1)
f(x)=0.5ex -x²
The x co-ordinate of B is approximately 2.15. A more exact estimate is to be made of this co-ordinate using iterations xn+1 = lng(x(n ))
Show that a possible form of g(x) is g(x)=4x.
What is it that i have to do? Do i have to find the range for a change in sign?? I have tried everything, and can't seem to get that answer.
they just want you to rearrange the original equation to x = ln4x
you want an equation for the root (i.e. where f(x) = 0)
0 = 0.5ex -x²
x² = 0.5ex
2x² = ex
ln(2x²) = x
hmm. i can't see how they get 4x from 2x².
2)
a)Sketch the graph of y=!2x+a! ,a>0.
b)explain how the graphs of y=1/x and y=!2x+a! show that there is only one soultion of the equation x!2x+a! - 1=0, and hence find the value of x.
x!2x+a! =1
the sketch involves a line through (0,a) with gradient 2. when x > -½a line has gradient 2. when x < -½a line has gradient -2.
the line |2x+a| is never negative. there is no solution below the x-axis. its probbaly best to draw another sketch of both lines.
|2x+a| = 1/x
x|2x+a| = 1
looking at the sketch, the required line is y=2x+a
so x(2x+a) = 1
2x² +ax -1 = 0
you want the positive root. using the quadratic formula x = ¼(-a +√(a²+8))