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Faculty of EngineeringCredit Hours Engineering Programs

EMAT 110: Calculus ICOMMProf. Salwa Ishak

Answer the following questions:

Ouestion l: ( 20 marks)a) (i) Evaluate the following limit

Final ExamFall2008Time: 3 hours

limh->0

sin(3+ h)2 -sin9 =f' Q)

where .f (*)=sinx2. Now 7/=(cosx')(zr),ro f'(3)=6cos9.

/(x)-8(ii) If lim" ' ' =10r+l X_l

1,$(/(,)-8)=o ,

, find tim/(x).

rjg/(,)= s

( using the Squeeze Theorem ) lim-r-+oo

(i) y =sin2 (cos.6t, ", )(ii) tan (, - y) =

(iii) Evaluate

0<sinax<1

sin'x ir!<_<_J; _J;

sino x0<lim------<0x+o Jx

-. sino xhm - -gx+o Vx

b) Find 4 :'dx

.Lsln'x-F

l+ x2

lt = 2sin("o, Gi, o, )(ror("or.,6t, ", ))(-rin Jsir-, )(#) {r.o.or)

tan(x-y) = -2t-t+ x'

, \ (t+ *r\ y, _ y(2*)(ii) (sec2 (x-l)).(1 - y' ) = #' r t (t+r')'

,[ I ^ I ^ \ 2xvv' l : "+sec' (*- y)l =sec' (r-y) +;*' ll+x' .l / t

(t+xr)'

Question 2: (20 marks)I tanx x<0

a)Let f(x\=) x' J.

lj.jl x>olx'-9(i) Does tE/(x) exist?

,l5p/=1, J1p f = -:, l13/ does not exist.

(ii) Find values of x where f (*)is discontinuous (Classifu the discontinuity)

* = -(2n +i+ n = 0,7,2,... infinite discontinuity., ,2x = 0 jump. x = 3 infinite discontinuity.

1_-nl /(r) =-+-a,\ / x'(x-3)

Ouestion 3: (20 marks)a) Show that the tangent line to the curve ! = x3 at any point (o,ot) meets the

curve again at a point where the slope is four times the slope at (a,a') .

a) y=x' , !' =3x' , yt =3aEquation of tangent line is y =3a2x-2a3

1^r^?x" =3a'x-,/.a"

Points of intersections (x +2a)(x - o)' =O

x=arx=-2aSlope of the tangent at the point (-Zr,to')Slope= Z(la')=I2a' ,4times slope at (a,a').

b) For what values of a,b and m does the function

satisfy the hypotheses of the Mean Value Theorem on the interval [O,Z] .

Then find "c" satisfying it.

ttq /(r)= 3 = /(0) ,a =3

trry/(x)=JT] f @)=f (t),m+b=5

[o x=0f' (*)=l-zx+3 o<x<l

l* 7<x<2m=1, b=4

/(x) continuous [O,Z] , differentiable (0,2) .so there is at least one c e (0, Z)

f, (*)=f (2)- {(o) -1where 2-0 2

, =1.(0,2)

Ouestion 4: (20 marks)tX

a) f@)=x+cot! , f' (*)=,-"ot'!r''. tx I x ISln--=-.Sln--*---:22'2 Jz

7t 3n.&-1' a

LL

5n 7r'a'a

LL

Global maximum ^t , =! is 3.7 , global minimum

b) Consider the function : /(x)= # , given that:

nt ,\ 6x2 t2x(1-2*t)f' (*)=;;1 ,, ,.f"(r)= ,. ,,(x'+1) (x'+l)

at x=L2

is 2.6

(i) Determine the intersection of /(x) with the coordinate axes.

(ii) Find the intervals where /(x) is increasing and decreasing.

(iii) Find the local extrema.(iv) Determine the intervals where /(x) is concave up and concave down.

(v) Find the inflection points.(vi) Obtain the vertical and horizontal asymptotes.(vii) Sketch the graph of /(x).

b) (D (0,-r),(r,o)(ii) Increasing (-oo,-l),(-1,0),(0,.o)(iii) No local extrerna.(iv) Concave up (-.o,-1),(0,0.8), concave down (-t,O),(0.4,-)

(v) Inflection points 10,-r;,[0.t,+]\ 3/(vi) Vertical asymptote x = -1, horizontal asymptote y = |(vii)

Ouestion 5: (20 marks)

a) Use Newton 's method with \ = 0.7 to find x3 r the third approximation to the

root of the given equation tan x = li- ;f (*)=t*r-Jt*' , .f' (*)=sec2 **-L

.ll - *'xz =0'64989

b) Express the limit as a definite integral on the given interval,then evaluate

itsvarue lg;I(, .+)'5

= txodx= 618.62

Ouestion 6: (20 marks)a) The radius of a sphere was measured and found to be 21 cm with a possible

error in measurement of at most 0.05 cm. What is the maximum error in usingthis value of the radius to compute the volume of the sphere ? ( volume of a

4sphere V =1nr3 )

3

4V -]nr', LV =4rr2Lr =4n(zr)'(O.os)=277.1cm3

3

2b) Evaruat. J (, -2Vl) d-

-l

(i) From the first principle of integration I given i i ="'7" ,.

(ii) By interpreting it in terms of areas.

(iii) By using The Fundamental Theorem Of Calculus.

2 2i -2iA)'=-,Ii =-,J (4)=:nnn

| --)-(i) .

Ar=1,r, =-l+ L,f G,)=-:*InnnIz=-1.5 , I=-3.5

r(*\=[-* x>0r \ / |.3, x<0

a1A=A,+A.=-- -2=-3.52

0,2(iii) I = It+ tr= llxax- I**=-3.5

" -i 0

Question 7: (20 marks)a) Evaluate the integral.

t

frl !ffia, = Q odd function.

4

3

(ii) Jlx'?-4ld*0

, = t?*'++)ax* J(,' -4)tu02

=?aJ

(iii)

u=3ax+bx3, du=3a+3bx2

.̂,

s(r)

b) If f (*)="'1}.ar, f'(*)o Vl+r'

cosr

s(x) = J [,.sin(r, )V, r, (r) = -.in0

o(;)=-t, r'(;)=-,

= g'(r)[

x[t +sin

l+ J , *r,...

(cos'

s'(r)

,)l

, -2tl3ax+ bx3

Ouestion 8 (20 marks)

a) State whether the following statements are TRUE or FALSE.3

(i) If f is continuous on [t,:] , tnen lf ' (v)dv = f (3)- f (t).I

(i) True

I

(ii) I-l-l --3(ii) False

(iii) If / has a local maximum or minimu m at c, then ,f ' (c) =g(iii) False

-) ( iv) *ilT;:::lir:.,t

s(x)= 2+(x-s)3 uut g does not have a rocar

(vi) True

(v) m f (x)=* , then ! =7 ,and y = -1 are horizontal asymptotes.

(v) False

(b) Two sides of a triangle are 4 m and 5 m in length and the angle between themis increasin g at a rate of 0.06 rad/s. Find the rate at which the area of the

triangle is increasing when the angle between the sides of fixed length i, I .J

A =;@)$)sind, # =rc"oro# =

' = 0.3m2 /s