講義文稿分析0607 s2 mid term test

MA1102CALCULUSMID-TERMTEST

SEMESTER2,2006/2007

MPSH1 7:30–9:00pm 27/02/2007 Totalmarks:60

Name: MatricNo.: ST/STM:

Letf(x)=√x,g(x)=x6andh(x)=1.

x

Writedownthesoff,gandh,respectively. (3marks)

Computeh?g?fandits . (3marks)

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MA1102CALCULUSMID-TERMTEST

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10

Usethelawsoflimits(l’H?opital’sruleisNOTallowed)toevaluatethefollowing.

x3?2x2?x+2

lim

x→1

2x2

—3x+1

. (3marks)

? ?

lim(√x2+x √x2 2x). (3marks)

x→∞

x→0

x

limx1. (3marks)

[Hint:Youmayusethefactthat“a?1<?a?≤aforanya∈R”withoutproof.]

lim

x→0

sin(3tanx)sin2x

. (3marks)

? ?

Usethe?,δ-definitionoflimittoshowthatlim(2x2 2x 1)=3. (6marks)

x→2

Showthattheequation√x+1= 1

(x?2)2

hasatleasttwopositiveroots.(Graphical

solutionisNOTacceptable) (6marks)

ax2+b, ifx≥π/2.

Letf(x)=(?cosx, ifx<π/2,

a π

Determine

b(b6=0)suchthatfiscontinuousat

. (3marks)

2

Determinethevaluesofaandbsuchthatfisdifferentiableatx=

π

.(3marks)

2

Attimet≥0,thepositionofaparticlealongthestraightlineiss(t)=t3?4t2+5t.

Findtheinstantaneousvelocityandaccelerationoftheparticle. (3marks)

Findtheintervalsonwhichtheparticlespeedsupandtheintervalsonwhichitslowsdown. (3marks)

Evaluatethefollowingderivatives.

dy

dx

Compute

forx2+2xy=2x+y2. (3marks)

(x,y)=(2,4)

d2y x

Compute

dx2x=1/2fory=√1?x2. (3marks)

Letr>1bearationalnumber.Provethat (6marks)

1 r r

2r?1≤x

+(1?x)

≤1 forall0≤x≤1.

Letf(x)beafunctiondefinedonR.Supposef(x+y)=f(x)f(y)foranyx,