A Taylor series expansion of function f (x) about some x-location x0 is given as in fig. Consider the function f (x) = exp(x) = ex. Suppose we know the value of f (x) at x = x0, i.e., we know the value of f (x0), and we want to estimate the value of this function at some x location near x0. Generate the first four terms of the Taylor series expansion for the given function (up to order (Δx)3 as in the above equation). For x0 = 0 and Δx = −0.1, use your truncated Taylor series expansion to estimate f (x0 + Δx). Compare your result with the exact value of e−0.1. How many digits of accuracy do you achieve with your truncated Taylor series?

A Taylor series expansion of function f (x) about some x-location x₀ is given as in fig. Consider the function f (x) = exp(x) = e^x. Suppose we know the value of f (x) at x = x₀, i.e., we know the value of f (x₀), and we want to estimate the value of this function at some x location near x₀. Generate the first four terms of the Taylor series expansion for the given function (up to order (Δx)³ as in the above equation). For x₀ = 0 and Δx = −0.1, use your truncated Taylor series expansion to estimate f (x₀ + Δx). Compare your result with the exact value of e^−0.1. How many digits of accuracy do you achieve with your truncated Taylor series?