MathDB

Let

C

be the part of the graph

y=\frac{1}{x}\ (x>0)

. Take a point

P\left(t,\ \frac{1}{t}\right)\ (t>0)

C

.
(i) Find the equation of the tangent

l

at the point

A(1,\ 1)

on the curve

C

.
(ii) Let

m

be the line passing through the point

P

and parallel to

l

. Denote

Q

be the intersection point of the line

m

and the curve

C

other than

P

. Find the coordinate of

Q

.
(iii) Express the area

S

of the part bounded by two line segments

OP,\ OQ

and the curve

C

for the origin

O

in terms of

t

.
(iv) Express the volume

V

of the solid generated by a rotation of the part enclosed by two lines passing through the point

P

and pararell to the

y

-axis and passing through the point

Q

and pararell to

y

-axis, the curve

C

and the

x

-axis in terms of

t

.
(v)

\lim_{t\rightarrow 1-0} \frac{S}{V}.

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