FP2: Maclaurin and Taylor seriesEdexcel International A Level Further Maths: Topic test

1. 1
   A function is defined by $f(x)=\mathrm{e}^x\sin x$.

   (a)

   Find $f''(x)$.

   [1 mark]

   • A$2\mathrm{e}^x\cos x$
   • B$\mathrm{e}^x(\sin x+\cos x)$
   • C$2\mathrm{e}^x\sin x$
   • D$\mathrm{e}^x(\cos x-\sin x)$

   (b)

   Find the coefficient of $x^3$ in the Maclaurin series of $f(x)$.

   [1 mark]

   • A$2$
   • B$\frac12$
   • C$\frac13$
   • D$\frac16$

   (c)

   Verify the coefficient of $x^3$ by multiplying the standard series for $\mathrm{e}^x$ and $\sin x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A function is defined by $g(x)=\cos x$, and its Taylor series is to be found in ascending powers of $\left(x-\frac{\pi}{3}\right)$.

   (a)

   Find the coefficient of $\left(x-\frac{\pi}{3}\right)$.

   [1 mark]

   • A$\frac{\sqrt3}{2}$
   • B$-\frac{\sqrt3}{2}$
   • C$-\frac12$
   • D$\frac12$

   (b)

   Find the coefficient of $\left(x-\frac{\pi}{3}\right)^2$.

   [1 mark]

   • A$-\frac12$
   • B$\frac14$
   • C$-\frac{\sqrt3}{4}$
   • D$-\frac14$

   (c)

   Use the series up to and including the term in $\left(x-\frac{\pi}{3}\right)^2$ to estimate $\cos1$, giving your answer to 3 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A function is defined by $f(x)=\ln(1+\sin x)$, for $-\frac{\pi}{2}<x<\frac{\pi}{2}$.

   (a)

   Show that $f''(x)=-\frac{1}{1+\sin x}$.

   [3 marks]

   (b)

   Hence find the Maclaurin series of $f(x)$ up to and including the term in $x^3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}+y=\mathrm{e}^x$, with $y=0$ and $\frac{dy}{dx}=1$ at $x=0$.

   (a)

   Use the Taylor series method to find $y$ as a series in ascending powers of $x$, up to and including the term in $x^5$.

   [6 marks]

   (b)

   It is given that the exact solution is $y=\frac12\left(\mathrm{e}^x-\cos x+\sin x\right)$. Using the standard Maclaurin series for $\mathrm{e}^x$, $\cos x$ and $\sin x$, show that this agrees with your series in part (a).

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A function is defined by $f(x)=\frac1x$, for $x>0$, and its Taylor series is to be found in ascending powers of $(x-2)$.

   (a)

   Find the coefficient of $(x-2)^2$.

   [1 mark]

   • A$\frac14$
   • B$-\frac18$
   • C$\frac18$
   • D$\frac{1}{16}$

   (b)

   Find the coefficient of $(x-2)^3$.

   [1 mark]

   • A$-\frac{1}{16}$
   • B$-\frac38$
   • C$\frac{1}{16}$
   • D$-\frac{1}{48}$

   (c)

   Use the series up to and including the term in $(x-2)^2$ to estimate $\frac{1}{2.1}$, giving your answer to 5 significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}=2\frac{dy}{dx}-xy$, with $y=1$ and $\frac{dy}{dx}=0$ at $x=0$.

   (a)

   Which expression is $\frac{d^3y}{dx^3}$?

   [1 mark]

   • A$2\frac{d^2y}{dx^2}-xy$
   • B$2\frac{d^2y}{dx^2}-y+x\frac{dy}{dx}$
   • C$2\frac{d^2y}{dx^2}-xy'$
   • D$2\frac{d^2y}{dx^2}-y-x\frac{dy}{dx}$

   (b)

   Find the value of $\frac{d^4y}{dx^4}$ at $x=0$.

   [1 mark]

   • A$-1$
   • B$-2$
   • C$0$
   • D$2$

   (c)

   Hence find the series solution for $y$ in ascending powers of $x$, up to and including the term in $x^4$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A function is defined by $f(x)=\sin2x$, and its Taylor series is to be found in ascending powers of $\left(x-\frac{\pi}{4}\right)$.

   (a)

   Find the series up to and including the term in $\left(x-\frac{\pi}{4}\right)^3$.

   [3 marks]

   (b)

   Find the coefficient of $\left(x-\frac{\pi}{4}\right)^4$, and use the series up to and including this term to estimate $\sin2.4$, giving your answer to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A function is defined by $f(x)=\ln\left(1+\mathrm{e}^x\right)$.

   (a)

   Find the Maclaurin series of $f(x)$ up to and including the term in $x^2$.

   [6 marks]

   (b)

   Find the Taylor series of $f(x)$ in ascending powers of $(x-\ln3)$ up to and including the term in $(x-\ln3)^2$, and use it to estimate $f(1.1)$, giving your answer to 5 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions