Module 2. Analitic geometry. Introduction to Calculus

Individual Home Task

Variant 5

1. For the triangle with the vertices А (–7;3), В (2;–1), С (–1;–5) 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 М ₁(1;–2;1), M ₂(2;3;4), of the plane Р ₁: x–2y+3z–4=0,
Р ₂: 2x+3y–4z+5=0 and лінії L ₁: ; L ₂: .

а) Write down the equation of the straight line passing through the points М ₁ and M ₂;

b) Find the acute angle between the lines L ₁ and L ₂;

c) Through the point M ₂ draw the plane parallel to the plane Р ₂;

d) Find the acute angle between the planes Р ₁ and Р ₂;

e) Find the distance from the point М ₁ to the plane Р ₂;

f) Through the point M ₂ draw the line parallel to the line L ₂;

g) Find the acute angle between the line L ₂ and the plane Р ₂;

h) Through the point М ₁ draw the line perpendicular to the plane Р ₁;

i) Write down the equation of the plane passing through the point М ₁ and perpendicular to the planes Р ₁ and Р ₂;

j) Write down the canonic equation of the line which is the concurent of two planes Р ₁ and Р ₂.

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 6

1. For the triangle with the vertices А (–8;–2), В (2;10), С (4;4) 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 М ₁(1;2;–6), M ₂(–3;3;4), of the plane Р ₁: x+y–z+4=0,
Р ₂: 2x-y+4z+5=0 and лінії L ₁: ; L ₂: .

a) Write down the equation of the straight line passing through the points М ₁ and M ₂;

b) Find the acute angle between the lines L ₁ and L ₂;

c) Through the point M ₂ draw the plane parallel to the plane Р ₂;

d) Find the acute angle between the planes Р ₁ and Р ₂;

e) Find the distance from the point М ₁ to the plane Р ₂;

f) Through the point M ₂ draw the line parallel to the line L ₂;

g) Find the acute angle between the line L ₂ and the plane Р ₂;

h) Through the point М ₁ draw the line perpendicular to the plane Р ₁;

i) Write down the equation of the plane passing through the point М ₁ and perpendicular to the planes Р ₁ and Р ₂;

j) Write down the canonic equation of the line which is the concurent of two planes Р ₁ and Р ₂.

3. Calculate the limits: а) ; b) ;

c) ; d) .

4. Calculate the limits: а) ; b) .

5. Investigate the function for continuity: .

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