Numerical methods Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Numerical methods questions. See exactly how to solve problems on mid-ordinate-rule, numerical-integration, simpsons-rule, eulers-method.

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Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
Use the mid-ordinate rule with 22 strips to estimate ∫0111+x2 dx\int_{0}^{1}\frac{1}{1+x^{2}}\,dx, giving your answer to 44 decimal places.

Worked solution

  1. Quote the mid-ordinate rule

    ∫abf(x) dx≈h(f(m1)+f(m2)+⋯+f(mn))\int_{a}^{b}f(x)\,dx\approx h\left(f(m_{1})+f(m_{2})+\cdots+f(m_{n})\right)

    Each strip is replaced by a rectangle whose height is the value of ff at the middle of that strip.

  2. Find hh, the mid-ordinates and the function values

    h=0.5,m1=0.25,m2=0.75,f(0.25)=0.94117647,f(0.75)=0.64000000h=0.5,\quad m_{1}=0.25,\quad m_{2}=0.75,\quad f\left(0.25\right)=0.94117647,\quad f\left(0.75\right)=0.64000000

    All the ingredients of the rule in one place.

  3. State the estimate to the required accuracy

    I≈0.5(0.94117647+0.64000000)=0.79058824≈0.7906I\approx0.5\left(0.94117647+0.64000000\right)=0.79058824\approx0.7906

    The question asks for 44 decimal places, so the estimate is rounded at the very end.

Answer
0.79060.7906
Question 2
2 markseasy
Use the mid-ordinate rule with 22 strips to estimate ∫021+x3 dx\int_{0}^{2}\sqrt{1+x^{3}}\,dx, giving your answer to 44 decimal places.

Worked solution

  1. Quote the mid-ordinate rule

    ∫abf(x) dx≈h(f(m1)+f(m2)+⋯+f(mn))\int_{a}^{b}f(x)\,dx\approx h\left(f(m_{1})+f(m_{2})+\cdots+f(m_{n})\right)

    Each strip is replaced by a rectangle whose height is the value of ff at the middle of that strip.

  2. Find the strip width hh

    h=b−an=2−02=1h=\frac{b-a}{n}=\frac{2-0}{2}=1

    The interval is divided into 22 strips of equal width.

  3. Find the mid-ordinates and the function values

    m1=0.5,m2=1.5,f(0.5)=1.06066017,f(1.5)=2.09165007m_{1}=0.5,\quad m_{2}=1.5,\quad f\left(0.5\right)=1.06066017,\quad f\left(1.5\right)=2.09165007

    All the ingredients of the rule in one place.

  4. State the estimate to the required accuracy

    I≈1(1.06066017+2.09165007)=3.15231024≈3.1523I\approx1\left(1.06066017+2.09165007\right)=3.15231024\approx3.1523

    The question asks for 44 decimal places, so the estimate is rounded at the very end.

Answer
3.15233.1523
Question 3
2 markseasy
Use the mid-ordinate rule with 22 strips to estimate ∫01e−x2 dx\int_{0}^{1}e^{-x^{2}}\,dx, giving your answer to 44 decimal places.

Worked solution

  1. Quote the mid-ordinate rule

    ∫abf(x) dx≈h(f(m1)+f(m2)+⋯+f(mn))\int_{a}^{b}f(x)\,dx\approx h\left(f(m_{1})+f(m_{2})+\cdots+f(m_{n})\right)

    Each strip is replaced by a rectangle whose height is the value of ff at the middle of that strip.

  2. Find hh, the mid-ordinates and the function values

    h=0.5,m1=0.25,m2=0.75,f(0.25)=0.93941306,f(0.75)=0.56978282h=0.5,\quad m_{1}=0.25,\quad m_{2}=0.75,\quad f\left(0.25\right)=0.93941306,\quad f\left(0.75\right)=0.56978282

    All the ingredients of the rule in one place.

  3. State the estimate to the required accuracy

    I≈0.5(0.93941306+0.56978282)=0.75459794≈0.7546I\approx0.5\left(0.93941306+0.56978282\right)=0.75459794\approx0.7546

    The question asks for 44 decimal places, so the estimate is rounded at the very end.

Answer
0.75460.7546
Question 4
2 markseasy
Use the mid-ordinate rule with 22 strips to estimate ∫121x dx\int_{1}^{2}\frac{1}{x}\,dx, giving your answer to 44 decimal places.

Worked solution

  1. Quote the mid-ordinate rule

    ∫abf(x) dx≈h(f(m1)+f(m2)+⋯+f(mn))\int_{a}^{b}f(x)\,dx\approx h\left(f(m_{1})+f(m_{2})+\cdots+f(m_{n})\right)

    Each strip is replaced by a rectangle whose height is the value of ff at the middle of that strip.

  2. Find the strip width hh

    h=b−an=2−12=0.5h=\frac{b-a}{n}=\frac{2-1}{2}=0.5

    The interval is divided into 22 strips of equal width.

  3. Find the mid-ordinates and the function values

    m1=1.25,m2=1.75,f(1.25)=0.80000000,f(1.75)=0.57142857m_{1}=1.25,\quad m_{2}=1.75,\quad f\left(1.25\right)=0.80000000,\quad f\left(1.75\right)=0.57142857

    All the ingredients of the rule in one place.

  4. State the estimate to the required accuracy

    I≈0.5(0.80000000+0.57142857)=0.68571429≈0.6857I\approx0.5\left(0.80000000+0.57142857\right)=0.68571429\approx0.6857

    The question asks for 44 decimal places, so the estimate is rounded at the very end.

Answer
0.68570.6857
Question 5
2 markseasy
Use the mid-ordinate rule with 22 strips to estimate ∫02ln⁡(1+x) dx\int_{0}^{2}\ln(1+x)\,dx, giving your answer to 33 decimal places.

Worked solution

  1. Quote the mid-ordinate rule

    ∫abf(x) dx≈h(f(m1)+f(m2)+⋯+f(mn))\int_{a}^{b}f(x)\,dx\approx h\left(f(m_{1})+f(m_{2})+\cdots+f(m_{n})\right)

    Each strip is replaced by a rectangle whose height is the value of ff at the middle of that strip.

  2. Find hh, the mid-ordinates and the function values

    h=1,m1=0.5,m2=1.5,f(0.5)=0.40546511,f(1.5)=0.91629073h=1,\quad m_{1}=0.5,\quad m_{2}=1.5,\quad f\left(0.5\right)=0.40546511,\quad f\left(1.5\right)=0.91629073

    All the ingredients of the rule in one place.

  3. State the estimate to the required accuracy

    I≈1(0.40546511+0.91629073)=1.32175584≈1.322I\approx1\left(0.40546511+0.91629073\right)=1.32175584\approx1.322

    The question asks for 33 decimal places, so the estimate is rounded at the very end.

Answer
1.3221.322

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