# IBDP Past Year Exam Papers – Maclauren Series

1. [M17/P3/TZ0]

Let the Maclaurin series for be  where  and  ${a}_{5}$ are constants.

(a) Find series for  , in terms of  and  ${a}_{5}$ , up to and including the  ${x}^{4}$ term

(i) by differentiating the above series for  ;
(ii) by using the relationship  .   [3 marks]

(b) Hence, by comparing your two series, determine the values of  and  ${a}_{5}$ .   [3 marks]

2. [N16/P3/TZ0]

(a) By successive differentiation find the first four non-zero terms in the Maclaurin series for . [11 marks]

(b) Deduce that, for n ≥ 2 , the coefficient of  ${x}^{n}$ in this series is  ${\left(–1\right)}^{n}\frac{1}{n\left(n–1\right)}$ .   [1 mark]

3. [M16]

Consider the functions  and  ${g}_{n}\left(x\right)={f}_{n}\left(x\right)\mathrm{tan}x$ .

(a)        Show that

(i)  $\frac{d{f}_{n}\left(x\right)}{dx}=n{g}_{n}\left(x\right);$

(ii)   $\frac{d{g}_{n}\left(x\right)}{dx}=\left(n+1\right){f}_{n+2}\left(x\right)–n{f}_{n}\left(x\right)$ .    [5 marks]

(b)     (i)   Use these results to show that the Maclauren series for the function  ${f}_{5}\left(x\right)$ up to and including the term in  ${x}^{4}$ is  $1+\frac{5}{2}{x}^{2}+\frac{85}{24}{x}^{4}$ .

(ii)   By considering the general form of its higher derivatives explain briefly why all coefficients in the Maclauren series for the function ${f}_{5}\left(x\right)$ are either positive or zero.

(iii)   Hence show that  $se{c}^{3}\left(0.1\right)>1.02535$ .    [14 marks]

4. [N15/P3/TZ0]

Let  $f\left(x\right)={e}^{x}\mathrm{sin}x$ .

(a) Show that  $f‘‘\left(x\right)=2\left(f‘\left(x\right)–f\left(x\right)\right)$ .   [4 marks]

(b) By further differentiation of the result in part (a) , find the Maclaurin expansion of  $f\left(x\right)$  , as far as the term in  ${x}^{5}$ .  [6 marks]

5. [M15/P3/TZ0]

The function $f$ is defined by .

By finding a suitable number of derivatives of  $f$ , determine the first non-zero term in its Maclaurin series.   [7 marks]

6. [M09/P3/TZ0]

The variables $x$ and  $y$ are related by  .
(a) Find the Maclaurin series for  $y$ up to and including the term in ${x}^{2}$ given that when  $x=0$ . [7 marks]
(b) Solve the differential equation given that  $y=0$ when  $x=\mathrm{\pi }$ . Give the solution in the form  $y=f\left(x\right)$ . [10 marks]

7. [N09/P3/TZ0]

The function  $f$ is defined by  $f\left(x\right)={e}^{\left({e}^{x}–1\right)}$ .

(a) Assuming the Maclaurin series for  ${e}^{x}$ , show that the Maclaurin series for  $f\left(x\right)$ is  [5 marks]

(b) Hence or otherwise find the value of  .  [5 marks]

8. [M10/P3/TZ0]

(a) Using the Maclaurin series for  ${\left(1+x\right)}^{n}$ , write down and simplify the Maclaurin series approximation for ${\left(1–{x}^{2}\right)}^{\frac{1}{2}}$ as far as the term in  ${x}^{4}$ . [3 marks]

(b) Use your result to show that a series approximation for arccos  $x$ is arccos  .    [3 marks]

(c) Evaluate  .   [5 marks]

(d) Use the series approximation for arccos  $x$ to find an approximate value for

${\int }_{0}^{0.2}arc\mathrm{cos}\left(\sqrt{x}\right)dx$ ,

giving your answer to 5 decimal places. Does your answer give the actual value of the integral to 5 decimal places?    [6 marks]

9. [N10/P3/TZ0]

(a) Using the Maclaurin series for the function ex , write down the first four terms of the Maclaurin series for   .   [3 marks]

(b) Hence find the first four terms of the series for  .  [3 marks]

(c) Use the result from part (b) to find an approximate value for   .    [3 marks]