Download Numerical methods

f(x) = 4 cosec x – 4x + 1, where x is in radians.
1.
(a)
Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3].
(2)
(b)
Show that the equation f(x) = 0 can be written in the form
x
1
1

sin x 4
(2)
(c)
Use the iterative formula
xn 1 
1
1
 , x0  1.25,
sin xn 4
to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places.
(3)
(d)
By considering the change of sign of f(x) in a suitable interval, verify that α = 1.291
correct to 3 decimal places.
(2)
(Total 9 marks)
f(x) = x3 + 2x2 – 3x – 11
2.
(a)
Show that f(x) = 0 can be rearranged as
 3x  11 
x 
,
 x2 
x  2.
The equation f(x) = 0 has one positive root α.
(2)
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 3x  11 
 is used to find an approximation to α.
The iterative formula x n 1   n
x

2
n


(b)
Taking x1 = 0, find, to 3 decimal places, the values of x2, x3 and x4.
(3)
(c)
Show that α = 2.057 correct to 3 decimal places.
(3)
(Total 8 marks)
3.
The diagram above shows part of the curve with equation y = –x3 + 2x2 + 2, which intersects the
x-axis at the point A where x = α.
To find an approximation to α, the iterative formula
xn 1 
2
2
( xn ) 2
is used.
(a)
Taking x0 = 2.5, find the values of x1, x2, x3 and x4.
Give your answers to 3 decimal places where appropriate.
(3)
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(b)
Show that α = 2.359 correct to 3 decimal places.
(3)
(Total 6 marks)
f(x) = 3xex – 1
4.
The curve with equation y = f (x) has a turning point P.
(a)
Find the exact coordinates of P.
(5)
The equation f (x) = 0 has a root between x = 0.25 and x = 0.3
(b)
Use the iterative formula
1
x n 1  e  xn
3
with x0 = 0.25 to find, to 4 decimal places, the values of x1, x2 and x3.
(3)
(c)
By choosing a suitable interval, show that a root of f(x) = 0 is x = 0.2576 correct to 4
decimal places.
(3)
(Total 11 marks)
f(x) = 3x3 – 2x – 6
5.
(a)
Show that f(x) = 0 has a root, α, between x = 1.4 and x = 1.45
(2)
(b)
Show that the equation f (x) = 0 can be written as
2
x  
x
2
, x  0 .
3
(3)
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(c)
Starting with x0 = 1.43, use the iteration
 2 2
xn 1    
 xn 3 
to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places.
(3)
(d)
By choosing a suitable interval, show that α = 1.435 is correct to 3 decimal places.
(3)
(Total 11 marks)
f(x) = ln(x + 2) – x + 1, x > –2, x 
6.
(a)
.
Show that there is a root of f(x) = 0 in the interval 2 < x < 3.
(2)
(b)
Use the iterative formula
xn+1 = 1n(xn + 2) + 1,
x0 = 2.5
to calculate the values of x1, x2 and x3 giving your answers to 5 decimal places.
(3)
(c)
Show that x = 2.505 is a root of f(x) = 0 correct to 3 decimal places.
(2)
(Total 7 marks)
f(x) = – x3 + 3x2 – 1.
7.
(a)
Show that the equation f(x) = 0 can be rewritten as
 1 
 .
x  
 3 x 
(2)
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(b)
Starting with x1 = 0.6, use the iteration
 1
x n 1  
 3  xn



to calculate the values of x2, x3 and x4, giving all your answers to 4 decimal places.
(2)
(c)
Show that x = 0.653 is a root of f(x) = 0 correct to 3 decimal places.
(3)
(Total 7 marks)
8.
y

O
x
4
P
The figure above shows part of the curve with equation
y = (2x – 1) tan 2x, 0  x <

4
The curve has a minimum at the point P. The x-coordinate of P is k.
(a)
Show that k satisfies the equation
4k + sin 4k – 2 = 0.
(6)
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The iterative formula
1
x n  1  (2  sin 4 x n ),
4
x0  0.3,
is used to find an approximate value for k.
(b)
Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places.
(3)
(c)
Show that k = 0.277, correct to 3 significant figures.
(2)
(Total 11 marks)
9.
f(x) = 2x3 – x – 4.
(a)
Show that the equation f(x) = 0 can be written as
2 1
x   
 x 2
(3)
The equation 2x3 – x – 4 = 0 has a root between 1.35 and 1.4.
(b)
Use the iteration formula
 2 1
xn 1     ,
 x 2
with x0 = 1.35, to find, to 2 decimal places, the values of x1, x2 and x3.
(3)
The only real root of f(x) = 0 is α.
(c)
By choosing a suitable interval, prove that α = 1.392, to 3 decimal places.
(3)
(Total 9 marks)
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f(x) = 3ex –
10.
(a)
1
2
ln x – 2, x > 0.
Differentiate to find f (x).
(3)
The curve with equation y = f(x) has a turning point at P. The x-coordinate of P is .
(b)
Show that  =
1
6
e–.
(2)
The iterative formula
xn + 1 =
1
6
e  xn , x0 = 1,
is used to find an approximate value for .
(c)
Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places.
(2)
(d)
By considering the change of sign of f (x) in a suitable interval, prove that  = 0.1443
correct to 4 decimal places.
(2)
(Total 9 marks)
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11.
y
O
A
B
x
C
f(x) =
1
x
– 1 + ln , x > 0.
2
2x
The diagram above shows part of the curve with equation y = f(x). The curve crosses the x-axis
at the points A and B, and has a minimum at the point C.
(a)
Show that the x-coordinate of C is
1
2
.
(5)
(b)
Find the y-coordinate of C in the form k ln 2, where k is a constant.
(2)
(c)
Verify that the x-coordinate of B lies between 4.905 and 4.915.
(2)
(d)
Show that the equation
2e
1 - 21x 
1
x
– 1 + ln
= 0 can be rearranged into the form x =
2x
2
.
(2)
The x-coordinate of B is to be found using the iterative formula
xn + 1 = 2e
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1 - 
1
2 xn
,
with x0 = 5.
8
(e)
Calculate, to 4 decimal places, the values of x1, x2 and x3.
(2)
(Total 13 marks)
12.
f(x) = x3 – 2 –
(a)
1
,
x
x  0.
Show that the equation f(x) = 0 has a root between 1 and 2.
(2)
An approximation for this root is found using the iteration formula
1

1 3
 ,
xn + 1 =  2 
x n 

(b)
with x 0 = 1.5.
By calculating the values of x1, x2, x3 and x4, find an approximation to this root, giving
your answer to 3 decimal places.
(4)
(c)
By considering the change of sign of f(x) in a suitable interval, verify that your answer to
part (b) is correct to 3 decimal places.
(2)
(Total 8 marks)
13.
The curve C with equation y = k + ln 2x, where k is a constant, crosses the x-axis at the point
1 
A  , 0 .
 2e 
(a)
Show that k = 1.
(2)
(b)
Show that an equation of the tangent to C at A is y = 2ex – 1.
(4)
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(c)
Complete the table below, giving your answers to 3 significant figures.
x
1
1.5
1 + ln 2x
2
2.10
2.5
3
2.61
2.79
(2)
(d)
Use the trapezium rule, with four equal intervals, to estimate the value of
3

 (1  ln 2 x) dx
1
.
(4)
(Total 12 marks)
14.
f(x) = x3 + x2 – 4x – 1.
The equation f(x) = 0 has only one positive root, .
(a)
Show that f(x) = 0 can be rearranged as
x=
 4 x 1 

 , x  –1.
 x 1 
(2)
The iterative formula xn + 1 =
(b)
 4 xn 1 

 is used to find an approximation to .
 xn 1 
Taking x1 = 1, find, to 2 decimal places, the values of x2, x3 and x4.
(3)
(c)
By choosing values of x in a suitable interval, prove that  = 1.70, correct to 2 decimal
places.
(3)
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(d)
Write down a value of x1 for which the iteration formula xn + 1 =
 4 xn 1 

 does not
x

1
n


produce a valid value for x2.
Justify your answer.
(2)
(Total 10 marks)
15.
f(x) = x +
(a)
ex
, x
5
.
Find f (x).
(2)
The curve C, with equation y = f(x), crosses the y-axis at the point A.
(b)
Find an equation for the tangent to C at A.
(3)
(c)
Complete the table, giving the values of
x

ex 
 x  
5

0
0.5
0.45
0.91

ex
 x 
5

1

 to 2 decimal places.

1.5
2
(2)
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(d)
Use the trapezium rule, with all the values from your table, to find an approximation for
the value of
2



0

ex
 x 
5


 dx

.
(4)
(Total 11 marks)
16.
y
C
R
2
O
The curve C has equation y = f(x), x 
0  x  2.
x
. The diagram above shows the part of C for which
Given that
dy
= ex – 2x2,
dx
and that C has a single maximum, at x = k,
(a)
show that 1.48 < k < 1.49.
(3)
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Given also that the point (0, 5) lies on C,
(b)
find f(x).
(4)
The finite region R is bounded by C, the coordinate axes and the line x = 2.
(c)
Use integration to find the exact area of R.
(4)
(Total 11 marks)
17.
A student tests the accuracy of the trapezium rule by evaluating I, where
1.5
 3

I =    x 4  dx .
0.5  x

(a)
Complete the student’s table, giving values to 2 decimal places where appropriate.
x
0.5
0.75
3
 x4
x
6.06
4.32
1
1.25
1.5
(2)
(b)
Use the trapezium rule, with all the values from your table, to calculate an estimate for
the value of I.
(4)
(c)
Use integration to calculate the exact value of I.
(4)
(d)
Verify that the answer obtained by the trapezium rule is within 3% of the exact value.
(2)
(Total 12 marks)
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18.
The curve C has equation y = f(x), where
f(x) = 3 ln x +
1
,
x
x > 0.
The point P is a stationary point on C.
(a)
Calculate the x-coordinate of P.
(4)
(b)
Show that the y-coordinate of P may be expressed in the form k – k ln k, where k is a
constant to be found.
(2)
The point Q on C has x-coordinate 1.
(c)
Find an equation for the normal to C at Q.
(4)
The normal to C at Q meets C again at the point R.
(d)
Show that the x-coordinate of R
(i)
satisfies the equation 6 ln x + x +
(ii)
lies between 0.13 and 0.14.
2
– 3 = 0,
x
(4)
(Total 14 marks)
19.
(a)
Sketch, on the same set of axes, the graphs of
y = 2 – e–x and y = x.
[It is not necessary to find the coordinates of any points of intersection with the axes.]
(3)
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Given that f(x) = e–x + x – 2, x  0,
(b)
explain how your graphs show that the equation f(x) = 0 has only one solution,
(1)
(c)
show that the solution of f(x) = 0 lies between x = 3 and x = 4.
(2)
The iterative formula xn + 1 = (2 – e  xn )2 is used to solve the equation f(x) = 0.
(d)
Taking x0 = 4, write down the values of x1, x2, x3 and x4, and hence find an approximation
to the solution of f(x) = 0, giving your answer to 3 decimal places.
(4)
(Total 10 marks)
20.
The curve with equation y = ln 3x crosses the x-axis at the point P (p, 0).
(a)
Sketch the graph of y = ln 3x, showing the exact value of p.
(2)
The normal to the curve at the point Q, with x-coordinate q, passes through the origin.
(b)
Show that x = q is a solution of the equation x2 + ln 3x = 0.
(4)
(c)
Show that the equation in part (b) can be rearranged in the form x =
1
3
ex .
2
(2)
(d)
 x2
Use the iteration formula xn + 1 = 13 e n , with x0 = 13 , to find x1, x2, x3 and x4. Hence
write down, to 3 decimal places, an approximation for q.
(3)
(Total 11 marks)
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21.
y
O
A
x
B
The diagram above shows a sketch of the curve with equation y = f(x) where the function f
is given by
f: x
ex 2 – 1,
x
.
The curve meets the x-axis at the point A and the y-axis at the point B.
(a)
Write down the coordinates of A and B.
(2)
(b)
Find, in the form f 1(x): x
. . ., the inverse function of f and state its domain.
(5)
(c)
Prove that the equation f(x) = x has a root  in the interval [3, 4].
(2)
(d)
Use the iterative formula
xn + 1 = f 1(xn), with x1 = 3.5,
to find  to 3 decimal places. Prove that your answer is correct to 3 decimal places.
(5)
(Total 14 marks)
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22.
(a)
Sketch the curve with equation y = ln x.
(2)
(b)
Show that the tangent to the curve with equation y = ln x at the point (e, 1) passes through
the origin.
(3)
(c)
Use your sketch to explain why the line y = mx cuts the curve y = ln x between x = 1 and
x = e if 0 < m <
1
.
e
(2)
1x
Taking x0 = 1.86 and using the iteration xn + 1 = e 3
(d)
n
,
calculate x1, x2, x3, x4 and x5 , giving your answer to x5 to 3 decimal places.
(3)
The root of ln x –
(e)
1
3
x = 0 is .
By considering the change of sign of ln x  13 x over a suitable interval, show that your
answer for x5 is an accurate estimate of , correct to 3 decimal places.
(3)
(Total 13 marks)
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