Numerical methods – Year 13 A level maths worksheet with answers to download as PDF

\(f(x)=x^3+2x-7\).

1. Show that \(f(x)=0\) has a root in the interval \([1,2]\). [2 marks]
2. By evaluating \(f(1.5)\) and \(f(1.6)\), show that the root lies between 1.5 and 1.6. [2 marks]
3. Explain why the change of sign of \(g(x)=\dfrac1{x-1}\) between \(x=0\) and \(x=2\) does not mean there is a root. [2 marks]

The equation \(x^3+2x-7=0\) can be rearranged as \(x=\sqrt[3]{7-2x}\).

1. Use \(x_{n+1}=\sqrt[3]{7-2x_n}\) with \(x_0=1.5\) to find \(x_1,\dots,x_5\) to 4 decimal places. [3 marks]
2. Describe the pattern of the values. [1 mark]
3. Write down the root to 2 decimal places. [1 mark]

\(f(x)=x^3+2x-7\) has a root near 1.5.

1. Find \(f\prime(x)\). [1 mark]
2. Use the Newton-Raphson formula with \(x_0=1.5\) to find \(x_1\) and \(x_2\) to 5 decimal places. [4 marks]
3. Hence state the root correct to 4 decimal places. [1 mark]

Use the trapezium rule with 4 strips to estimate \(\displaystyle\int_1^2\dfrac1x\,dx\).

1. Complete a table of ordinates to 4 decimal places. [2 marks]
2. Calculate the estimate to 4 decimal places. [3 marks]
3. The exact value is \(\ln2=0.6931\). State whether the estimate is too large or too small and explain why. [2 marks]

The equation \(x=\cos x\) has one root \(\alpha=0.7391\) (4 d.p.). Consider \(x_{n+1}=\cos x_n\) with \(x_0=1\).

1. Calculate \(x_1,\dots,x_5\) to 4 decimal places (radians). [3 marks]
2. Find \(g\prime(x)\) where \(g(x)=\cos x\) and evaluate \(g\prime(\alpha)\). [2 marks]
3. Explain why the iteration converges. [1 mark]

The equation \(e^x=3x\) has two positive roots. Let \(f(x)=e^x-3x\).

1. Show that one root lies in \([0.6,0.7]\). [2 marks]
2. Show that the other root lies in \([1.5,1.6]\). [2 marks]
3. Explain how a sketch of \(y=e^x\) and \(y=3x\) shows there are exactly two roots. [2 marks]

A square sheet of card of side 30 cm has squares of side \(x\) cm cut from each corner. The sides are folded up to make an open box (a pizza tray).

1. Show that the volume of the box is \(V=x(30-2x)^2\). [2 marks]
2. The volume must be 1000 cm\(^3\). Show that a suitable value of \(x\) lies between 1 and 1.5. [3 marks]
3. Use Newton-Raphson on \(f(x)=x(30-2x)^2-1000\) with \(x_0=1.2\) to find \(x\) to 3 decimal places. [4 marks]
4. Give one reason why \(x=10\) is also a solution but not practical. [1 mark]

A customer borrows 1000 pounds and repays it in 5 equal annual instalments of 250 pounds. The annual interest rate \(r\) (as a decimal) satisfies \(f(r)=250\,\dfrac{1-(1+r)^{-5}}{r}-1000=0\).

1. Show that \(r\) lies between 0.07 and 0.08. [3 marks]
2. By evaluating \(f(0.079)\) and \(f(0.0795)\), find the rate to 3 decimal places. [4 marks]
3. Express the rate as a percentage and comment on whether it is a sensible rate for a loan. [1 mark]

The depth of a river is measured every 2 m across its width.

1. Use the trapezium rule with 6 strips to estimate the cross-sectional area of the river. [4 marks]
2. The water flows at an average speed of 0.8 m/s. Estimate the volume of water passing per second. [2 marks]
3. State whether the trapezium rule would give an over- or under-estimate if the bed were smooth and convex downward, and why. [1 mark]

The fuel flow rate of a delivery van is recorded every 10 minutes during a 60-minute journey.

1. Use the trapezium rule to estimate the total fuel used. [4 marks]
2. The van tank held 20 litres. Estimate the percentage of the tank used. [2 marks]
3. Explain what the area under the rate-time graph represents. [1 mark]

A spherical tank of radius 1.5 m is partly filled with water to a depth \(h\) m. The volume of water is \(V=\dfrac{\pi h^2(4.5-h)}3\) cubic metres. The tank must contain exactly 3 m\(^3\).

1. Show that \(h^2(4.5-h)=\dfrac9\pi\). [2 marks]
2. Show that \(h\) lies between 0.89 and 0.9. [3 marks]
3. Use Newton-Raphson on \(f(h)=h^2(4.5-h)-\dfrac9\pi\), with \(h_0=1\), to find \(h\) to 3 decimal places. [4 marks]

A biologist models the time \(x\) (hours) at which a culture reaches a target size by \(x+\ln x=3\).

1. Show that the equation has a root in \([2,3]\). [2 marks]
2. Show that the equation can be rearranged as \(x=3-\ln x\). [1 mark]
3. Use \(x_{n+1}=3-\ln x_n\) with \(x_0=2\) to find \(x_1,\dots,x_6\) to 4 decimal places. [3 marks]
4. Write the root to 2 decimal places and explain why the iteration converges. [3 marks]

A playground designer models the profile of a slide by \(f(x)=x^3-2x+2\) and wants the point where it meets the ground, \(f(x)=0\).

1. Show that \(f(x)=0\) has a root in \([-2,-1.5]\). [3 marks]
2. Use Newton-Raphson with \(x_0=0\) and write down \(x_1\), \(x_2\), \(x_3\), \(x_4\). [4 marks]
3. Explain what has gone wrong and suggest a better starting value. [2 marks]

\(f(x)=x^3-4x-3\).

1. Show that \(f(x)=0\) has a root \(\alpha\) in the interval \([2.3,2.31]\). [2 marks]
2. The equation can be written as \(x=\sqrt[3]{4x+3}\). Use \(x_{n+1}=\sqrt[3]{4x_n+3}\) with \(x_0=2\) to find \(x_1,x_2,x_3,x_4\) to 4 decimal places. [3 marks]
3. Use Newton-Raphson with \(x_0=2.3\) to find \(\alpha\) correct to 4 decimal places, showing \(x_1\) and \(x_2\). [4 marks]
4. By considering the sign of \(f(2.3025)\) and \(f(2.3035)\) prove that \(\alpha=2.303\) to 3 decimal places. [3 marks]

The equation \(x^3-x-1=0\) has a root \(\alpha\approx1.3247\). It can be rearranged in two ways: (A) \(x=x^3-1\) and (B) \(x=\sqrt[3]{x+1}\).

1. Use each formula with \(x_0=1.5\) to find \(x_1,x_2,x_3\) (A) and \(x_1,\dots,x_5\) (B). [4 marks]
2. Find the gradient function of each rearrangement and evaluate it at \(\alpha\). [4 marks]
3. Hence explain which rearrangement is suitable and why the other fails. [2 marks]
4. Describe the pattern of values for (B) and sketch the staircase diagram in words. [2 marks]

The integral \(I=\displaystyle\int_0^1e^{-x^2}dx\) cannot be found exactly by elementary methods. Its value to 4 decimal places is \(0.7468\).

1. Use the trapezium rule with 2 strips to estimate \(I\). [3 marks]
2. Use the trapezium rule with 4 strips to estimate \(I\). [3 marks]
3. Calculate the error of each estimate. Comment on how the error changes when the number of strips doubles. [3 marks]
4. State whether you expect the estimate to be too large or too small, giving a reason. [1 mark]

Let \(f(x)=e^x-3x\), with roots near 0.62 and 1.51.

1. Write down the Newton-Raphson iteration for \(f\). [2 marks]
2. Starting with \(x_0=0.5\), find \(x_1,x_2,x_3\) to 5 decimal places. [3 marks]
3. Starting with \(x_0=2\), find \(x_1,x_2,x_3,x_4\) to 5 decimal places. [3 marks]
4. Starting with \(x_0=1\), find \(x_1\) and \(x_2\) and explain the difficulty. [4 marks]

A circular window of radius 5 dm has a segment cut off by a chord, as shown by the shaded region. The angle at the centre is \(\theta\) radians and the segment has area 12 dm\(^2\).

1. Show that \(\theta-\sin\theta=0.96\). [3 marks]
2. Show that \(\theta\) lies between 1.9 and 2.0. [3 marks]
3. Use Newton-Raphson from \(\theta_0=1.8\) to find \(\theta\) to 3 decimal places. [4 marks]
4. Hence find the length of the chord. [2 marks]

To find \(\sqrt a\) Newton-Raphson is applied to \(f(x)=x^2-a\).

1. Show that \(x_{n+1}=\dfrac12\left(x_n+\dfrac a{x_n}\right)\). [2 marks]
2. Show that \(x_{n+1}-\sqrt a=\dfrac{(x_n-\sqrt a)^2}{2x_n}\). [3 marks]
3. Starting with \(x_0=2.5\), use the formula to estimate \(\sqrt7\) to 6 decimal places. Explain why the number of correct digits roughly doubles. [4 marks]

The trapezium rule with \(n\) strips is applied to \(\displaystyle\int_0^1x^2\,dx=\dfrac13\).

1. Show that the estimate is \(T_n=\dfrac13+\dfrac1{6n^2}\). [5 marks]
2. Find \(T_4\) and \(T_8\) and the ratio of their errors. [2 marks]
3. Explain why the trapezium rule always overestimates this integral. [1 mark]