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Problem 6. We consider an investment of $1000 over one year with annual interest rate r. If this rate is compounded n times over the year, the final value of an investment of $1 is given by an(r) := (1 + )". With continuously compounded interest, it is given by b(r) = e. The purpose of this problem is to illustrate whether the interest calculation method significantly impacts the result or not. (a) Find the first Taylor polynomial for both an(r) and b(r). What is the error between both ? (b) Find the second Taylor polynomial for both an(r) and b(r). What is the error between both ? (c) Find the third Taylor polynomial for both an(r) and b(r). What is the error between both ? (d) In the sequel, we will consider the approximation of an(s) and b(r) by second-order Taylor polynomials. Explain why this is reasonable. (e) We consider interest compounded monthly. On an investment of $1000, for which values of the interest rate is the approximation by continuously compounded interest leading to a difference of less than $1 ? (f) We consider the interest rate r = 5%. On an investment of $1000, how often does the interest have to be compounded in order for the continuous approximation to yield a difference of less than $1 ? Problem 6. We consider an investment of $1000 over one year with annual interest rate r. If this rate is compounded n times over the year, the final value of an investment of $1 is given by an(r) := (1 + )". With continuously compounded interest, it is given by b(r) = e. The purpose of this problem is to illustrate whether the interest calculation method significantly impacts the result or not. (a) Find the first Taylor polynomial for both an(r) and b(r). What is the error between both ? (b) Find the second Taylor polynomial for both an(r) and b(r). What is the error between both ? (c) Find the third Taylor polynomial for both an(r) and b(r). What is the error between both ? (d) In the sequel, we will consider the approximation of an(s) and b(r) by second-order Taylor polynomials. Explain why this is reasonable. (e) We consider interest compounded monthly. On an investment of $1000, for which values of the interest rate is the approximation by continuously compounded interest leading to a difference of less than $1 ? (f) We consider the interest rate r = 5%. On an investment of $1000, how often does the interest have to be compounded in order for the continuous approximation to yield a difference of less than $1

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