TUTORIAL FAM0035 JAN 2015.pdf

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FAM0035 CALCULUS 2014

1 JAN 2014

TUTORIAL

CALCULATING LIMITS

Find

1. lim→ − 1 − 1 2. lim→ √ + 9 − 3

3. lim→ + − 2

− 3 + 2 4. lim→ 1

− 1 − 2 − 1

5. lim→| − 2| − 2

6. lim→3 − − 2

5 + 4 + 1

7. lim→ + 1 − 8. lim→ +

3 −

9. lim→2 − 1 − 2

2 − 3 10. lim→

√ + 3 − 2 + 5

Answes ! 2, ,−3,

, −1 , " , # , −$, −2 ,

"

"#RI$ATI$#S %"IFF. FORMULAS& TRIGO& FUNCTIONS ' C(AIN RUL#)

Differentiate the function

1. % & 2 + 3' − 2 2. () & 1) − 3

)' ) +5)

3. ) & 1 − 4. ) &

3 − 2 + 1

5.Suppose that *5 & 1, *5 & , -5 & −3 and -5 & 2 Find the following values

a) *-5 b) * -. 5 c) - *. 5

6. ( & ' + 3 − 2" 7. ) & /0 +

8. * & 2 − 3' + + 1" 9. ) & i2 + 1




2 JAN 2014

10. ) & 4 11. ) & i/0

12. ) & i2 13. ) & 6 + √

14. ) & iii 15. ) & + /0 , where is constant

16. ) & i 17. ) & 7 + 1

− 18

18. ) & + 1 − 3": 19.- & 2; i ; + <= , where >,<,; areconstants

20. - & 5 − i 2 21.?

? 5 − '@

22. ) & 7

A + − 1

8

' 23. ) & B/

24. *C & iC1 + /0C

25. ; & iC /02C

26. ) & iD − 2 27. ) & 4 + 3' + 1E

Answer :

1. % & 14 − 4 − , 2. () & 5 + FGH + '

GI , 3. ) & IJEIKEII , 4. ) & E

IELI ,

5. a) -16 b) − F , c) 20 6. 1#' + 3 − 2'2 + 3 7. −3 i +

8. 2 − 3 + + 1'2A − 12 − M 11. /0/0 /0 − i

12. 2B/ 2 /02 13.√ L

'√ L√ 14. /0/0ii /0i 15. −3i/0

16. N OPQJRSK + i

17. − JILKIIEH 18. 3 + 1 −3":2 − 151 − 3':

19. 2>; /0 ; 2; i ; + <=E 20. −2/02B/5 − i 2 21. M5 − '15 − 4

22. 4 IT + −

' + 1 +

I 23. B/ B/ 24.QU VW

LOPQWI

26. 2DiD − 2 /0D − 2 25. 2C/02C/0C − 2i 2Ci C 27.'LXLH 4 + M




1 JAN 2014

TUTORIAL 2

DERIVATIVES (IMPLICIT, LOGARITHMIC, EXPONENTIAL FUNCTIONS)

Find

by implicit differentiation

1. + = 6 2. 2 + = +

3. − = − 4. = − 1 + 1

5. = tan 6. + tan = 0

7. sin 1 = 1 − 8. − + = 7

9. 2√ + = 3 10. − 2 + = , is constant

11. + = 1 + 12. 1 + = sin

13. sin = sin 14. + = 1 +

15. tan − = 1 + 16. se = tan

Answer :

1. ! "", 2.

#$"$#, 3.

$%"&$&$%"& , 4.

#"#&, 5. 's 6.

!()*&$

7.$&

*,-./"$()*-.

/ , 8.$&&$, 9. −2 10.

$&$, 11.

%$%"&&$ , 12.

#$& ()*&4()*&

13.*,&4$()*&4*,&$()*& 14.

&√ "$##$&√ " 15.

"#"&4& *5(&$#"&#"& *5(&$"# 16.

*5(& $*5(*5(89,$89,




2 JAN 2014

Differentiate the function.

1. : = ;n + 10 2. : = 's;n

3.

: = ;'<

4.

: = ;n √ >

5. :? = #"@,A#$@,A 6. B = ;n- + − 1

7. : = ;n1 + 8. = ;n sin

9. = 1089,C 10. DE = ;n F&$G&F&"G& H I is constant

11. = ;n1 + 12. = ;'<$ 'sK

13. = 2L& 14. = ;n

15. = ;nse + tan 16. : = 11 + ;n

17. : = ;n ;n ;n 18. = 1 +

19. = M's N + N , is constant 20. O = √

21.

:? = sinA

+ *,A 22.

= P89, √ ,

Q is constant

23. = + $ − $ 24. :? = sin*,& A4

25. = 1 + $ 26. : =

Answer : 1.

&"#R 2. ! *,@, 3.

"# @ , 4.

# 5.

A#$@,A& 6.

# &$# 7.

S/#"S/ + ;n1 +

8.()*"*,

* , 9. 1089,C ;n 10se T 10.$F&GF$G 11.

SL @,#"SL#"SL 12.

!U*,U"()*U ()*U@,

13. 22L& 3&;n 2;n 3 14.#$@,

% 15. se 16. − ##"@,& 17.

#@, @,@, 18.

S L#"&

19. M's N − sinN + + N 20. V√ + #√ W 21. A 'sA + *,A 's ? 22.

P√ se √ P89, √

23.$

SL$SXL& 24. 2sin2?sin*,& A4 's*,& A4 *,& A 25.SX&L#$ #"SX&L 26. + 2




1 JAN 2014

TUTORIAL 3

INDETERMINATE FORM & L’HOPITAL’S RULE)

Find the limit. Use L’Hopital’s rule where appropriate.

1. lim→ + − 6

− 2 2. lim→sin

3. lim→

4. lim→ − 1 − − 1

2

5. lim→cos − cos

6. lim→

1 − + ln 1 + cos

7. lim→ − 1 − 1

ln 8. lim→ −

9. lim→ sincos − 1 10. lim→

− 1 − 1

11. lim→ln

− 1 12. lim→ 1 + − 1

13. lim→sin!"#n$ 14. lim→

%ln &

15. lim→2 − sin 2 + cos 16. lim→

17. lim→ ln 18. lim→ '

Answer :5, (, (,) ,

% − &, −

*' , , 1, -2,

, 1,

,

, , 0 , 0, 0, 0, 0




1 JAN 2014

TUTORIAL 4

DERIVATIVES (RELATED RATES)

1.

A man starts walking north at 1.2 m/s from a point P. Five minutes later a woman startwalking south at 1.6 m/s from a point 200 m due east of P. At what rate are the people

moving apart 15 min after the woman start walking? [ 2.79 ⁄ ]

2.

!f two resistors with resistan"es and are "onne"ted in parallel as in the figure

then the total resistan"e measured in ohms is given #$

=

+

!f and are in"reasing at rates of 0.3 Ω ⁄ and 0.2 Ω ⁄ respe"tivel$ how fast is

"hanging when = 80 Ω and = 100 Ω %0.132 Ω ⁄ &

'.

A 1' ft ladder is leaning against a house when its #ase starts to slide awa$. ($ the time

the #ase is 12 ft from the house the #ase is moving at the rate of 5 ft/se".

a)

*ow fast is the top of the ladder sliding down the wall? %+12 ft/s&

#)

At what rate is the area of the triangle formed #$ the ladder wall and ground

"hanging? %+5,.5 ft/s&

")

At what rate is the angle - #etween the ladder and the ground "hanging? %+1 rad/s&

.

A girl flies a kite at a height of '00 ft the wind "arr$ing the kite from her a rate of 25ft/s. *ow fast must she let out the string when the kite is 500 m awa$ from her? %20

ft/s&

5.

At noon ship A is 100 km west of ship (. hip A is sailing south at '5 km/hr and ship (

is sailing north at 25 km/hr. *ow fast is the distan"e #etween the ships "hanging at

.00 pm? %55. km/hr&

6.

A runner sprints around a "ir"ular tra"k of radius 100 m at a "onstant speed of m/s.

he runners friend is standing at a distan"e of 200 m from the "enter of the tra"k.

*ow fast is the distan"e #etween the friends "hanging when the distan"e #etween themis 200m? %6.3 m/s&

. and falls from a "onve$or #elt at the rate of 10 ⁄ onto the top of a "oni"al pile.

he height of the pile is alwa$s '/3 of the #ase diameter. *ow fast is the radius

"hanging when the pile is m high? %15 32⁄ ⁄ &




2 JAN 2014

3. A "u#e is e4panding so that its volume is in"reasing at a "onstant rate of 12.3 !.

*ow fast is the surfa"e area "hanging at the instant when the volume is 27 ?

%1".1 !&




1 JAN 2014

TUTORIAL 5

DERIVATIVES

(OPTIMIZATION)

1.

A closed 3 dimensional box is to be constructed in such a way that its volume is

4500 . It is also specified that the length of the base is 3 times the width of

the base. ind the dimensions of the box! which must satisfies these conditions and

will result in the minimum possible surface area of the box. "10! 30! 15#

$.

A farmer wants to fence an area of 1.5 million s%uare feet in a rectangular field and

divide it in half with a fence parallel to one of the sides of the rectangle. &ow can

he do this as to minimi'e the cost of the fence( "1000 ft by 1500 ft#

3.

A rectangular storage container with an open top is to have a volume of 10 . )he

length of its base is twice the width. *aterial for the base costs +*10.00 per

s%uare meter. *aterial for the sides cost +*,.00 per s%uare meter. ind the cost

of materials for the cheapest such container. "+*1,3.54#

4.

ind the dimensions of the rectangle of the largest area that can be inscribed in a

circle of radius r.

5.

A piece of wire 10 m long is cut into two pieces. -ne piece is bent into a s%uare and

the other is bent into an e%uilateral triangle. &ow should the wire be cut so that

the total area enclosed is a/ maximum b/ minimum( "use all of the wire for the

s%uare! 40√ 3 9 + 4√ 3 m for the s%uare#

,.

)he top and bottom margins of a poster are each , cm and side margins are each 4

cm. If the area of the printed material on the poster is fixed at 34 ! find the

dimensions of the poster with the smallest area. " $4 cm! 3, cm#

.

ind two positive integers such that the sum of the first number and four times the

second number is 1000 and the product of the numbers is as large as possible. "500

and 1$5#




2 JAN 2014

.

or insurance purposes! a manufacturer plans to fence in a 10,800 rectangular

storage area ad2acent to a building by using the building as one side of the enclosed

area. )he fencing parallel to the building faces a highway and will cost +*3 per

foot installed! whereas the fencing for the other two sides costs +*$ per foot

installed. ind the amount of each type of fence so that the total cost of the fence

will be a minimum. hat is the minimum cost( " 1$0 and 0! +*$0#

.

A box with an open top is to be constructed from a s%uare piece of cupboard! 3 ft

wide! but cutting out a s%uare from each of the four corners and bending up the

sides. ind the largest volume that such a box can have. "2 #




1 JAN 2014

TUTORIAL 6

INTEGRATION

(Basic Integration and Integration by Substitution)

Evaluate

1. ∫ −+

dx x

x x4

2512

2. [ ]∫ + dx x x cos2sin4

3. ( )∫ + dx x x x tansecsec 4. ∫ dx x

x2

cos

sin

5. ∫−

−

5

1)31( dx x 6. ( )dx x x∫ −+

4

1

252

7. dx x∫−

2

1 8. ( )dx x∫ +

8

234

9. dt t ∫

2

1 4

3 10. dx

x∫

4

1

1

11. ( )∫ +

1

03 dx x x 12. dt

t

t ∫

−9

4

3

13. ∫ −

−2

3

2

1

1dx

x

x 14. ∫

−

−

−

1

2

21

dx x

x

15. Solve the IVP

i) 2)1(,

3==

y xdx

dy

ii) 2

1

3,1sin =

+=

π

yt dt

dy

16. ( )∫ + dx x x102

12 17. dx x x∫ +12

18.( )∫

+

dx x

x25

4

1

5 19. ∫

+

dx x

x

13

2

20. dt t t ∫ +1272

21. ∫−

dx x

x

254

22. ( )∫ + dx x 9sin 23. ∫ dx x2cos

24. ( )∫ dx x x2

3cos 25. ( )∫ dx x x322

sec

26. ∫ dt t t 2sin2cos3

27. ( )[ ]∫ dx x x cossincsc2

28.( )

∫ dx x

x2sec

29. ∫ dx x

x2

5sin




2 JAN 2014

30. dx x x

+∫ 23

1sin 31. ∫ π

0

2)cos( dx x x

32. ∫ 2

0)sin(sincos

π

dx x x 33. dx x

x∫

+12

34. ∫ + dxee x x 1 35. ∫ dt t e t sincos

36. ∫ +

dxe

e x

x

1 37. ∫ dx

x x ln

1

38. θ θ

θ π

d ∫ 8

0 22cos

2sin 39. ∫

+

+dx

x x

x

4

2

2

40. ∫ dx x

x

sin

2sin 41.

( )∫

+dx

x

x2

2ln

Answers :

1) C x x

x++−

3

2

3

12

2 2) C x x ++− sin2cos4 3) C x x ++ sectan

4) C x +sec 5) -30 6) 21 7)2

5 8) 138 9)

8

7 10) 2 11)

5

17 12)

3

2013)

2

7−

14)6

5 15) i)

4

5

4

3)( 3

4

+= x x y ii)3

1cos)( π

−++−= t t t y

16)( )

C x

++

11

1112

17) ( ) ( ) ( ) C x x x ++++−+ 2325271

321

541

72 18)

( ) C

x+

+

−

11

5

19) ( ) C x ++213

132

20) ( ) C t ++2

32

12721

1 21) C x +−−

254

5

1

22) ( ) C x ++− 9cos 23) C x +2sin2

1

24) ( ) C x +2

3sin6

1 25) C x +

3tan

3

1

26) C t +− 2cos8

1 4 27) C x +− )cot(sin

28) C x +tan2 29) C x

+

5cos

5

1

30) C x +

+−2

3

1cos3

2 31) 0 32) 1cos1−

33) ( ) C x ++

1ln2

1 2

34) ( ) C e

x++

23

13

2

35) C e

t +−

cos

36) ( ) C e

x++

1ln

37) C x +lnln 38) [ ]122

1− 39) C x x ++ 4

2 40) C x +sin2 41)

( )C

x+

+

3

2ln3




1 JAN 2014

TUTORIAL 7

INTEGRATION BY PARTS

1.

cos5

2.

cos 3

3. ln(2 + 1) 4. sec 2

5. (ln ) 6. sin 3

7. sin 3

8. ln

9.

10. cos ⁄

11. cosln(sin) 12. (ln )

13. cos (1 + sin ) 14. cos1 + sin

15.

ln(1 + )

16.

!

17. "#n 18. $ cos

19. ( + 1)$

20. $

Answer : 1)% sin5 +

% cos5 + & 2) sin 3 +

' cos 3 sin3 + &

3)

(2 + 1

)ln

(2 + 1

) + &

4)

"#n 2

ln*sec2

* + &

5)

(ln

)

2 ln + 2 +& 6) (2 sin 3 3 cos 3) + & 7) 8) ln 2 9) 10) ,- + . 3/ 30

11) sin (ln sin 1) + & 12)% (ln 2)

% ln 2 + % 13) sin + 4 $

+ &

14) "#n(sin) + & 15) ln(1 + ) 2 + 2 "#n + & 16) !(6 + . + . +

3) + & 17) "#n

+ ln( + 1) + & 18)

$(cos + sin ) + & 19) . + 3

20) $

$ + &




2 JAN 2014

TRIGONOMETRIC INTEGRALS

1. sin cos 2. sin 3 ⁄

3. cos ⁄

4. cos% sin

5. sec 6. "#n sec

7. sin5 sin 2 8. sin3 cos

Answer : 1) % cos% cos + & 2) 3) 4) % sin% sin + ' sin' + &

5) "#n +"#n + & 6)

sec sec + & 7)

sin 3

sin7 + &

8) cos6

cos 2 + &




1 JAN 2014

TUTORIAL 8

Integration Involving Inverse Trigonometric Functions and Strategy for

Integration

1.

+ 1 2.

1 + 1

. √ 1 − !. cos

1 + sin ⁄

". 1 + 2

#. 1 + −

$. cos(ln)

8. tan ln(cos )

%. ( + 1) ln( + 1) 1&. sectan

1 + sec

Ans'ers (

1. tan + ! ) 2.

) .

sin() + ! ) !.

) ".

ln

#. ln2 − " ) $.

sin(ln ) + ! ) 8. −#ln(cos)$ + ! ) %. lnln( + 1) + ! ) 1&. ln%1 + sec % + !




2 JAN 2014

Integration Involving *artial Fractions

1. 1 − & − ' 2. 11 + 1

2 + − '

. 2 − − * − !. − + &

". & − 1, − ' + '

#. + + 2* −

$. 2 + &( − 1)

8. 2 − 1, + '( + 1)( − &)

%.

( + 1)*

1&. 2 − 1

(' − 1)(

+ 1)

11. * + & + + ( + 1)( + &)

12. * − 2 + 2 − 2 + 1

1. + -* + 1, + + - + 1,

Ans'er ( 1+ ln .

/. + ! 2+ ln(2 − 1) + &ln( + ') + ! + ln .(/*)0

* . + !

!+0 − & + ln% + &% + ! "+ & + 12 ln% − 2% −

+ ! #+ + * + ln .()0(/)

0 . + !

$+ &ln − ln% − 1% − + ! 8+ * + ln% − &% + ln% + 1% + ! %+ / − (/)0 +ln% + 1% + ! 1&+ − "

* ln%' − 1% + " ln( + 1) + *

" tan + !

11+ &tan + ln( + &) + ! 12+

0 − 2 +

ln( + 1) + !




1 JAN 2014

TUTORIAL 9

AREAS, VOLUMES AND ARC LENGTH

AREAS

Sketch the region enclose !" the gi#en c$r#es% &in the 're' o( the region%

)% = , =

*%

= + 1, = − 1, = −1, = 2

+%

= , =

%

= 1 − , = − 1

-%

= + 4 , = 12 −

Ans.er / )0 *0

+0

0

-0

VOLUMES

&in the #ol$1e o( the soli o!t'ine !" rot'ting the region !o$ne !" the

gi#en c$r#es '!o$t the s2eci(ie line% Sketch the region, the soli 'n t"2ic'l

isk, .'sher or t"2ic'l shell% Use '22ro2ri'te 1etho to sol#e 3isk, .'sher or

c"linric'l shell 1etho0

)%

= , = 1, = 0 4 '!o$t the 56'5is%

*%

= √ − 1 , = 2, = 5, = 0 4 '!o$t the 56'5is%

+% = , = 1 4 '!o$t = 1

%

= , = √ 4 '!o$t = 2

-%

= , = 4 '!o$t = −1

7%

= , = 4 '!o$t the 56'5is

8%

= , = 1, = 2, = 0 4 '!o$t the "6'5is

%

= , = 0, = 1 4 '!o$t the "6'5is

9%

= − , = 0 4 '!o$t = 3

Ans.er / )0 *0 +0 0 -0 70 80 2 0 90

ARC LENGTH%




2 JAN 2014

)%

&in the length o( 'rc o( the gi#en c$r#e (ro1 2oint A to 2oint :%

12 = 4 + 3 ; 12 , 1! , " #

24 , 2!

*%

&in the length o( the gi#en c$r#e%

i0

= − 1 $ , 1 % % 3

ii0

= & +

' , 1 % % 3

iii0

= ()* , 0 % %

i#0 =

- − 3 , 0 % % .

#0

3 = 2 − 1 $ , 2 % % 5

Ans.er / )0 *0 i0 ii0 iii0 ()/√ 2 + 1 i#0 12 #0 /5√ 5 − 2√ 2




1 JAN 2014

TUTORIAL 10

SERIES

By using the appropriate test, determine whether the given series onverge

or diverge!

1! 23 + 1

"! 13 + 2 − 1

⁄

#! + 1 + 3

$! √ √ + 3

%! l n

&! 1 + 2 + 3

'! 1 − 2

(! 47

Answers ) *, *, 1 *, 1 *, *, + , S *, -S +

.! 3!

10! + 4!4!!4

11! 1" + 1

1"! ! 1$3

1#! 47 − 1

1$! % 1$$&

1%! 3'

1&! 7 ()*

1'! + 1

1(! −1 24 + 1

1.! −1 1ln

"0! −1 + 13 + 1

Answer ) 0 +, , +, * , *, 0 +, *, S *, p/series +, *, +, + AST, * *T

"1! 4 − 2 + -./ + − .

""! 3 − 2 + 4/ − + 2

"#! + 1 − 1

"$! + '√ / + 0

"%! + 2 + 1

"&! 12 +




2 JAN 2014

"'! √ + 1

"(! 1√ − 1

".! + 1

Answers ) + + * + + + + * *

POWER SERIES

ind the radius o onvergene and interva2 o onvergene!

1! + 1

)

"! '

#! 3!

)

$!

3

+ 1

)

%!

−1

+ 1

)

&! −1 + 1

'! −1 + 2 2

(! − '

.! 3 − 2 3

10! 2 − 3 + 3

11! 4 + 1

Answers ) 13 1 −151 "3 * − * 5 *6 #3 + −5 + $3 − 5 8 %3 1 −1516 &3 1 −25$6 '3 2 −45$6 (3 1 45 -6 .3 3 − 5 *

103 *

5 /

113 9 −

5 $6

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TUTORIAL FAM0035 JAN 2015.pdf