05.-Basic-Engineering-Correlation-Calculus-v3.docx

8/20/2019 05.-Basic-Engineering-Correlation-Calculus-v3.docx

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Basic Engineering Correlation (Calculus Reviewer)

1. The depth of water in cylindrical tank 4 m in diameter is increasing at the rate of 0.7 m/min. Find the rate at

which the water ows into the tank.

a. 6.4

b. .!c. 1.!

d. 8.8

. The "ol#me of the sphere is increasing at the rate of 6 cm $ / hr. %t what is its s#rface area increasing &in cn/hr'

when the radi#s is !0cm(

a. 0.$

b. 0.24

c. 0.4

d. 0.!

$. Find the height of aright circ#lar cylinder of ma)im#m "ol#me* which can be inscribed in a sphere of radi#s 10

cm.

a. 1.+1 cm.b. 11.55 cm.

c. 1!.11 cm.

d. 14.1 cm.

4. ,nd the area in the ,rst -#adrant bo#nded by the parabola y 2 = 4x, x = 1 and x $

a. .!!

b. !.!!

c. 5.55

d. .!!!

!. Find the ma)im#m point of y = x + 1/x

a. &1*'

b. &*$'

c. (!1" !2)

d. &* !/'

6. is the concept of ,nding the deri"ati"e of composite f#nctions.

a. ogarithmic di2erentiation

b. 3mplicit di2erentiation

c. Trigonometric di2erentiation

d. C#ain Rule

7. Find the area bo#nded by the c#r"e de,ned by the e-#ation x 2 = 8y and its lat#s rect#m.

a. /$

b. $2%$c. 16/$

d. 11/$

+. 3f y ) ln). Find

a. 1/)

b. 1%&

c. 1/)

d. 1/)



. 5ar % mo"es d#e east at $0 kph* at the same instant car is mo"ing $0 o 8 with the speed 60 kph. The distance

from % to is $0 km. Find how fast is the distance between them separating after 1 ho#r

a. $+ kph

b. $6 kph

c. 45 '#

d. 40 kph

10. % bo) is to be constr#cted from a piece of 9inc 0 s-. in. by c#tting e-#al s-#area from each corner and t#rning

#p the 9inc to form the side. :hat is the "ol#me of the largest bo) that can be so constr#cted(

a. 52.5 cu. in.

b. 6.4 c#. 3n

c. !7.!0 c#. 3n

d. !.! c#. in.

11. Find the coordinates of the "erte) of the parabola y = x 2 - 4x + 1 by making #se of the fact that at the "erte)*

the slope of the tangent is 9ero.

a. &* $'

b. &$* '

c. &1* $'d. (2" !$)

1. ;i"en the f#nction f(x) = x 3 - 6x +2. Fnd the ,rst deri"ati"e at x

a. $ x 2 - 5

b. +

c.

d. 7

1$. 3f the ,rst deri"ati"e of the f#nction is constant* then the f#nction is.

a. inear

b. *ogarit#mic

c. in#soid

d. 8)ponential

14. <sing the two e)isting corner sides of an e)isting wall* what is the ma)im#m rectang#lar area that can be

fenced by a fencing material $0 ft. long(

a. !0 s-. ft.

b. 225 s+.,t.

c. 00 s-. ft.

d. 16 s-. ft.

1!. The "elocity of a body is gi"en by v(t) sin(xt)* where the "elocity is gi"en in meters per second and = t = is

gi"en in seconds. The distance co"ered in meters between t 1/4 and 1/ second is close to

a. 0.!1 m

b. 0.!1 mc. 0.!1 m

d. 0.2551 m

16. >i2erentiate y e x cos x 2

a. e x (cos x 2 ! 2 x sin x 2)

b. -2xe x sin x 2

c. e x sin x 2

d. e x cos x 2 x sin x 2



17. Three sides of a trape9oid are each + cm. long. ?ow long is the fo#rth side when the area of the trape9oid has

the greatest "al#e(

a. 10

b. +

c. 1

d. 1

1+. >i2erentiate y = sec(x 2 + 2)

a. cos& x 2 @ 'cot& x 2 @ '

b. x cos& x 2 @ '

c. cos&) @ '

d. 2 x sec( x 2 - 2)tan(&2 - 2)

1. % stat#e $ m high is standing on a base of 4 m high. 3f an obser"erAs eye is 1.! m abo"e the gro#nd* how far

sho#ld he stand from the base in order that the angle s#btended by the stat#e is a ma)im#m.

a. $.41 m

b. 4.41 m

c. $.!1 m

d. $.1 m

0. :hat is the area of the largest rectangle that can be inscribed in a semicircle of radi#s 10(

a. B !0

b. 100

c. 1000

d. B !0

1. Find the partial deri"ati"e with recpect to ) of the f#ncyion xy 2 - 5y + 6

a. )y

b. )y !y

c. y 2 !

d. y 2

. 3n the c#r"e @ 1) )$* ,nd the critical points.

a. &*1+' C &* 14'

b. &*1+' C &*14'

c. &*1+' C &*14'

d. (2"18) / (!2"!14)

$. % man on a wharf $.6 m abo"e sea le"el is p#lling a rope tied to a raft at 0.60 m/sec. ?ow fast is the raft

approaching the wharf when there are 6 m of rope o#t(

a. 0.! m/s

b. !0.5 m%sec

c. 0.6! m/sec

d. 0.+!m/sec

4. Find of y $sin )

a. $ cos 4)

b. sin )

c. 6 cos )

d. cos 2&

!. 3f the distance ) from the point of depart#re at a time t is de,ned by the e-#ation ) 16t @ !000t @ !000*

what is the initial "elocity(

a. 000



b. 5000

c. 0

d. $000

6. Find the slope of the tangent to the c#r"e x 2 + y 2 - 6x + 10y + 5 = 0 at the point &1*0'

a. D

b. 2%5c.

d. 1/!

7. >i2erentiate y arc sin cos )

a.

b. 1

c.

d. !1

+. 8"al#ate the limit ln)/) as ) approaches positi"e in,nity.

a. 0

b. 1

c. 1d. innit

. >etermine the diameter of a closed a closed cylindrical tank ha"ing a "ol#me of 11.$ c#. m. to obtain minim#m

s#rface area.

a. 1.

b. .6+

c. 1.64

d. 2.44

$0. >i"ide the n#mber 10 into two parts s#ch that the prod#ct of one and the s-#are of the other is ma)im#m.

a. $0 and 0

b. 60 and 60

c. 40 and 80

d. !0 and 70

$1. 8"al#ateE im & x 'tan

a. b e

b. e2π

c.

d. 0

$. :ater is r#nning into a hemispherical bowl ha"ing a radi#s of 10 cm. at a constant rate of $ c#. cm/ min. :hen

the water is ) cm. deep* the water le"el is rising at the rate of 0.014 cm./min. :hat is the "al#e of )(

a.

b. 4c. $

d. !

$$. Find the area bo#nded by the line x y @ 10 0* the )a)is* the ya)is and x 10

a. !0

b. 5

c. 100

d. !



$4. Find the area bo#nded by the y - axis and x = 4 = y2/3

a. 12.8

b. !.6

c. !6.+

d. $0.6

$!. % triangle has "ariable sides )* y* 9 s#bGect to the constaint s#ch that the perimeter is ,)ed to 1+ cm. :hat isthe ma)im#m possible area for the triangle(

a. 14.0$ s-.cm.

b. 15.5 s+. cm.

c. 17.1! s-. cm.

d. 1+.71 s-. cm.

$6. The cost of a prod#ct is a f#nction of the -#antity ) of the prod#ctE 5&)' ) 400) @ !0. Find the -#antity for

which the cost is minim#m.

a. 2000

b. $000

c. !000

d. 0

$7. Find the slope of the line tangent to the c#r"e y )$ ) @ 1 at ) 1.

a. 1/$

b. 1

c. 1/4

d. 1/

$+. :ater is r#nning o#t in a conical f#nnel at the rate of 1 c#. 3n. per second. 3f the radi#s of the base of the f#nnel

is 4 inches and the altit#de in + inches* ,nd the rate at which the water le"el is dropping when it is inches from

top.

a. in./sec

b. in./sec

c. !1%in.%sec.

d. in./sec

$. :hat is the area between y 0* y = 3x2* x = 0 and x = 2(

a. 4

b. 6

c. 8

d. 1

40. 3f y = (t 2 + 2)2 and t = x 1/2, daterine

a. x5/2 + x1/2

b. 2(x + 2)

c. 3/2

d. !etter "

41. Find the area between the c#r"e y cosh x and the )a)is from x = 0 and x = 1

elect oneE

a. 1.667 s-. #nits

b. 1.$$$ s-. #nits

c. 1.1! s-.#nits

d. 1.15 s+. units

4. Find the second deri"ati"e of y by implicit di2erentiation from the e-#ation 4) @ +y $6.



a. !%4$

b. 16/y$

c. $)y

d. 64)

4$. Find the area in s-. #nits bo#nded by the parabolas x 2 - 2y = 0 and x 2 + 2y - 8 = 0

a. .7b. 4.7

c. 10.

d. 11.7

44. :hat is the second deri"ati"e of a f#nction y !)$ @ ) @ 1(

a. $0&

b. 1+

c. $0

d. !)

4!. #va!$ate t%e !iit &f !i(x 2 + 3x - 4) as x a''r&ac%es 3.

a. 54

b. 14

c. 18

d. 2

46. The rate of change of f#nction y with respect to ) e-#als y and y + when x 0. Find y when x ln

a.

b. !

c.

d. 5

47. 3f y 4 cos ) @ sin )* what is the slope of the c#r"e when ) radians(

a. !4.4

b. .1

c. .1

d. $.!

4+. >i2erentiate y = !&10& x 2 + 1'2

a. 4 x(x 2 + 1)

b. log e(x)(x 2 + 1)

c. 3one o, t#e c#oices

3one o, t#e c#oices

d. x(x 2 + 1)

4. ;i"en a cone of diameter ) and altit#de of h. :hat percent is the "ol#me of the largest cylinder which can be

inscribed in the cone to the "ol#me of the cone(

a. .1b. 2.25

c. .+6

d. .!1

!0. Find the minim#m distance from the point &4*' to the parabola y +)

a. 4 B $

b. 2 $

c. B $

d. B



!1. Find the area enclosed y the c#r"e x 2 + 8y @ 16 0* the ) a)is* the ya)is and the line x - 4 = 0

a. +.67 s-. #nits

b. .67s-. #nits

c. 10. s+. units

d. 7.67 s-. #nits

!. Find the e-#ation of the normal to x 2 + y 2 = 1 at t%e '&int (2,1).

a. 2x +3y = 3

". y = 2x

c. x + y = 1

d. x = 2y

!$. * '&ster is t& c&ntain 300 c. s. &f 'rinted atter it% arins &f 10 c. at t%e t&' and "&tt& and 5 c at

eac% side. ind t%e &vera!! diensi&ns if t%e t&ta! area &f t%e '&ster is ini$.

a. 22.24, 44.5

". 2.6, 4.8

c. 25.55, 46.

d. 20.45, 35.6

54. ind t%e e$ati&n &f t%e n&ra! t& ix 2 + y 2 = 5 at the point &* 1'

a. & 2

b. ) @ y 1

c. ) @$y $

d. y )

!!. Find the e-#ation of the c#r"e at e"ery point of which the tangent line has a slope of ).

a. y = -x 2 +

b. y = x 2 + C

c. x = -y 2 +

d. 1x = y 2 +

!6. The radi#s of spheres is r inches at time t seconds. Find the radi#s when the rates of increase of the s#rface

area and the radi#s are n#merically e-#al.

a. H in

b. 1/4H in

c. H in

d. 1%8 in

!7. iven a c&ne &f diaeter x and a!tit$de &f %. %at 'ercent is t%e v&!$e &f t%e !arest cy!inder %ic% can "e

inscri"ed in t%e c&ne t& t%e v&!$e &f t%e c&ne

a. 0.56

b. 0.44

c. 0.65

d. 0.46

!+. %e area enc!&sed "y t%e e!!i'se (iae) is rev&!ved a"&$t t%e !ine x = 3. %at is t%e v&!$e enerated

a. 365.1

". 360.1

c. 30.3

d. 355.3

!. f y = 2x + sin 2x, nd x if y = 0

a. π/2



". 3π/2

c. π/4

d. 2π/3

60. % Iorman window is in the shape of a rectangle s#rmo#ntedby a semicircle. :hat is the ratio of the width of

the rectangle to the total height so that it will yield a window admitting the most light for a gi"en perimeter(

a. 1b. /$

c.

d. J

61. Find the area bo#nded by the parabola* x 2 = 4y, and y 4.

a. $$.1

b. 21.$$

c. 1$.$

d. $1.$

6. The area bo#nded by the c#r"e y = 2x 1/2* the line y = 6 and the ya)is is to be re"ol"ed at y = 6. >etermine the

centroid of the "ol#me generated.

a. 1.4b. 0.!6

c. 1.8

d. 1.0

6$. Find the "ol#me generated if the area between y cosh x and x - axis from x = 0 to x = 1 is is re"ol"ed abo#t

the x - axis.

a. $.4$ c#. <nits

b. 4.42 cu. 6nits

c. $.+$ c#. <nits

d. .+$ c#. <nits

64. :hat is the area bo#nded by the c#r"e y = x 3* the )a)is and the line x = -2 and ) 1(

a. !.4

b. .4!

c. !.4

d. 4.25

6!. Find the appro)imate increase by the #se of di2erentials* in the "ol#me of the sphere if the radi#s increases

from to .0! in one second.

a. .1

b. 2.51

c. .+6

d. .!

66. The integral of cos x w#th respect to ) isa. csc x +

b. sec x +

c. sin x +

d. sin x + C

67. 8"al#ateE im

a. in,nity

b. 1

c. 0



d.

6+. The distance of a body tra"els is a f#nction of time t and is de,ned byE x(t) = 18t + 7t 2.:hat is its "elocity at

t$(

a. 1+

b. !4

c. $6d. 2

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05.-Basic-Engineering-Correlation-Calculus-v3.docx