Numerical methodsAQA A-Level Maths: Topic test

1. 1
   Let $f(x)=x^3-7x+5$, which is continuous for all real $x$. Some values are $f(-2)=11$, $f(-1)=11$, $f(0)=5$, $f(1)=-1$, $f(2)=-1$, $f(3)=11$ and $f(4)=41$.

   (a)

   Which of these intervals must contain a root of $f(x)=0$?

   [1 mark]

   • A$0\le x\le1$
   • B$1\le x\le2$
   • C$3\le x\le4$
   • D$-1\le x\le0$

   (b)

   Given only $f(-2)=11$ and $f(3)=11$, which conclusion about roots in the interval $-2\le x\le3$ is correct?

   [1 mark]

   • AThere are no roots, because $f(-2)$ and $f(3)$ have the same sign
   • BThere is exactly one root, because both values are positive
   • CThe test is inconclusive: there may be no roots or an even number of roots
   • DThere must be a root, because the two values are equal

   (c)

   Show that $f(x)=0$ has a root between $x=2$ and $x=3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The equation $e^{-x}=x-1$ has a root $\alpha$ close to $1.3$. It can be rearranged as $x=1+e^{-x}$, giving the iteration $x_{n+1}=1+e^{-x_n}$ with $x_0=1$.

   (a)

   What is $x_2$, correct to 3 decimal places?

   [1 mark]

   • A$1.368$
   • B$1.255$
   • C$1.285$
   • D$1.000$

   (b)

   Which statement correctly describes how the iterates approach $\alpha$?

   [1 mark]

   • AThey increase steadily towards $\alpha$ in a staircase pattern, because $g'(\alpha)>0$
   • BThey diverge from $\alpha$, because $|g'(\alpha)|>1$
   • CThey decrease steadily towards $\alpha$ in a staircase pattern, because $0<g'(\alpha)<1$
   • DThey alternate either side of $\alpha$ and converge in a cobweb pattern, because $-1<g'(\alpha)<0$

   (c)

   Find $x_4$, correct to 4 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\displaystyle\int_0^2\ln\left(1+x^2\right)dx$. The function $y=\ln\left(1+x^2\right)$ is increasing for $x\ge0$.

   (a)

   Use the trapezium rule with 4 strips of equal width to estimate $I$, giving your answer to 3 decimal places.

   [3 marks]

   (b)

   Use rectangles of width $0.5$ to find a lower bound and an upper bound for $I$, giving each to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The equation $x^4-3x-1=0$ has a positive root $\alpha$. Let $f(x)=x^4-3x-1$.

   (a)

   Show that $\alpha$ lies between $1.5$ and $1.6$. Starting with $x_0=1.5$, use the Newton-Raphson method twice to find $x_1$ and $x_2$ to 4 decimal places.

   [6 marks]

   (b)

   (i) Explain why the Newton-Raphson method fails when $x_0=\left(\frac34\right)^{1/3}$. (ii) Show that $f(x)=0$ can be rearranged as $x=(3x+1)^{1/4}$, and use $x_{n+1}=(3x_n+1)^{1/4}$ with $x_0=1.5$ to find $x_1$ and $x_2$ to 4 decimal places.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The integral $I=\displaystyle\int_0^3 10e^{-x/2}\,dx$ is estimated using the trapezium rule with 3 strips of equal width.

   (a)

   What is the estimate, correct to 3 significant figures?

   [1 mark]

   • A$15.5$
   • B$31.7$
   • C$12.2$
   • D$15.9$

   (b)

   Which statement about the estimate is correct?

   [1 mark]

   • AIt is an overestimate, because the curve is convex so each chord lies above the curve
   • BIt is an underestimate, because the curve is decreasing
   • CIt is an underestimate, because the curve is convex so each chord lies below the curve
   • DIt is exact, because the strips have equal width

   (c)

   Find the exact value of $I$ and hence the percentage error in the estimate, correct to 2 significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $f(x)=\dfrac{x-1}{x-3}$ for $x\ne3$.

   (a)

   In which interval do the values of $f$ change sign because of a root of $f(x)=0$?

   [1 mark]

   • A$2\le x\le4$
   • B$4\le x\le6$
   • C$0\le x\le2$
   • D$-2\le x\le0$

   (b)

   Why does the sign change of $f$ between $x=2$ and $x=4$ not indicate a root?

   [1 mark]

   • A$f(2)$ and $f(4)$ are both undefined
   • B$f$ is not continuous at $x=3$, which lies in the interval
   • C$f$ has a repeated root in the interval
   • DThe interval is too wide

   (c)

   Show that $f(0)$ and $f(5)$ have the same sign, even though the equation $f(x)=0$ has a root between $0$ and $5$. Explain how this happens.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The concentration $C$ mg per litre of a drug in a patient's blood, $t$ hours after an injection, is modelled by $C=12te^{-t/2}$. The concentration is $5$ mg per litre at two times, and $T$ is the later of these. Let $f(t)=12te^{-t/2}-5$.

   (a)

   Show that $T$ lies between $4.9$ and $5$.

   [3 marks]

   (b)

   Show that $f(t)=0$ can be rearranged as $t=2\ln\left(\dfrac{12t}{5}\right)$. Use the iteration $t_{n+1}=2\ln\left(\dfrac{12t_n}{5}\right)$ with $t_0=5$ to find $t_1$ and $t_2$ to 3 decimal places.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $f(x)=x^3-2x+2$. The equation $f(x)=0$ has exactly one real root $\alpha$.

   (a)

   Show that $\alpha$ lies between $-2$ and $-1.5$. A student uses the Newton-Raphson method with $x_0=0$. Find $x_1$ and $x_2$ and describe what happens to the sequence.

   [6 marks]

   (b)

   Starting with $x_0=-1.5$, use the Newton-Raphson method twice to find $x_1$ and $x_2$ to 4 decimal places. By evaluating $f(-1.775)$ and $f(-1.765)$, show that $\alpha=-1.77$ to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions