math

Please solve each question in the space provided. You should give the details of your solutions and not just the final results.

Q-1: [3+2 marks] Let g(s) = s 2 + 4 and h(s) = s 2 — 4. a) Describe the domain of h(s). b) Find h(g (s)) , simplify your answer.

CL) DOMOkibil h (5),

5 2 —N > C

S

D ov\koiAo c34 k(s) s I

or s s c--, IrR

li,Cg Cs))

MST129 - Applied Calculus 2021-2022 / Fall 3

Q-2: [2+3 marks] Solve for x the following equations:

a) 1+:e-x = 3' b) 2 2 x— 6 • 2' + 8 = 0.

3 Z

A

b.) De-z- 6. ozx (-me o

MST129 - Applied Calculus

2021-2022 / Fall 4

Q-3: [3+2 marks] a) At what points on the graph of f (x) = (x — 1) (x 2 — 8x + 7) the tangent

lines are horizontal.

b) Find an equation of the tangent line to the curve y = 1-x3 at x = 0.

d 0 d 2

(x 2 - + - 4 1-)

L> 2 -2 CZ-±)( g)1

_g x -62x+8 oZz a z

2-8z 4-2X 2- -110X_ -. S 3X 2 le

Fo tA5-kys hAdA irowvcrivvi-- vJ yvt_itor- c‘g-KAic cift,;v01 /4A-ve. 31 O •

3x z_igz --€ 1.6'=0 Cz z - 67 z_

3 (z--f)Cz-)--0

oti -1C=-0

:= al

lS 1,1,014 zovvtrof,

MST129 - Applied Calculus 2021-2022 / Fall 5

b)

d fix)1 dz g (z)

x 0

d _ goo.)

(q(2C))L

h-z)=022c-i 9 (x):=1- z5

= dz (zx -1)(1 -z; (2x _z_)

1-z3Diz

oz 32c_ Lcozg_-1)

-z' 3

- z -

26- — 2,r3

qX — 3X 4 c2 26 6 — p2, X-3

04- Hite duviktakive. 0

2c 3 — 3z C -t- 2 -

co) 3- 3ae c2 Co) — 2(0)'K +

=

= (k k(x4-1(0)(Ci t

J-

z (x-+ I/0 (a. tlz-4

( X \ - - A )

—1 /c1/AA k—> 0

Q-4: [5 marks] Let f (x) = . Use the definition of the derivative to find r (x)

4,,AA4 Jo( h) - i(x)

kix

\fT- . \izkft)

z-nf- t4. \r-z7 \ i242c fx t kL.

_

4;\AA ift

/CAM

k-->o -t-4 x÷ k Cx-t-ii0)(j ta:70q)

d.

MST129 - Applied Calculus 2021-2022 / Fall

--t- -A- -V

Q-5: [3+2 marks] Let g(t) t 21+1 42+1)2' a) Find the intervals on which g is increasing or decreasing. b) Find the local maximum and minimum of f, if any.

ivt_s 6zp5voYiov, 9 ( t i

9C-i) Lci (624/) 2 - cf2,_ c-c+j)? (-t 2+1)

-F- (-Ea+)2 (-Ea 4- 11-

a) 'co 'C4t*a \Acz_ ‘‘‘Nc-Rxvo-ts iirtc^ce_0./)7iVq \r\pe vie e8 c() 'C/I:v\c) (1-k c_76(--icc4 0,0 -Ws t

i-Age 6 €, NA‘rci-

.9 1 06) al-Lb C6 z) tt[( CL-4- 1-)1

Cta -1-1 ) 2- 2

Fi. i4c)vizg z,e71,0s d 4-vv d_u‘',tvcd,i CS-Cvi CalvvoAvors)

(-( (-Ez+V

G) w5k"--futkcK lr•k iyvic-trVok.ks lc-04kt' s: —OD 0 1 too

(-11 0) ( 0 i) L i, 00) -

co AclAs 1‘ ow

--11 3CC.4_1)+2-t C-C-1-15 1)L,

(La + 1)3

9 et) \vicreases

(0, 1) ct) de_crect se s

(--i(0)(1-( 00 )

0 G

2

MST129 - Applied Calculus 2021-2022 / Fall 7

19) c i Vice we, Vvw\re c_s \'1)\fk.e_

CilAil''CO-1 1> 01 \IS of C-E) wc (GAN c culcdre \ cal VVt ax; \Mat ON\ (i LAA:uMWtq

c Y i • P °I t

(--oz SC —

(e'-1)1 --f-1)

( )2- 9 co)

(i)z

C-0 ha

-7K-- lociLl 141 a A i'vw? CO- 1

i

-C4 (c)cal tAAA vIA mot (0

C)

Q-6: [5 marks] For the given shape (half a circle on top of a rectangle), find x such

that its circumference is 14 and area is maximum.

Obi e c elfikaMOAA

= z Tr7(- covowv\s/vAc eJwatio-,„

-f- 02z T- 2C1 2x

sw\it'vg opyvAct&A'c-- k-ke a-6,ecH e vci,k ov\

c k z tTrx =1(.( lq-2z-rr

02_ -9 - -2_1"

2, c 1-1` eq/ Qa h'64-1(1 6:6aywif

A = 02,z x

gzzz p-z2-

viz ,,,2 2_

a(- --2-jr

Cal6ct(aVNI9 A/ am d t k

Ai

/4'1 = sk‘1/\.c 'c\kB 1-\(\e (M-s( co-( poi Fs c5E '•

/ t i — (-(X- -Trz

= - R1 - _

MST129 - Applied Calculus 2021-2022 / Fall

1h

k 01 /4AAci suu6sH-4e_

8

covv.SAASio ti

Q-7: [3+2 marks] a) let f(x) x. Using the second derivative test find the local extrema of f. e 2x

b) Find IL' in terms of x if e2x +e2Y = 4. dx

t(2c) cam -6e frCe.W9-(ilett-eAA 73_ e -ozz

covtcut-tcui.vn Pae (FY4. pcx ) d S-gz) e ± (3 - qx) Cax. ,

- -Ozz dz

--c e"°Zz ,z( 3 -4 iz)e"`Z

- ciz --c)e-02x Se_COVIA C/1/; voAive

qz)C-c2zi= 84- c z cc(z_)ecz

Cv2-__q)c-°21z

c4(11z-q) oacqz

ge-̀ 2 z

(02g ot 4-ndivi9 tht

Clag -_16x)e-2

oo

6 ccaco1 c6--1'titg (j z d'x-

L( 3 -

-0 0 2g -142( () e ozz

-16X -a - 28 oo

iv11.-e rya_ oo, 1 • 5)

—2 since exist - a locctA ex hie VV\ WA\

C-Ine_AA f(z) has a ( occtif .e)( --revv‘A.Wvii aizA 5-

2_

0-0

f t -k — —

MST129 - Applied Calculus 2021-2022 / Fall 9

2z

ay

2x-

dy

-e

e t e zy ='t

e ar- t e ay 0/3

f az\ 87cL e cf-Tc

,d5 CC 293,

-> c'zz ei2Y d9 ( x_

26 2̀. (JO 0

icy) —

7 1_31. 75./w ifm( 7•-z)

x a-

02. -eme. 4( z 3) - -(AAC z a )

dlc _zz_-1/6(x3)_14,\Aci+2_-L))

2- 3 2_ ( e-a

x - (02-X 3Y E--

z.

e 2x (2_x3)312

Q-8: [5 marks] Using the logarithmic differentiation, find -̀ 1- of y = / dx )

/ctA (t3)=_ /cvl 6-oz- 4w (02- 3.1z -CAA/1 ii-t-xaD k

MST129 - Applied Calculus 2021-2022 / Fall 10