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A function has derivatives of all orders for all . Some values of are given in the table below. It is known that .

  1. Use a trapezoidal sum with four subintervals to approximate .

  2. Use your approximation from part (a) to estimate the value of . Justify your estimate with your work.

  3. Use Euler’s Method, starting with , with four steps of equal size, to approximate . Show your work.

    At , the slope is . Since the step size is , then 

    Starting at , the next point is: 

  4. If and , write a third-degree Taylor polynomial about and use it to approximate .

  5. How close were your estimates of in parts (b) through (d)? Explain why this happened.