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Answers to the third edition of advanced mathematics
( 1)

lim(x->; ∞)( 1 + 1/x)^ 10 = 1

lim(x->; ∞)( 1 + 1/ 10)^x->; +∞

A: A.

(2)

f(x)=∫( 1-& gt; X) SGD/T.

f'(x) = sinx/x

∫(0->; 1) xf(x) dx

=( 1/2)∫(0->; 1)dx^2

=( 1/2)x^2.f(x)]|(0->; 1)-( 1/2)∫(0->; x^2.f'(x) dx

=( 1/2)x^2.f(x)]|(0->; 1)-( 1/2)∫(0->; 1) xsinx dx

=( 1/2)f( 1)+( 1/2)∫(0-& gt; 1) x dcosx

= 0+( 1/2)[xcosx]|(0-& gt; 1)-( 1/2)∫(0->; 1) cosx dx

=( 1/2)cos 1-( 1/2)[sinx]|(0-& gt; 1)

=( 1/2)cos 1-( 1/2)sin 1

A: B.

(3)

f(x) = x^p +( 1-x)^p

=px^(p- 1 -p( 1-x)^(p- 1)

f'(x)=0

px^(p- 1)-p( 1-x)^(p- 1)=0

x= 1-x

x= 1/2

f '(x)| x = 1/2+& lt; 0,f '(x)| x = 1/2-& gt; 0

X= 1/2 (minimum)

Minimum f(x)

=f( 1/2)

= ( 1/2)^p +( 1/2)^p

=( 1/2)^(p- 1)

A: C.