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C3 Help.Teach me!!

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.

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

3)
a)Given that sinx=3/5, use an appropriate double angle formula to find the exact value of sec2x.

b)Prove that cot2x + cosec2x = cotx (x≠(n&#960:wink:/2, neZ)

Thanks for any help :smile: WIll reward you justly :smile:
Reply 1
Srathmore


3)
a)Given that sinx=3/5, use an appropriate double angle formula to find the exact value of sec2x.

b)Prove that cot2x + cosec2x = cotx

Thanks for any help :smile: WIll reward you justly :smile:


sec2x=1/cos2x
sec2x=1/(1-2sin²x)
........=1/(1-(18/25))
........=1/(7/25)
........=25/7

b)
cot2x+cosec2x=cotx

LHS:
cos2x/sin2x + 1/sin2x
(cos2x+1)/sin2x
[(2cos²x-1)+1]/(2sinxcosx)
(2cos²x)/(2sinxcosx)
(2cos²x)/(2sinxcosx)
cosx/sinx
=cotx=RHS
Reply 2
3.a)

sin x = 3/5

sec 2x = 1/(cos 2x)

= 1/(1-2sin²x)
=1/(1-2(3/5)²)
=1/(7/25) = 25/7

b) LHS = cot 2x + cosec 2x

= cos 2x/sin 2x + 1/sin 2x

= (cos 2x + 1)/sin 2x

= ((2cos²x -1) +1)/(2sinxcosx)

= (2cos²x)/(2sinxcosx)

= cos x/ sin x

= cot x
Reply 3
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
= 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 +&#8730;(a²+8))
Reply 4
1) f(x) = 0

When ex - 2x² = 0

ex = 2x²

x = ln(2x²)

i can't see how they get 4x from 2x²


ln(2x²) = ln(4x)

So g(x) = 4x as required