12 Linearization and Newton-Raphson Method: Problem 1 (9 points) Use linearization to approximate 1 49.3. Let f(x) = Va. Then, the linearization of f(x) at x = 49 is 149(x) = yo + m(x - 49), where: m = and yo = Use the linearization to compute the approximation. V49.3 ~ 149 (49.3) = (Enter your result as a fraction, or a decimal with at least 9 significant figures.)\fUse a linear approximation (i.e., linearization) to approximate sin(28): Let f(a:) = sin(as). Find the equation of the line tangent to _f(a';) at a "nice" point near 28. Then, use the tangent line to approxim: sin(28) = | Hint: Convert degrees to radians in your calculations. The number v6 can be thought of as the solution of the equation a - 6 = 0. Newton's method is an iterative numerical method to find roots (zeroes) of an equation of of the form f() = 0. To implement Newton's method, one starts with an initial approximation co and then computes a sequence of approximations via the formula Ck+1 = g(k), where g(x) = x - f(z) fl (20) To find v6, the iteration function is: g( a) = With the starting value x0 = 1, the first few iterates are: C1 = C2 = C4= 05 =Sometimes Newton's method does not work. This exercise illustrates an example. /5 Use Newton's method to find a root of p(m) = &, with the starting value Lo = 5>, and observe what happens with the first few iterations. For your calculations, use exact arithmetic (not decimals). For example, enter g as sqrt (5) /5. 3 Ty = Ty = T3 T3 T4 = Newton's method may fail in many different ways. A detailed analysis of Newton's method, and related methods, is a huge subject and well beyond the scope of our course. Use Newton's Method to approximate v 12 to 6 significant figures starting with x = 3.5. Compare with the value obtained from a calculator. V12 ~