final and answers 2008 calculus1 ahmedawad
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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
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