HW4 - 13.2(a The function can be plotted 40 0-2-1-40-80-120...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
13.2 (a) The function can be plotted -120 -80 -40 0 40 -2 -1 0 1 2 (b) The function can be differentiated twice to give 2 4 24 45 ) ( " x x x f = Thus, the second derivative will always be negative and hence the function is concave for all values of x . (c) Differentiating the function and setting the result equal to zero results in the following roots problem to locate the maximum 12 8 9 0 ) ( ' 3 5 + = = x x x f A plot of this function can be developed -400 -200 0 200 400 -2 -1 0 1 2 A technique such as bisection can be employed to determine the root. Here are the first few iterations: iteration x l x u x r f ( x l ) f ( x r ) f ( x l ) × f ( x r ) ε a 1 0.00000 2.00000 1.00000 12 -5 -60.0000 2 0.00000 1.00000 0.50000 12 10.71875 128.6250 100.00% 3 0.50000 1.00000 0.75000 10.71875 6.489258 69.5567 33.33% 4 0.75000 1.00000 0.87500 6.489258 2.024445 13.1371 14.29% 5 0.87500 1.00000 0.93750 2.024445 -1.10956 -2.2463 6.67%
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The approach can be continued to yield a result of x = 0.91692. 13.3 First, the golden ratio can be used to create the interior points, 2361 . 1 ) 0 2 ( 2 1 5 = = d 2361 . 1 2361 . 1 0 1 = + = x 7639 . 0 2361 . 1 2 2 = = x The function can be evaluated at the interior points 1879 . 8 ) 7639 . 0 ( ) ( 2 = = f x f 8142 . 4 ) 2361 . 1 ( ) ( 1 = = f x f Because f ( x 2 ) > f ( x 1 ), the maximum is in the interval defined by x l , x 2 , and x 1 .where x 2 is the optimum. The error at this point can be computed as % 100 % 100 7639 . 0 0 2 ) 61803 . 0 1 ( = × = a ε For the second iteration, x l = 0 and x u = 1.2361. The former x 2 value becomes the new x 1 , that is, x 1 = 0.7639 and f ( x 1 ) = 8.1879. The new values of d and x 2 can be computed as 7639 . 0 ) 0 2361 . 1 ( 2 1 5 = = d 4721 . 0 7639 . 0 2361 . 1 2 = = x The function evaluation at f ( x 2 ) = 5.5496. Since this value is less than the function value at x 1 ,
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

HW4 - 13.2(a The function can be plotted 40 0-2-1-40-80-120...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online