IBDP Past Year Exam Papers – Maclauren Series

1. [M17/P3/TZ0]

Let the Maclaurin series for tan x be  tan x= a1x + a3x3 + a5x5 + ...... where  a1 , a3  and  a5 are constants.

(a) Find series for  sec2 x , in terms of  a1 , a3 and  a5 , up to and including the  x4 term

(i) by differentiating the above series for  tan x ;
(ii) by using the relationship  sec2 x= 2 + tan2 x .   [3 marks]

(b) Hence, by comparing your two series, determine the values of  a1 , a3 and  a5 .   [3 marks]

2. [N16/P3/TZ0]

(a) By successive differentiation find the first four non-zero terms in the Maclaurin series for fx=x+1ln1+x  x . [11 marks]

(b) Deduce that, for n ≥ 2 , the coefficient of  xn in this series is  1n1nn1 .   [1 mark]

3. [M16]

Consider the functions  fnx=secnx, x<π2 and  gnx=fnxtanx .

(a)        Show that

           (i)  dfnxdx=ngnx;

          (ii)   dgnxdx=n+1fn+2xnfnx .    [5 marks]

(b)     (i)   Use these results to show that the Maclauren series for the function  f5x up to and including the term in  x4 is  1+52x2+8524x4 .

        (ii)   By considering the general form of its higher derivatives explain briefly why all coefficients in the Maclauren series for the function f5x are either positive or zero.

       (iii)   Hence show that  sec30.1>1.02535 .    [14 marks]

4. [N15/P3/TZ0]

Let  fx=exsinx .

(a) Show that  fx=2fxfx .   [4 marks]

(b) By further differentiation of the result in part (a) , find the Maclaurin expansion of  fx  , as far as the term in  x5 .  [6 marks]

5. [M15/P3/TZ0]

The function f is defined by fx=excosx + x  1 .

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  dxdy  ytanx=cosx .
(a) Find the Maclaurin series for  y up to and including the term in x2 given that when  x=0 . [7 marks]
(b) Solve the differential equation given that  y=0 when  x=π . Give the solution in the form  y=fx . [10 marks]

7. [N09/P3/TZ0]

The function  f is defined by  fx=eex1 .

(a) Assuming the Maclaurin series for  ex , show that the Maclaurin series for  fx is  1 + x +x2 + 56x3 +........  [5 marks]

(b) Hence or otherwise find the value of  limx0fx  1fx  1 .  [5 marks]

8. [M10/P3/TZ0]

(a) Using the Maclaurin series for  1+xn , write down and simplify the Maclaurin series approximation for 1x212 as far as the term in  x4 . [3 marks]

(b) Use your result to show that a series approximation for arccos  x is arccos  xπ2  x  16x3  340x5 .    [3 marks]

(c) Evaluate  limx0π2arccosx2  x2x6 .   [5 marks]

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

                                      00.2arccosxdx ,    

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   e x22 .   [3 marks]

(b) Hence find the first four terms of the series for  0xe u22du .  [3 marks]

(c) Use the result from part (b) to find an approximate value for   12π01e x22dx .    [3 marks]