- [10 Pts] Convert the base 10 floating-point number
`123.875`

to base 2 (i.e. to binary form). - [10 Pts] Suppose you have a decimal machine with 4 decimal digits
allocated to the mantissa. Also, assume that such a computer
correctly rounds to the nearest machine-representable number.
What
**positive**real numbers*x*have the property*fl(1.0+x) = 1.0*? - [10 Pts] Show how
*log x - log (1/x)*can be computed without a serious loss of significant figures. Note: the logarithm base does not matter. - [20 Pts] Suppose you apply Newton's method to solve the nonlinear
equation
3 x + 3x = 6x - 1 .

- (a) Compute the first two Newton iterates which approximate
a solution to the nonlinear equation using the initial guess
x = 0 . 0

- (b) Would Newton's method converge to a solution of the nonlinear
equation if the initial guess is
x = 1 ? 0

Explain why or why not.

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- (a) Compute the first two Newton iterates which approximate
a solution to the nonlinear equation using the initial guess
- [15 Pts] Using
**Lagrange**interpolation, determine the polynomial of least degree that assumes the following table of values.x y 0 0 1/2 1 -1 -1 - [15 Pts] How accurately can one compute
*cos x*by**quadratic**interpolation on the closed interval [0,1] using a table of known values (equally) separated by*h= 0.001*? - [10 Pts] Compute an approximation to the definite integral
2 / dx | --- | 2 / x 1

using**lower sums**with stepsize*h=1/2*. - [10 Pts] Complete the following Matlab function
**trap.m**which could be used to approximate the definite integral of an arbitrary function*f(x)*on the closed interval*[a,b]*using the**Trapezoid Rule**with**constant stepsize***h*. You may assume the existence of a Matlab function**func.m**defined by the declaration`function [f] = func(x),`

which can be used to evaluate the function

*f(x)*at any real number*x*.[I] = trap(a,b,n) % Inputs: % a is the left endpoint of the interval % b is the right endpoint of the interval % n is the no. of subintervals in the mesh % Output: % I is the approximation to the definite % integral h= (b-a)/n;