Please label each part.

1. Find the third degree Taylor Polynomial 73(x) centered at a for the following functions (a) f(x) = 2+x at a = -1 (b) f(x) = tanx at a = 0 (c) f(x) = ex/2 at a = 2 (d) f(x) = In(1 + 2x) at a = 1 2. Consider the Taylor polynomials of f(x) = Va centered at x = 9. (a) Find Ti(x), T2(x), and T3(x). (b) Use Desmos to plot f(x) as well as the above Taylor polynomials on the same set of axes. (c) Use Ti(x), T2(x) and T3(x) to approximate v9.1. (d) Find the error in these approximations. The error is defined to be |V9.1 - T,(9.1)|. (Recal that (x - yl is the "number line distance" between a and y.) (e) The Taylor Polynomial Error Bound guarantees that If (x) - In(2)| SK Z - anti (n + 1)!' where K = maximum of [fo+(u)| on the closed interval between a and x, (provided fo+ exists and is continuous). What does the Error Bound guarantee about th size of |V9.1 - 73(9.1) |? How does it compare to the actual error? 3. Division requires a larger computational budget than multiplication (computers are just like us we hate long division!), which can cause issues in responsiveness of user interfaces, framerates in videogames, and the time needed to run simulations for important engineering problems. (a) Find the linearization, L(x), of f(x) = - for x near 1. (b) It can often save significant time to use the above linearization to approximate S = c () ~ CL(I), since multiplication is computationally cheaper than division. For numbers a ~ 1 the loss i accuracy is negligible compared to the gain in efficiency. 2 MATH 141 Homework 17 Approximate 17 1.1 and 2.5 0.97 using the above idea. (c) What does the Taylor Polynomial Error Bound guarantee regarding the errors in using th above approximations? (d) How do these guarantees compare to the actual errors