m14/5/mathl/hp3/eng/tz0/se mathematics higher

M14/5/MATHL/HP3/ENG/TZ0/SE

MATHEMATICSHIGHER LEVELPAPER 3 – CALCULUS

Thursday 15 May 2014 (afternoon)

INSTRUCTIONS TO CANDIDATES

Do not open this examination paper until instructed to do so.Answer all the questions.Unless otherwise stated in the question, all numerical answers should be given exactly or

correct to three significant figures.A graphic display calculator is required for this paper. A clean copy of the Mathematics HL and Further Mathematics HL formula booklet is required

for this paper.The maximum mark for this examination paper is [60 marks].

2214-7208 3 pages

1 hour

© International Baccalaureate Organization 2014

22147208


2214-7208

– 2 –

Please start each question on a new page. Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working. For example, if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.

1. [Maximum mark: 16]

Consider the functions f and g given by e e( )2

x x

f x−+

= and e e( )2

x x

g x−−

= .

(a) Show that ( ) ( )f x g x′ = and ( ) ( )g x f x′ = . [2]

(b) Findthefirstthreenon-zerotermsintheMaclaurinexpansionof ( )f x . [5]

(c) Hencefindthevalueof 20

1 ( )limx

f xx→

− . [3]

(d) Findthevalueoftheimproperintegralg xf x

x( )( )[ ]

∞

∫ 20d . [6]


(a) Consider the functions 2( ) (ln )f x x= , 1x > and ( )( ) ln ( )g x f x= , 1x > .

(i) Find ( )f x′ .

(ii) Find ( )g x′ .

(iii) Hence, show that ( )g x is increasing on ] [1, ∞ . [5]

(b) Considerthedifferentialequation

d 2 2 1(ln ) , 1d (ln )y xx y xx x x

−+ = > .

(i) Findthegeneralsolutionofthedifferentialequationintheform ( )y h x= .

(ii) Showthattheparticularsolutionpassingthroughthepointwithcoordinates( )2e, e

is given by 2

2

e(ln )

x xyx

− += .

(iii) Sketchthegraphofyoursolutionfor 1x > ,clearlyindicatinganyasymptotesandanymaximumorminimumpoints. [12]


2214-7208

– 3 –


2 3Each term of the power series 1 1 1 11 2 4 5 7 8 10 11

x x x+ + + +…× × × ×

has the form

1( ) ( )

nxb n c n×

, where ( )b n and ( )c n arelinearfunctionsofn .

(a) Find the functions ( )b n and ( )c n . [2]

(b) Find the radius of convergence. [4]

(c) Findtheintervalofconvergence. [6]


The function f isdefinedby ( )2 3 2e 2 , 1( )

, 1

x x x x xf x

ax b x

− − + + ≤= + >

, where a and b are constants.

(a) Findtheexactvaluesofa and b if f iscontinuousanddifferentiableat 1x = . [8]

(b) (i) Use Rolle’s theorem, applied to f , to prove that 4 3 22 4 5 4 1 0x x x x− − + + =hasarootintheinterval ] [1, 1− .

(ii) Henceprovethat 2x4 − 4x3 − 5x2 + 4x + 1 = 0 hasatleasttworootsintheinterval ] [1, 1− . [7]

N14/5/MATHL/HP3/ENG/TZ0/SE

MATHEMATICSHIGHER LEVELPAPER 3 – CALCULUS

INSTRUCTIONS TO CANDIDATES


correct to three significant figures.A graphic display calculator is required for this paper. A clean copy of the Mathematics HL and Further Mathematics HL formula booklet is required

for this paper.The maximum mark for this examination paper is [60 marks].

8814-7208 4 pages

Thursday 13 November 2014 (afternoon)

1 hour


88147208


8814-7208

– 2 –



(a) Use the integral test to determine the convergence or divergence of

10 5

1 nn.

=

∞

∑ . [3]

(b) Let S xn

n

nn

=+×=

∞

∑ ( ).

12 0 5

1

.

(i) Use the ratio test to show that S is convergent for − < <3 1x .

(ii) HencefindtheintervalofconvergenceforS . [11]


8814-7208

– 3 –

Turn over


(a) Use an integrating factor to show that the general solution for ddxtxt t

t− = − >2 0, is

x ct= +2 , where c is a constant. [4]

The weight in kilograms of a dog, t weeks after being bought from a pet shop, can be modelled by the following function:

w tct t

tt

( ) =+ ≤ ≤

− >

2 0 5

16 35 5.

(b) Given that w t( ) iscontinuous,findthevalueofc . [2]

(c) Write down

(i) the weight of the dog when bought from the pet shop;

(ii) an upper bound for the weight of the dog. [2]

(d) Provefromfirstprinciplesthatw t( ) is differentiable at t = 5 . [6]


Consider the differential equation ddyx

f x y= ( , ) where f x y y x( , ) = − 2 .

(a) Sketch, on one diagram, the four isoclines corresponding to f x y k( , ) = where k takes the values − −1 0 5 0, . , and 1. Indicate clearly where each isocline crosses the y axis. [2]

A curve, C, passes through the point ( , )0 1 andsatisfiesthedifferentialequationabove.

(b) Sketch C on your diagram. [3]

(c) State a particular relationship between the isocline f x y( , ) .= −0 5 and the curve C, at their point of intersection. [1]

(d) UseEuler’smethodwithastepintervalof0.1tofindanapproximatevaluefory on C, when x = 0 5. . [4]


8814-7208

– 4 –


In this question you may assume that arctan x is continuous and differentiable for x∈ .

(a) Considertheinfinitegeometricseries

1 2 4 6− + − +…x x x x <1.

Show that the sum of the series is 1

1 2+ x. [1]

(b) Hence show that an expansion of arctan x is arctan x x x x x= − + − +…

3 5 7

3 5 7 [4]

(c) fisacontinuousfunctiondefinedon[ , ]a b and differentiable on ] , [a b with ′ >f x( ) 0 on ] , [a b .

Use the mean value theorem to prove that for any x , y ∈ [ , ]a b , if y x> then f y f x( ) ( )> . [4]

(d) (i) Given g x x x( ) arctan= − , prove that ′ >g x( ) 0 , for x > 0 .

(ii) Use the result from part (c) to prove that arctan x x< , for x > 0 . [4]

(e) Use the result from part (c) to prove that arctan x x x> −

3

3, for x > 0 . [5]

(f) Hence show that 163 3

63

< <π . [4]

m15/5/mATHL/HP3/eng/TZ0/Se

MathematicsHigher levelPaper 3 – calculus

© International Baccalaureate Organization 20153 pages

Instructions to candidates

yy Do not open this examination paper until instructed to do so.yy Answer all the questions.yy Unless otherwise stated in the question, all numerical answers should be given exactly or

correct to three significant figures.yy A graphic display calculator is required for this paper.yy A clean copy of the mathematics HL and further mathematics HL formula booklet is

required for this paper.yy The maximum mark for this examination paper is [60 marks].

1 hour

Thursday 21 May 2015 (afternoon)

2215 – 7208

– 2 –



The function f is defined by f (x) = e–x cos x + x − 1 .

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


(a) Show that yxf x x= ∫

1 ( )d is a solution of the differential equation

x yxy f xd

d+ = ( ) , x > 0 . [3]

(b) Hence solve x yxy xd

d+ =

−12 , x > 0 , given that y = 2 when x = 4 . [5]


(a) Show that the series 12

2 n nn ln=

∞

∑ converges. [3]

(b) (i) Show that ln ( ) ln ln ( )nn

n+ +

= +1 1 1 .

(ii) Using this result, show that an application of the ratio test fails to determine

whether or not 1

2 n nn ln=

∞

∑ converges. [6]

(c) (i) State why the integral test can be used to determine the convergence or

divergence of 1

2 n nn ln=

∞

∑ .

(ii) Hence determine the convergence or divergence of 1

2 n nn ln=

∞

∑ . [8]


– 3 –


(a) Use l’Hôpital’s rule to find limx

xx→∞

−2e . [4]

(b) Show that the improper integral x xx2

0e d−∞

∫ converges, and state its value. [8]


(a) The mean value theorem states that if f is a continuous function on [a , b] and

differentiable on ] a , b [ then ′ =−−

f c f b f ab a

( ) ( ) ( ) for some c ∈ ] a , b [ .

(i) Find the two possible values of c for the function defined by f (x) = x3 + 3x2 − 2 on the interval [−3 , 1] .

(ii) Illustrate this result graphically. [7]

(b) (i) The function f is continuous on [a , b] , differentiable on ] a , b [ and f ′(x) = 0 for all x ∈ ] a , b [ . Show that f (x) is constant on [a , b] .

(ii) Hence, prove that for x ∈ [0 , 1] , 2 1 2 2arccos arccosx x+ −( ) = π . [9]



Mathematics

Higher level

Paper 3 – calculus




correct to three significant figures.A graphic display calculator is required for this paper.A clean copy of the mathematics HL and further mathematics HL formula booklet is

required for this paper.The maximum mark for this examination paper is [60 marks].

1 hour

Wednesday 18 November 2015 (afternoon)

8815 – 7208

N15/5/MATHL/HP3/ENG/TZ0/SE– 2 –

Please start each question on a new page. Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working. For example, if gra hs are use to fin a so ution ou shou s etch these as art o our ans er. here an ans er is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.


The function f : \ → \ is efine as f xx

xx

:,,

→−

<≥

11

00

.

By considering limits, prove that f is

(a) continuous at x = 0 ; [2]

(b) not differentiable at x = 0 . [3]


Let f (x) = ex sin x .

(a) Show that f '' (x) = 2 ( f ' (x) − f (x)) . [4]

(b) urther i erentiation o the resu t in art a fin the ac aurin e ansion o f (x) , as far as the term in x5 . [6]


(a) Prove by induction that n! > 3n , for n ≥ 7 , n ∈ ] . [5]

(b) Hence use the comparison test to prove that the series 1

2!

r

r r

∞

=∑ converges. [6]

N15/5/MATHL/HP3/ENG/TZ0/SE– 3 –


Consider the function f xx

( ) =+1

1 2 , x ∈ \ .

(a) Illustrate graphically the inequality, 5 41

01 0

1 1( )d5 5 5 5r r

r rf f x x f= =

< <∑ ∑∫ . [3]

(b) se the ine ua it in art a to fin a o er an u er oun or π . [5]

(c) Show that 1 21

22

0

1 ( 1)( 1)1

n nnr r

r

xxx

−−

=

+ −− =

+∑ . [2]

(d) Hence show that π 21 11

200

( 1)ð 4 ( 1) d2 1 1

r nnn

r

x xr x

−−

=

−= − −

+ +∑ ∫ . [4]


The curves y = f (x) and y = g (x) both pass through the point (1 , 0) an are efine the

differential equations ddyx

x y= − 2 and ddyx

y x= − 2 respectively.

(a) Show that the tangent to the curve y = f (x) at the point (1 , 0) is normal to the curve y = g (x) at the point (1 , 0) . [2]

(b) Find g (x) . [6]

(c) Use Euler’s method with steps of 0.2 to estimate f (2) to 5 decimal places. [5]

(d) Explain why y = f (x) cannot cross the isocline x − y2 = 0 , for x > 1 . [3]

(e) (i) Sketch the isoclines x − y2 = −2 , 0 , 1 .

(ii) On the same set of axes, sketch the graph of f . [4]








1 hour

Wednesday 18 May 2016 (morning)

2216 – 7208

– 2 –



The function f is defined by f (x) = ex sin x , x ∈ .

(a) By finding a suitable number of derivatives of f , determine the Maclaurin series for f (x) as far as the term in x3 . [7]

(b) Hence, or otherwise, determine the exact value of lim sinx

x x x xx→ 0

− −e 2

3 . [3]

(c) The Maclaurin series is to be used to find an approximate value for f (0.5) .

(i) Use the Lagrange form of the error term to find an upper bound for the absolute value of the error in this approximation.

(ii) Deduce from the Lagrange error term whether the approximation will be greater than or less than the actual value of f (0.5) . [7]


A function f is given by f (x) = 0

ln (2 sin )dx

t t+∫ .

(a) Write down f '(x) . [1]

(b) By differentiating f (x2) , obtain an expression for the derivative of 2

0

ln (2 sin )dx

t t+∫

with respect to x . [3]

(c) Hence obtain an expression for the derivative of 2

ln (2 sin )dx

x

t t+∫ with respect to x . [3]


– 3 –


(a) Given that f (x) = ln x , use the mean value theorem to show that, for 0 < a < b , b a

bba

b aa

-< <

-ln . [7]

(b) Hence show that ln(1.2) lies between m1 and n

1 , where m , n are consecutive positive integers to be determined. [2]



xy

xy= - where y > 0 and y = 2 when x = 0 .

(a) Show that putting z = y2 transforms the differential equation into dd

zx

xz x+ =2 2 . [4]

(b) By solving this differential equation in z , obtain an expression for y in terms of x . [9]


Consider the infinite series 0

nn

S u∞

=

= ∑ where ( 1)π

π

sin dn

nn

tu tt

+

= ∫ .

(a) Explain why the series is alternating. [1]

(b) (i) Use the substitution T = t - π in the expression for un+1 to show that | un+1 | < | un | .

(ii) Show that the series is convergent. [9]

(c) Show that S < 1.65 . [4]









1 hour

Friday 18 November 2016 (morning)

8816 – 7208

– 2 –




xx

y x++

=

21 2

2 , given that y = 2 when x = 0 .

(a) Show that 1 + x2 is an integrating factor for this differential equation. [5]

(b) Hence solve this differential equation. Give the answer in the form y = f (x) . [6]


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

(b) Deduce that, for n ≥ 2 , the coefficient of xn in this series is ( )( )

--

1 11

n

n n. [1]

(c) By applying the ratio test, find the radius of convergence for this Maclaurin series. [6]


– 3 –

Turn over


(a) Using l’Hôpital’s rule, find

1arcsin( 1)

lim 1x

x

x→ ∞

+

. [6]

Consider the infinite spiral of right angle triangles as shown in the following diagram.

1

1

1

1

a3

a2

a1

θ3θ2

θ1

The nth triangle in the spiral has central angle θn , hypotenuse of length an and opposite side of length 1 , as shown in the diagram. The first right angle triangle is isosceles with the two equal sides being of length 1 .

(b) (i) Find a1 and a2 and hence write down an expression for an .

(ii) Show that θn n=

+arcsin

( )11

. [3]

Consider the series 1

nn

θ∞

=∑ .

(c) Using a suitable test, determine whether this series converges or diverges. [6]


– 4 –


(a) State the mean value theorem for a function that is continuous on the closed interval [a , b] and differentiable on the open interval ]a , b[ . [2]

Let f (x) be a function whose first and second derivatives both exist on the closed interval [0 , h] .

Let g x f h f x h x f x h xh

f h f hf( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )= − − − ′ −−

− − ′( )2

2 0 0 .

(b) (i) Find g(0) .

(ii) Find g(h) .

(iii) Apply the mean value theorem to the function g(x) on the closed interval [0 , h] to show that there exists c in the open interval ]0 , h[ such that g'(c) = 0 .

(iv) Find g'(x) .

(v) Hence show that − − ′′ +−

− − ′( ) =( ) ( ) ( ) ( ) ( ) ( )h c f c h ch

f h f hf2 0 0 02 .

(vi) Deduce that f h f hf h f c( ) ( ) ( ) ( )= + ′ + ′′0 02

2

. [9]

(c) Hence show that, for h > 0

12

2

− ≤cos ( )h h. [5]









1 hour

Monday 8 May 2017 (afternoon)

2217 – 7208

– 2 –



Use l’Hôpital’s rule to determine the value of

lim sinln( )x

xx x→ +0

2

1.


Let the Maclaurin series for tan x be

tan x = a1 x + a3 x3 + a5 x

5 + …

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 = 1 + tan2 x . [3]

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


– 3 –

Turn over


Use the integral test to determine whether the infinite series 2

1lnn n n

∞

=∑ is convergent or

divergent.


(a) Consider the differential equation

ddyx

f yx

x=

>, 0 .

Use the substitution y = vx to show that the general solution of this differential equation is

d Constantvf v v

x( )

ln−

= +∫ . [3]

(b) Hence, or otherwise, solve the differential equation

ddyx

x xy yx

x=+ +

>2 2

2

3 0, ,

given that y = 1 when x = 1 . Give your answer in the form y = g(x) . [10]


– 4 –


Consider the curve yx

=1

, x > 0 .

(a) By drawing a diagram and considering the area of a suitable region under the curve, show that for r > 0 ,

11

1 1r

rr r+

<+

<ln . [4]

(b) Hence, given that n is a positive integer greater than one, show that

(i) 1 11 r

nr

n

=∑ > +ln( ) ;

(ii) 1 11 r

nr

n

=∑ < + ln . [6]

Let Ur

nnr

n

= −=∑ 1

1ln .

(c) Hence, given that n is a positive integer greater than one, show that

(i) Un > 0 ;

(ii) Un + 1 < Un . [4]

(d) Explain why these two results prove that {Un} is a convergent sequence. [1]



y Do not open this examination paper until instructed to do so. y Answer all the questions. y Unless otherwise stated in the question, all numerical answers should be given exactly or

correct to three significant figures. y A graphic display calculator is required for this paper. y A clean copy of the mathematics HL and further mathematics HL formula booklet is

required for this paper. y The maximum mark for this examination paper is [50 marks].


3 pages

Thursday 16 November 2017 (afternoon)

1 hour



8817 – 7208



The function f is defined by

f xx xax b x

( ),

,=

− <+ ≥

2 2 1

1

where a and b are real constants.

Given that both f and its derivative are continuous at x = 1 , find the value of a and the value of b .


Consider the differential equation d

d

yx

xx

y x++

=2 1

where y = 1 when x = 0 .

(a) Show that x2 1+ is an integrating factor for this differential equation. [4]

(b) Solve the differential equation giving your answer in the form y = f (x) . [6]


(a) Use the limit comparison test to show that the series 21

12n n

∞

= +∑ is convergent. [3]

Let ( )

21

32

n

n

xS

n

∞

=

-=

+∑ .

(b) Find the interval of convergence for S . [9]

– 2 – N17/5/MATHL/HP3/ENG/TZ0/SE


The mean value theorem states that if f is a continuous function on [a , b] and differentiable

on ]a , b[ then ′ =−−

f c f b f ab a

( )( ) ( )

for some c ∈ ]a , b[ .

The function g , defined by g x x x( ) cos= ( ) , satisfies the conditions of the mean value

theorem on the interval [0 , 5π] .

(a) For a = 0 and b = 5π , use the mean value theorem to find all possible values of c for the function g . [6]

(b) Sketch the graph of y = g(x) on the interval [0 , 5π] and hence illustrate the mean value theorem for the function g . [4]


Consider the function f (x) = sin(p arcsin x) , -1 < x < 1 and p ∈ .

(a) Show that f '(0) = p . [2]

The function f and its derivatives satisfy

(1 - x2) f (n + 2)(x) - (2n + 1) xf (n + 1)(x) + (p2 - n2) f (n)(x) = 0 , n ∈

where f (n)(x) denotes the n th derivative of f (x) and f (0)(x) is f (x) .

(b) Show that f (n + 2)(0) = (n2 - p2) f (n)(0) . [1]

(c) For p ∈ \ {±1 , ±3} , show that the Maclaurin series for f (x) , up to and including the x5 term, is

pxp p

xp p p

x+−( )

+−( ) −( )1

3

9 1

5

2

3

2 2

5

! !. [4]

(d) Hence or otherwise, find limsin( arcsin )

x

p xx→0

. [2]

(e) If p is an odd integer, prove that the Maclaurin series for f (x) is a polynomial of degree p . [4]

– 3 – N17/5/MATHL/HP3/ENG/TZ0/SE

m14/5/mathl/hp3/eng/tz0/se mathematics higher

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