Trigonometric inequalities

Trigonometric equation.

Solve for x in the following equations

Vectors and coordinate

 

1) Find the vector form, Cartesian form, parametric form of the line passing through the point A (2, 1, 3) which is also parallel to the vector i-2j+3k.

2) Find the vector form, Cartesian form, parametric form of the line passing through the line A (2, 0, 5) and B (3, 4, 8).

3) Show that the lines and are parallel.

4) Find the acute angle between the following lines r=2j+3k + t(3i+4j+5k) and

r=-2i+5j+3k+s(-i+2j+k).

5) Find the point of intersection of the lines and .

6) Find the value(s) of k, such that the lines and are perpendicular.

7) Find the centroid of triangle, if vertices A(1,2,-4), B(3,0,-2) and C(-3,6,4).

8) Given two points A (3,-6, 4) and B (3, 2, 0). Find the point M, if it satisfies: .

9) Diameter of the sphere passes through the points A (1,-2, 5) and B (3, 4, 3), find the equation of a sphere

10) Point М (4; 4; -5) belongs to the sphere with center in point (1; -3; 0). Write the equation of this sphere.

11) Find the parametric equations of the line passing through the points A and B .

12) Find the components of a vector, which is perpendicular to the vectors .

 

Indefinite and definite Integration.

1. Find indefinite integral the following functions

2. Using the substitution u = 1 + 2 x, or otherwise, find

3. Use the substitution u = 2x + 3 to find

4. Use the substitution method to find the exact value of the integral.

5. By using the substitution method, find

6. Use the substitution method to evaluate

7. Use integration by parts to find

8. Use integration by parts to find

9. Show that

10. Evaluate .

11. Use integration by parts to find the exact value of

12. Show that

13. Use integration by parts to find

14. Find . 15. Find 16. Find

17. Find the area of the region bounded by , the x- axis, and the line x=3

18. Find the area of the region bounded by , the x- axis, and the line x= -2 and x= 0

19. Find the area of the region bounded by , the x- axis, and the line x= 4 and x= 5

20. The part of the line y=x+1 between x=0 and x=3 is rotated about the x -axis. Find the volume of this solid of revolution

21. A curve is defined by . If this curve is rotated about the x -axis, find the volume of the solid of revolution formed.

22. The part of the curve between the x values 2 and 3 is rotated about the x-axis. Find the volume of the solid formed in this way.

 

Differentiation

1. Given the function find the values for which:

(a)

(b) Find the stationary points and the points of inflection for

(c) Sketch the graph of

2. Given the function find the values If any) for which:

(d)

(e) Find the stationary points and the points of inflection for

(f) Sketch the graph of

3. Given the function find the values for which:

(g)

(h) Find the stationary points and the points of inflection for

(i) Sketch the graph of

4. Find the local maximum, local minimum points and points of inflection for the function (if they exist)

5. Sketch the graph of by finding the turning points and points of inflection

5. Vectors and coordinate

a) AA₁ +A₁C-?

b) AD + A₁B₁ + A₁C-?

c) Which vectors are coplanar?

1) BB₁, A₁C, A₁C₁ ; 2) B₁D₁, C₁C, A₁A; 3) BC, A₁D₁, AD;

 

d) Find the area of the triangle with vertices (1,6,3), (0,10,1) and (5,8,3).

e) Find the vector equation of the plane containing the vectors and which also includes the point (1,2,0).

f) Which vectors are parallel?

 

Trigonometric inequalities.

b) Find domain for .

7. Analytic Geometry

 

1. Find the angle between the line and plane .

2.

 

 

Polynomials.