Pure Mathematics Year 2 (A Level) Unit Test 9: Numerical Methods
1
a Show that the equation f(x) = 0 can be written as , where a
and b are constants to be found. (2 marks)
b Let x0 = 1.5. Use the iteration formula , together with your
values of a and b from part a, to find, to 4 decimal places, the values of x1, x2, x3
and x4. (2 marks)
A root of f(x) = 0 is α.
c By choosing a suitable interval, prove that α = −2.782 to 3 decimal places. (3 marks)
2 , –40 < x < 20, x is in radians.
a Show that the equation g(x) = 0 can be written as
(3 marks)
b Using the formula ,
find, to 3 decimal places, the values of x1, x2 and x3. (2 marks)
3 , where x is in radians.
a Show that f(x) = 0 has a root α between x = 1.9 and x = 2.0. (2 marks)
b Using x0 = 1.95 as a first approximation, apply the Newton–Raphson procedure
once to f(x) to find a second approximation to α, giving your answer to 3
decimal places. (5 marks)
4
a By drawing an appropriate sketch, show that there is only one solution to the
equation g(x) = 0 (2 marks)
−x
b Show that the equation g(x) = 0 may be written in the form x = 2e + 1 (2 marks)
c Let x0 = 1.5. Use the iterative formula to find to 4 decimal places
the values of x1, x2, x3 and x4. (2 marks)
d Using x0 = 1.5 as a first approximation, apply the Newton–Raphson procedure
once to g(x) to find a second approximation to α, giving your answer to 4
decimal places. (4 marks)
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