Further numerical methodsAQA A-Level Further Maths: Topic test

1. 1
   The integral $I=\int_0^2\frac{1}{1+x^2}\,dx$ is to be estimated using Simpson's rule with four strips of equal width.

   (a)

   Which row gives the coefficients of $y_0,\ y_1,\ y_2,\ y_3,\ y_4$, in that order, inside the square bracket of Simpson's rule?

   [1 mark]

   • A$1,\ 2,\ 2,\ 2,\ 1$
   • B$1,\ 4,\ 2,\ 4,\ 1$
   • C$1,\ 2,\ 4,\ 2,\ 1$
   • D$4,\ 1,\ 2,\ 1,\ 4$

   (b)

   Find the estimate of $I$ given by Simpson's rule, to 4 decimal places.

   [1 mark]

   • A$1.1071$
   • B$1.1038$
   • C$2.2103$
   • D$1.1051$

   (c)

   The exact value of $I$ is $\tan^{-1}2=1.1071$ to 4 decimal places. Find the percentage error in the Simpson's rule estimate, to 2 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A population $P$ of bacteria, in thousands, at time $t$ hours satisfies $\frac{dP}{dt}=0.2P\left(1-\frac{P}{50}\right)$, with $P=10$ when $t=0$. Euler's method with step length $h=1$ is used.

   (a)

   Find the estimate of $P$ when $t=1$.

   [1 mark]

   • A$11.6$
   • B$12$
   • C$1.6$
   • D$11.8$

   (b)

   Find the estimate of $P$ when $t=2$, to 2 decimal places.

   [1 mark]

   • A$13.20$
   • B$13.92$
   • C$13.38$
   • D$13.46$

   (c)

   Explain why the true value of $P$ when $t=1$ is greater than the Euler estimate.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The integral $I=\int_1^3\ln x\,dx$ is to be estimated using Simpson's rule. The exact value is $3\ln3-2=1.2958$ to 4 decimal places.

   (a)

   Use Simpson's rule with four strips of equal width to estimate $I$. Give your answer to 4 decimal places.

   [3 marks]

   (b)

   Use Simpson's rule with two strips to estimate $I$. By comparing the errors, state which of the two Simpson's rule estimates is more accurate and by roughly how much.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve satisfies $\frac{dy}{dx}=\frac1x$ with $y=0$ when $x=1$. The exact solution is $y=\ln x$, so $y(2)=\ln2=0.6931$ to 4 decimal places.

   (a)

   Use Euler's method with step length $h=0.25$ to estimate $y(2)$, showing the value of $y$ at each step. Find the percentage error in your estimate, to 2 significant figures.

   [6 marks]

   (b)

   Because $\frac{dy}{dx}$ depends only on $x$, $y(2)=\int_1^2\frac1x\,dx$. Use Simpson's rule with four strips to estimate this integral, and by comparing the errors in the two methods explain why Simpson's rule is more accurate than Euler's method here.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The integral $I=\int_1^4\sqrt{x}\,dx$ is to be estimated using the mid-ordinate rule with three strips of equal width. The exact value is $\frac{14}{3}$.

   (a)

   At which values of $x$ are the ordinates evaluated?

   [1 mark]

   • A$1,\ 2,\ 3$
   • B$2,\ 3,\ 4$
   • C$1.5,\ 2.5,\ 3.5$
   • D$1,\ 2.5,\ 4$

   (b)

   Find the mid-ordinate estimate of $I$, to 3 decimal places.

   [1 mark]

   • A$4.677$
   • B$4.646$
   • C$4.146$
   • D$5.146$

   (c)

   Explain why the estimate is greater than the exact value.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A curve satisfies $\frac{dy}{dx}=xy$ with $y=2$ when $x=1$. Estimates of $y$ are found using a step length $h=0.1$. The exact solution is $y=2e^{(x^2-1)/2}$.

   (a)

   Use Euler's method to estimate $y$ when $x=1.2$.

   [1 mark]

   • A$2.2$
   • B$2.4$
   • C$2.464$
   • D$2.442$

   (b)

   The first step gives $y=2.2$ when $x=1.1$. Use the improved Euler method $y_{r+1}=y_{r-1}+2hf(x_r,y_r)$ to estimate $y$ when $x=1.2$.

   [1 mark]

   • A$2.442$
   • B$2.484$
   • C$2.42$
   • D$2.684$

   (c)

   Use the exact solution to find the error in each of the estimates of $y(1.2)$ found in parts (a) and (b), and state which method is more accurate.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The speed $v$ of a train, in m s⁻¹, is measured every $10$ seconds as it leaves a station. The readings at $t=0,\ 10,\ 20,\ \ldots,\ 60$ seconds are $0,\ 4.2,\ 9.1,\ 12.6,\ 14.0,\ 14.8$ and $15.0$.

   (a)

   Use Simpson's rule to estimate the distance travelled by the train in the first 60 seconds.

   [3 marks]

   (b)

   Use the mid-ordinate rule with three strips of width $20$ s, using the readings at $t=10,\ 30$ and $50$, to find a second estimate of the distance. Find the difference between the two estimates as a percentage of the Simpson's rule estimate, to 2 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A curve satisfies $\frac{dy}{dx}=\sin x$ with $y=0$ when $x=0$, where $x$ is in radians. The exact solution is $y=1-\cos x$, so $y\left(\frac{\pi}{2}\right)=1$.

   (a)

   Use Euler's method with step length $h=\frac{\pi}{4}$ to estimate $y\left(\frac{\pi}{4}\right)$. Then use the improved Euler method $y_{r+1}=y_{r-1}+2hf(x_r,y_r)$ to estimate $y\left(\frac{\pi}{2}\right)$, and find the percentage error in this estimate, to 3 significant figures.

   [6 marks]

   (b)

   Use Simpson's rule with two strips to estimate $\int_0^{\pi/2}\sin x\,dx$, which equals $y\left(\frac{\pi}{2}\right)$, and find the percentage error. Show that the improved Euler estimate in part (a) is equal to the mid-ordinate rule estimate with one strip.

   [6 marks]

   Total for question 8: 12 marks

End of questions