k
x
1. (a) Consider the power series
k .
k 1 2
(i) Find the radius of convergence.
(ii) Find the interval of convergence.
(10)
(1) k 1 k
(b) Consider the infinite series .
k 1 2k 1
2
(i) Show that the series is convergent.
(ii) Show that the sum to infinity of the series is less than 0.25.
(5)
(Total 15 marks)
dy
2. Given that – 2y2 = ex and y = 1 when x = 0, use Euler’s method with a step length of 0.1 to find an
dx
approximation for the value of y when x = 0.4. Give all intermediate values with maximum possible
accuracy.
(Total 8 marks)
dy
3. Consider the differential equation x 2 y 2 where y = 1 when x = 0.
dx
(a) Use Euler’s method with step length 0.1 to find an approximate value of y when x = 0.4.
(7)
(b)Write down, giving a reason, whether your approximate value for y is greater than or less than the actual
value of y.
(1)
(Total 8 marks)
4. (a) Show that the solution of the differential equation
dy
= cos x cos2 y,
dx
π
given that y = when x = π, is y = arctan (1 + sin x).
4
(5)
(b) Determine the value of the constant a for which the following limit exists
arctan(1 sin x) a
lim 2
π
x π
2 x
2
and evaluate that limit.
(12)
(Total 17 marks)
IB Questionbank Mathematics Higher Level 3rd edition 1
dy π
5. Given that + 2y tan x = sin x, and y = 0 when x = , find the maximum value of y.
dx 3
(Total 11 marks)
6. Solve the differential equation
dy
x2 y 2 3xy 2 x 2
dx
given that y = –1 when x = 1. Give your answer in the form y = f(x).
(Total 11 marks)
7. The function f is defined by
1
f (x) = ln .
1 x
(a) Write down the value of the constant term in the Maclaurin series for f (x).
(1)
(b) Find the first three derivatives of f (x) and hence show that the Maclaurin series for f (x) up to and
x2 x3
including the x3 term is x .
2 3
(6)
(c) Use this series to find an approximate value for ln 2.
(3)
(d) Use the Lagrange form of the remainder to find an upper bound for the error in this approximation.
(5)
(e) How good is this upper bound as an estimate for the actual error?
(2)
(Total 17 marks)
8. (a) (i) Find the first four derivatives with respect to x of y = ln(1 + sin x)
(ii) Hence, show that the Maclaurin series, up to the term in x4, for y is
1 2 1 3 1 4
y=x– x x x ...
2 6 12
(10)
(b) Deduce the Maclaurin series, up to and including the term in x4, for
(i) y = ln(1 – sin x);
(ii) y = ln cosx;
(iii) y = tan x.
(10)
tan( x 2 )
(c) Hence calculate lim .
x 0 ln cos x
(4)
(Total 24 marks)
IB Questionbank Mathematics Higher Level 3rd edition 2
dy
9. The variables x and y are related by – y tan x = cos x.
dx
π
(a) Find the Maclaurin series for y up to and including the term in x2 given that y =
2
when x = 0.
(7)
(b) Solve the differential equation given that y = 0 when x = π. Give the solution in the form
y = f(x).
(10)
(Total 17 marks)
10. (a) Find the first three terms of the Maclaurin series for ln (1 + ex).
(6)
2 ln(1 e x ) x ln 4
(b) Hence, or otherwise, determine the value of lim .
x 0 x2
(4)
(Total 10 marks)
1
11. (a) Find the value of lim cot x .
x 0 x
(6)
(b) Find the interval of convergence of the infinite series
( x 2) ( x 2) 2 ( x 2) 3
2 3 ... .
3 1 3 2 3 3
(10)
(c) (i) Find the Maclaurin series for ln(1 + sin x) up to and including the term in x3.
(ii) Hence find a series for ln(1 – sin x) up to and including the term in x3.
(iii) Deduce, by considering the difference of the two series, that
π π2
ln 3 ≈ 1 .
3 216
(12)
(Total 28 marks)
12. Solve the differential equation
dy
(x – 1) + xy = (x – 1)e–x
dx
given that y = 1 when x = 0. Give your answer in the form y = f(x).
(Total 13 marks)
xn
13. The exponential series is given by ex = n! .
n 0
(a) Find the set of values of x for which the series is convergent.
(4)
(b) (i) Show, by comparison with an appropriate geometric series, that
IB Questionbank Mathematics Higher Level 3rd edition 3
2x
ex – 1 < , for 0 < x < 2.
2 x
n
2n 1
, for n
+
(ii) Hence show that e < .
2n 1
(6)
(c) (i) Write down the first three terms of the Maclaurin series for 1 – e–x and explain why you are able
to state that
x2
1 – e–x > x , for 0 < x < 2.
2
n
2n 2
(ii) Deduce that e > 2 , for n
+
.
2 n 2 n 1
(4)
(d) Letting n = 1000, use the results in parts (b) and (c) to calculate the value of e correct to as many
decimal places as possible.
(2)
(Total 16 marks)
ln x
14. (a) Find the value of lim .
x 1 sin 2 x
(3)
1 e x 2
(b) By using the series expansions for e x2
and cos x evaluate lim .
x 0 1 cos x
(7)
(Total 10 marks)
15. (a) Using the Maclaurin series for (1 + x)n, write down and simplify the Maclaurin series approximation
1
for (1 x 2 ) 2
as far as the term in x4.
(3)
(b) Use your result to show that a series approximation for arccos x is
π 1 3 5
arccos x ≈ x x3 x .
2 6 40
(3)
π
arccos(x 2 ) x 2
(c) Evaluate lim 2 .
x 0 x6
(5)
(d) Use the series approximation for arccos x to find an approximate value for
0. 2
arccos(
0
x )dx ,
giving your answer to 5 decimal places. Does your answer give the actual value of the integral to 5
decimal places?
(6)
(Total 17 marks)
IB Questionbank Mathematics Higher Level 3rd edition 4
