taylor series and understand the approximation of functions using polynomials. Accurate calculation and exact expressions are critical in higher mathematical analysis. **Tip:** Review the Taylor series expansion formula and practice deriving and evaluating the polynomials for various functions to strengthen your understanding. **References:** - Taylor Series - Polynomial Approximations - Error Analysis in Taylor Series