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Edexcel IAL Mathematics P3.6 numerical methods

Numerical methods questions link sign-change evidence with iteration: evaluate a stated function, choose a valid interval and round successive approximations carefully.

Syllabus
First assessment 2019
Course
Mathematics YMA01
Level
A2

Exam points

  • Evaluate f(a) and f(b) to prove a sign change across a stated interval.
  • Use a narrow interval and named function to justify a root to 3 or 4 d.p.
  • Apply recurrence formulae from x1 to calculate iterates and rounded roots.

Covered syllabus points

P3.6.1 - Locating roots using sign changesP3.6.2 - Iteration and approximate equation solving

P3.6 - Numerical methods question 1

[Maximum number: 7]
Figure 1

Figure 1

Figure 1 shows a sketch of part of the curve with equation y=f(x) where

f(x)=2x2+3x−4ex−1x2x∈Rx≠0\mathrm{f}(x)=\frac{2 x^{2}+3 x-4}{\mathrm{e}^{x}}-\frac{1}{x^{2}} \quad x \in \mathbb{R} \quad x \neq 0

Question (a)

(a)

Show that f(x)=0 has a root α\alpha in the interval [1,2]

[ 2 ]

Question (b)

(b)

Show that the equation f(x)=0 can be written in the form

x=ex+4x22x+33x=\sqrt[3]{\frac{\mathrm{e}^{x}+4 x^{2}}{2 x+3}}

Using the iteration formula

xn+1=exn+4xn22xn+33 with x1=1x_{n+1}=\sqrt[3]{\frac{\mathrm{e}^{x_{n}}+4 x_{n}^{2}}{2 x_{n}+3}} \quad \text { with } x_{1}=1

find, to 4 decimal places,

[ 2 ]

Question (c)

(c)

the value of x3x_{3}

[ 3 ]

P3.6 - Numerical methods question 2

[Maximum number: 7]
Figure 1

Figure 1

Figure 1 shows a sketch of part of the curve with equation

y=6ln⁡(2x+3)−12x2+4x>−32y=6 \ln (2 x+3)-\frac{1}{2} x^{2}+4 \quad x>-\frac{3}{2}

The curve cuts the negative x-axis at the point P, as shown in Figure 1.

Question (a)

(a)

Show that the x coordinate of P lies in the interval [-1.25,-1.2]

The curve cuts the positive x-axis at the point Q, also shown in Figure 1.
Using the iterative formula

xn+1=12ln⁡(2xn+3)+8 with x1=6x_{n+1}=\sqrt{12 \ln \left(2 x_{n}+3\right)+8} \quad \text { with } \quad x_{1}=6
[ 2 ]

Question (b)

(b)

find, to 4 decimal places, the value of x2x_{2}

[ 2 ]

Question (c)

(c)

find, by continued iteration, the x coordinate of Q. Give your answer to 4 decimal places.

The curve has a maximum turning point at M, as shown in Figure 1.

[ 3 ]
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