Module 2. Analitic geometry. Introduction to Calculus

Individual Home Task

Variant 9

1. For the triangle with the vertices А(1;–1), В(6;4), С(–2;2) find :

a) length of the side АВ;

b) equation of the line АМ which parallel to the side ВС;

c) the equation of the hieght ВF;

d) the equation of the median AD;

e) the interior angle ;

f) the coordinates of the points N and K, which divides the greater sige of the triangular into three equal parts;

g) area of the treangular АВС.

2. There are given the points М₁(–3;4;2), M₂(1;5;–6), of the plane Р₁: x–y+3z+1=0, Р₂: 2x+3y–z+3=0 and лінії L₁: ; L₂: .

а) Write down the equation of the straight line passing through the points М₁ and M₂;

b) Find the acute angle between the lines L₁ and L₂;

c) Through the point M₂ draw the plane parallel to the plane Р₂;

d) Find the acute angle between the planes Р₁ and Р₂;

e) Find the distance from the point М₁ to the plane Р₂;

f) Through the point M₂ draw the line parallel to the line L₂;

g) Find the acute angle between the line L₂ and the plane Р₂;

h) Through the point М₁ draw the line perpendicular to the plane Р₁;

i) Write down the equation of the plane passing through the point М₁ and perpendicular to the planes Р₁ and Р₂;

j) Write down the canonic equation of the line which is the concurent of two planes Р₁ and Р₂.

3. Calculate the limits: а) ; b) ;

c) ; d) .

4. Calculate the limits: а) ; b) .

5. Investigate the function for continuity: .

Module 2. Analitic geometry. Introduction to Calculus

Individual Home Task

Variant 10

1. For the triangle with the vertices А(-2;1), В(3;6), С(5;2) find :

a) length of the side АВ;

b) equation of the line АМ which parallel to the side ВС;

c) the equation of the hieght ВF;

d) the equation of the median AD;

e) the interior angle ;

f) the coordinates of the points N and K, which divides the greater sige of the triangular into three equal parts;

g) area of the treangular АВС.

2. There are given the points М₁(4;0;–3), M₂(–2;5;4), of the plane Р₁: 3x–y+z–5=0,

Р₂: x+2y–z+2=0 and лінії L₁: ; L₂: .

а) Write down the equation of the straight line passing through the points М₁ and M₂;

b) Find the acute angle between the lines L₁ and L₂;

c) Through the point M₂ draw the plane parallel to the plane Р₂;

d) Find the acute angle between the planes Р₁ and Р₂;

e) Find the distance from the point М₁ to the plane Р₂;

f) Through the point M₂ draw the line parallel to the line L₂;

g) Find the acute angle between the line L₂ and the plane Р₂;

h) Through the point М₁ draw the line perpendicular to the plane Р₁;

i) Write down the equation of the plane passing through the point М₁ and perpendicular to the planes Р₁ and Р₂;

j) Write down the canonic equation of the line which is the concurent of two planes Р₁ and Р₂.

3. Calculate the limits: а) ; b) ;

c) ; d) .

5. Calculate the limits: а) ; b) .

6. Investigate the function for continuity: