This is a long question that works you through an example of linear and quadratic approximations. Take your time. Round your answers to two digits. Consider the following power function f(a): f(ac) = 724 We want to find the first and second order approximations around To = -1 and compare the remainders. Part 1 First, let us find the first-order approximation and express it in "distance from pivotal point." In other words, find a and b such that for ac close to x0 = -1 f(x) ~ atb. (x - -1). The coefficient a is equal to 7 The coefficient b is equal to -28 Next, we can rewrite that expression into standard polynomial form. In other words, find new coefficients a and b such that for x close to Co = -1 f(x) ~ a t bx. The coefficient a is equal to -21 The coefficient b is equal to -28Part 2 Second, we find the second-order approximation and express it in "distance from pivotal point." In other words, find a, b, and c such that for x close to to = -1 f(x) ~ atb. (2 - -1)+ . C. (x - -1)2. The coefficient a is equal to 7 The coefficient b is equal to -28 The coefficient c is equal to 84 Next, we rewrite that second-order expression into standard polynomial form. In other words, find new coefficients a, b, and c such that for x close to co = -1 f( x ) ~ at ba + ca?. The coefficient a is equal to 49 X The coefficient b is equal to 140 X The coefficient c is equal to 84 X Part 3 Now we can use these approximations to evaluate the function f(x) for a close to To = -1 Specifically, use both approximations to find approximate values for x = -0.9The coefficient a is equal to 49 X The coefficient b is equal to 140 X The coefficient c is equal to 84 X Part 3 Now we can use these approximations to evaluate the function f(x) for a close to do = -1 Specifically, use both approximations to find approximate values for x = -0.9 The linear approximation yields an approximate value of 4.2 The quadratic approximation yields an approximate value of -8.96 XA common difficulty students have with approximations is to distinguish the point at which the linear function approximates a given function and the point at which the linear function is evaluated to approximate the function. This question hopefully helps you make this distinction. Complete your calculations. Then round your your answers to two digits before entering them. Consider the following function f(x): f(x) = 3Inx +4.x Part 1 First, consider x0 = 1 as the pivotal point. Find the linear approximation to f(x) at do = 1 and express it as L() = A+ B . (x - x0) The coefficient A is equal to 4 The coefficient B is equal to 7 Next, use the linear approximation to approximate the functional value at x = 1.1. f(1.1) ~ L(1.1) = 4.7J(x) = 3Inc +4 . Part 1 First, consider x0 = 1 as the pivotal point. Find the linear approximation to f(x) at To = 1 and express it as L(x) = A + B. (x -To) The coefficient A is equal to 4 The coefficient B is equal to 7 Next, use the linear approximation to approximate the functional value at x = 1.1. f(1.1) ~ L(1.1) = 4.7 Part 2 Second, reverse the points. Consider 0 = 1.1 as the pivotal point. Find the linear approximation to f(x) at co = 1.1 and express it as 1(a) = a +b. (2 -20) The coefficient a is equal to 1.64 X The coefficient b is equal to 4.64 X Next, use the linear approximation to approximate the functional value at x = 1. f(1) ~ Z(1) 6.28 XThis question emphasizes that different functions can have the same linear approximations, though possibly at different points. Indeed, the question asks you to find the points where a given linear function approximates certain functions. While this is not how linear approximations are usually used, working backward in this manner solidifies your understanding of linear approximations. Round your answers to two digits. Consider the following linear function L(m): L(m) = 5 - a: + 17 This linear function is both the linear approximation to at) 2 93:3 222': -l 35. at point 5:0 2 4 X . and the linear approximation to 9(3) : 7 - 1n(2m) 9:1: + 24. at point 5:0 = 0.2 X . Let's apply your understanding of linear approximations. Round your answers to two digits. Find the firstorder {linear} approximation for ffml for a: close to m. = 1for: at point To = 0.2 X Let's apply your understanding of linear approximations. Round your answers to two digits. Find the first-order (linear) approximation for f(x) for a close to do = 1 for: f(ac) = am where m is a parameter. Use the approximation to find an approximate value for (a) 0.9527 0.38 X (b) v1.01 1 X Finish review