Challenging Recreational Mathematics: ▪ Japan Today's Calculation Of Integral 2012

Τρίτη 13 Μαρτίου 2012

▪ Japan Today's Calculation Of Integral 2012

770. Find the value of

such that:

771. (1) Find the range of

for which there exist two common tangent lines of the curve

and the parabola

other than the

-axis.

(2) For the range of

found in the previous question, express the area bounded by the two tangent lines and the parabola

in terms of

.

772. Given are three points

in the coordinate space. Find the volume of the solid of a triangle

generated by a rotation about

-axis.

773. For

find the value of

by which

is minimized.
774. Find the real number

such that

775. Let

be negative constant. Find the value of

and

such that

holds for any real numbers

.
776. Evaluate

777. Given two points

on the parabola

in the

plane. Note that the

coodinate of

is less than that of

.

(a) If the origin

is the midpoint of the lines　egment

, then find the equation of the line

.

(b) If the origin

divides internally the line segment

by 2:1, then find the equation of

.

(c) If the origin

divides internally the line segment

by 2:1, find the area of the figure bounded by the parabola

and the line

.

778. In the

space with the origin

, Let

be the surface and inner part of the sphere centered on the point

with radius 2 and let

be the surface and inner part of the sphere centered on the point

with radius 2. For three points

in the space, consider points

defined by

(1) When

move every cranny in

respectively, find the volume of the solid generated by the whole points of the point

.

(2) Find the volume of the solid generated by the whole points of the point

for which for any

belonging to

and any

belonging to

,

belongs to

.

(3) Find the volume of the solid generated by the whole points of the point

for which for any

belonging to

and any

belonging to

,

belongs to

.

779. Consider parabolas

and

in the coordinate plane. When

and

have two intersection points, find the maximum area enclosed by these parabolas.

780. Let

be integer. Given a regular

-polygon

with side length 4 on the plane

in the

-space.Llet

be a circumcenter of

. When the center of the sphere

with radius 1 travels round along the sides of

, denote by

the solid swept by

.

Answer the following questions.

(1) Take two adjacent vertices

of

. Let

be the intersection point between the perpendicular dawn from

to

, prove that

.

(2) (i) Express the area of cross section

in terms of

when

is cut by the plane

.

(ii) Express the volume

of

in terms of

.

(3) Denote by

the line which passes through

and perpendicular to the plane

. Express the volume

of the solid by generated by a rotation of

around

in terms of

.

(4) Find

781. Let

be the tangent lines passing through the point

on the line

and touch the parabola

. Note that the slope of

is greater than that of

.

(1) Exress the slope of

in terms of

.

(2) Denote

be the points of tangency of the lines

and the parabola

.

Find the minimum area of the part bounded by the line segment

and the parabola

.

(3) Find the minimum distance between the parabola

and the line

.

782. Let

be the part of the graph

. Take a point

on

.

(i) Find the equation of the tangent

at the point

on the curve

.

(ii) Let

be the line passing through the point

and parallel to

. Denote

be the intersection point of the line

and the curve

other than

. Find the coordinate of

.

(iii) Express the area

of the part bounded by two line segments

and the curve

for the origin

in terms of

.

(iv) Express the volume

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

and pararell to the

-axis and passing through the point

and pararell to

-axis, the curve

and the

-axis in terms of

.

(v)

783. Define a sequence

.

(1) Find

.
(2) Let

for

.
Prove that

for each

.

784. Define for positive integer

, a function

In the coordinate plane, denote by

the area of the figure enclosed by

, the

-axis and the line

and denote by

the area of the rectagle with four vertices

and

.

(1) Find the local maximum

.

(2) When

moves in the range of

, find the value of

for which

is maximized.

(3) Find

and

.

(4) For each

, prove that there exists the only

such that

.

Note that you may use

785. For a positive real number

, find the minimum value of

786. For each positive integer

, define

(1) Find

.

(2) Express

interms of

Then prove that

is a polynpmial with degree

by induction.

(3) Let

be real number. For

, express

in terms of

.

(4) Find

.

If necessary, you may use

for a positive integer

.

787. Take two points

on the

-plane. Let

be the figure by which the whole points

on the plane satisfies

and the figure formed by

.

Answer the following questions:

(1) Illustrate

.

(2) Find the volume of the solid generated by a rotation of

around the

-axis.

788. For a function

,

answer the following questions:

(1) Find

.

(2) Sketch the graph of

.

(3) Let

be a mobile point on the curve

and

be a point which is on the tangent at

on the curve

and such that

. Note that the

-coordinate of

is les than that of

. Find the locus of

.

789. Find the non-constant function

such that

790. Define a parabola

by

on the coordinate plane. Let

be real numbers with

. Denote by

the tangent lines drawn from the point

to the parabola

.

(1) Find the equations of the tangents

.

(2) Let

be positive real number. Find the pairs of

such that the area of the region enclosed by

is

.

791 Let

be the domain in the coordinate plane determined by two inequalities:

Denote by

the volume of the solid by a rotation of

about the

-axis and by

, by a rotation of

about the

-axis.

(1) Find the values of

.

(2) Compare the size of the value of

and 1.

792. Answer the following questions:

(1) Let

be positive real number. Find

(2) Evaluate

793. Find the area of the figure bounded by two curves

.

794. Define a function

for

.

Find the maximum and minimum value of

in

.

795. Evaluate

796. Answer the following questions:

(1) Let

be non-zero constant. Find

(2) Find the volume of the solid generated by a rotation of the figures enclosed by the curve

, the

-axis and the lines

about the

-axis.

797. In the

-space take four points

.

Find the volume of the part satifying

in the tetrahedron

.

798. Denote by

the graphs of the cubic function

, the line

.

(1) Find the range of

such that

and

have intersection point other than the origin.

(2) Denote

by the area bounded by

and

. If

move in the range found in (1), then find the value of

for which

is minimized.

799. Let

be positive integer. Define a sequence

by

(1) Find

and

.
(2) Find the general term

.
(3) Let

. Prove that

.
800. For a positive constant

, find the minimum value of

801. Answer the following questions:
(1) Let

be a function such that

is continuous and

for some

.
Prove that

.

(2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance

at time

. During the period, prove that there is an instant　for which the absolute value of the acceleration of the car is more than or equal to

802. Let

and

are positive constants. Denote by

the volume of the solid generated by a rotation of the figure enclosed

by the curve

, the line

and the

-axis around the

-axis, and denote by

that of

the solid by a rotation of the figure enclosed by the curve

, the line

and the

-axis around the

-axis.

Find the ratio

803. Answer the following questions:
(1) Evaluate

(2) Find

804. For

, find the minimum value of

805. Prove the following inequalities:
(1) For

,

(2)

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