Does equation 3x3 - 5x2 - 4x - ex = -4 have at least one solution in the interva

Question

Does equation 3x3 - 5x2 - 4x - ex = -4 have at least one solution in the interval (0,2) Yes. No. Can not tell. If the Bisection method is employed to find a root of a function in the interval (-5,-3), how many iterations would be needed to obtain an approximate solution with error less than 10-6? 20 . 21. 22. The trapezoid formula for numeric integration is more accurate than the rectangle formula. True . False. Can not tell. In numeric differentiation, the forward difference method provides better approximate than central difference method. True . False. Can not tell..In order for a polynomial to fit all n given data points, the order of the polynomial should be n-1. n. n+1.

Explanation / Answer

1.equation is

3x^3 - 5x^2- 4x - exp (x)+4 =0
let ,
f(x) =3x^3 - 5x^2- 4x - exp (x)+4
at x= 0 ,
f(0)= -1+4=3
at ,
x=2,
f(2) = 24-20-8-exp(2) + 4 = -exp (2)
so ,
f(0)* f(2) <0 ..hence one root exists between (0,2).

2. In case of Bisection method,
no of iteration required to achieve a given error(e),
n = [log( |b-a|) - log (e)]/log 2
so ,
here,
n = [log( 2) - log (10^-6)]/log 2 = 20.93
so,
n~21 iterations required.

3.
Trapezoidal rule is less accurate than rectangular formula.
so , it's False.

because, The size of the error in the Midpoint Rule( rectangular formula)
is about half that in the Trapezoidal Rule

4. It's False.
Central difference method gives better approximate than forward one because the order of error in the latter in one degree higher than the former.

5.We can see , that for finding a straight line we need two points. Therefore...given two points we can fit them into a polynomial of degree 1 (y=mx+c).
so , for fitting n values in a polynomial,
we need a (n-1) order polynomial.

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