Homework 1 MATH 455 Instructions 1 Calculus Review functions i f x ex2 Written assignments are graded based on quality of work and fully worked out solutions Points will be deducted for incomplete reasoning and disorganized work even if your answers are correct a Find the 2nd degree T2 x and 5th degree T5 x Taylor polynomials centered at x0 0 for the following ii f x 1 1 x iii f x cos 2x iv f x sin2 x b Let f x x 2 and carry out the following calculations i Calculate the Taylor polynomial of degree 4 about the point x 1 ii Use your result in a to approximate f 0 9 and f 1 1 iii Use the Taylor remainder to nd an error formula for the Taylor polynomial Give error bounds for each of the two approximations in part b Which of the two approximations in b do you expect to be closer to the correct value iv Use a calculator to compare the actual error in each case with your error bound from part c How do the actual error and error bound compare 2 Root nding and Fixed Points i 2 3 2 i g x 3 x a Use the Bisection Method to nd p3 for f x px cos x 0 on 0 1 b Let f x 3 x 1 x 1 2 x 1 Use the Bisection Method on the following intervals to nd p3 c Find all xed points for the following functions ii 5 4 5 2 ii g x x 6 3x 2 iii g x x2 4x 2 3 Newton s and Secant Methods a Let f x x2 6 and p0 1 Use Newton s Method to nd p2 b Let f x x3 cos x and p0 1 Use Newton s Method to nd p2 Could p0 0 be used Explain c Let f x x2 6 With p0 3 and p1 2 nd p3 using the Secant Method d Let f x x4 7x3 18x2 20x 8 Does Newton s Method converge quadratically to the root r 2 e Show that Newton s Method applied to f x ax b converges in one step 4 Programming use Matlab a Write a Matlab program which performs Bisection Method and use it for the following problems i Find an approximation to within 10 5 to a value in 0 5 1 5 with ex 2 cos ex 2 ii Find an approximation to within 10 5 to the rst positive value of x with x tan x 1 MATH 455 Homework 1 b Write a Matlab program to perform Fixed Point Iteration to nd the solution to the following equations i ex x 7 ii ex sin x 4 c Write a Matlab program which performs Newton s Method Apply your program to nd both roots of the function f x 14xe x 2 12e x 2 7x3 20x2 26x 12 on the interval 0 3 For each root print the sequence of iterates the errors en xn r and both error ratios en 1 e2 n and en 1 en that converges to a nonzero limit 2 G I e 1 x 2 e 1 x Tz x Tf x 1 x2 1 x iv sin x E 1 2s 2x then use taylor polynomials from iii b f x x 2 i Ty x 1 2 x 1 3 x 1 4 x 1 5 x 14 ii f 0 9 1 2345 f 1 1 0 8265 iii TAylaw Remainder 1n x MIx XM n 1 wherem is max of If r t far t in X0 XY If tll 3 x z is max at t 0 9 So M 7 9 r0 Ry x 6 9 x 1 Error bonds at each X 0 9 and X 1 1 are the same Ry x 1 1 25410 4 However on expect more necurate results at X 1 1 ble the 5th order derivative issmaller here b 0 2 3 2 f 2 1 1420 3 2 t 0 either Converga to Po 14 5 14 30 thus new internal is X 1 arX 42 ar X 1 2 4 P 9 82 1 they fl 9 8 20 And new crital is 1 8 4 looks like were headed to root at X 12 1 thus f al 30 And new interval is 118 in Pz z d i g x x 3 x X 13 ii 3x 2 x X 4 3x2 2x 0 3x2 3x 4 3 X2 X 2 3 X 2 X 1 X 2 1 c For the secant method n 1 Pn Pu Pn f pm f in A qn 1 with Po 3 P 2 8 2 2 And f 3 3 4z 2 f z 2 3 f z 7 3 I 5 2 2 P2 F 12 5 Y3 42 f 42 E f P2 P P2 f p 2 5 f b 2 43 7 d To Address the convergence using Newton s method we need r 2 to determine if Newton s method enjoys good Convergence or in that ease Newton s method is linearly ecurargent is a multiple roat a simple ret in thatease f 2 e t 2 4x3 21x2 36x x x 2 2 Since this rot is not simple convergence is linear 2 Note f x aX h has a but initial starting point Xo is my at X bla If for hearten s method then X Xo Xo b I CXo axo b C x I Ha f x CX b This meutan s method conces in one step when