Transcript Slide 1
2003 Paper 1 No 7
© Annie Patton
Diferentiate
x
1+4x with respect to x.
Leaving Certificate Higher 6c (i) paper 1 2003 Start clicking when you want to see the answer.
1 2 Let m=1+4x
dm dx
4
y
m
2 1
dy dm
1 2
m
1 2 2 1
m
1 2 1 4
x dy dx
dy dm dm dx
1
x
4 2
x
© Annie Patton
3 Show that the equation x -4x-2=0 has a root between 2 and 3.
Taking x =2 as the first approximation to this root ,use the Newton-Raphson 1 method to find x , the third approximation. Give your answer correct to two 3 decimal places.
Leaving Certificate 2003 Higher Level Paper 1 no 6(b)
Start clicking when you want to see the answer.
3 f(x)=x -4x-2 f(2)=8-8-2=-2 f(3)=27-12-2=13 Goes from positive to negative, hence a root between 2 and 3.
f(2)=-2
2
f'(x)=3x -4 f'(2)=12-4=8
u =2 2 -2 8 =2+ 1 4 =2.25
f(2.25) u =2.25 3 f'(2.25) .390625
=2.25 11.1875
=2.22
© Annie Patton
1 The function f(x)= 1-x is defined for x (i)Prove that the graph of f has no turning points and no points of inflection.
Leaving Certificate Higher 6c (i) paper 1 2003 Start clicking when you want to see the answer.
' f (x)= (1-x)0-1(-1) = (1-x) 2 1 (1-x) 2 ' At the turning points f (x)=0 So 1 =0 1 0 (1-x) 2 No turning points 2 d y dx 2 2 = (1-x) 3 0 Hence no points of inflection.
© Annie Patton
1 The function f(x)= 1-x is defined for x (ii) Write down a reason that justifies the statement "f is increasing at ever value of x '
f (x)= 1 (1-x)
2 Leaving Certificate Higher 6c(ii) paper 1 2003 Start clicking when you want to see the answer.
as 1> is always positive and
2
(1-x) is always positive.
Therefore
f '
(x) is always positive.
Hence f(x) is always increasing.
© Annie Patton
1 The function f(x)= 1-x is defined for x Given that y=x+k is a tangent to the graph of f where k is a real number, find the two possible values of k.
Leaving Certificate Higher 6c(iii) paper 1 2003 Start clicking when you want to see the answer.
' The slope of y=x+k is 1. Hence f (x)=1 (1 1
x
) 2 (1
x
) 2 1 1
x
x
2 1
x
2 2
x
0 2) 0
x
0
or x
2 1 At x=0 f(x)= 1-0 =1 So if (0,1) is on the line y=x+k 1=0+k k=1 © Annie Patton
1 The function f(x)= 1-x is defined for x Given that y=x+k is a tangent to the graph of f where k is a real number, find the two possible values of k.
Leaving Certificate Higher 6c(iii) paper 1 2003 Start clicking when you want to see the answer.
The second point is at x=2.
At x=2 1 f(x)=y= 1-2 =-1 Hence the point (2,-1) is on the line y=x+k -1=2+k k=-3 © Annie Patton