Higher_Mathematics_Part_1

vertex of the cone).

If a = b = c we have a right circular cone

Fig. 2.30

x2 + y2 = z2 .

11.4. Sphere

A sphere is a surface formed by rotation of a circle around its diameter. An equation of a sphere with a centre at the point M0 (x0 , y0 , z0 ) and a radius R

looks as follows

(x − x0 )2 + ( y − y0 )2 + (z − z0 )2 = R2 .

If the centre of the sphere is the origin we have the following standard equation of the sphere

x2 + y2 + z2 = R2.

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11.5. Ellipsoid

A set of all points in space satisfying the equation

the plane Охz about the axis Оz.

We have to note that each point of any one-sheeted hyperboloid is passes by a straight line lying in this hyperboloid entirely.

11.7. Hyperboloid of two sheets

A hyperboloid of two sheets (Fig. 2.33) is a surface described by the equation

x2 + y2 − z2 = −1. a2 b2 c2

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11.8. Elliptic paraboloid

A surface is called an elliptic paraboloid (Fig. 2.34), if its standard equation looks as follows:

If p = q we have a circular paraboloid x2 + y2 = 2 pz formed by rotation of

the parabola x2 = 2 pz lying in the plane Охz about the axis Оz. The point (0; 0; 0) is the vertex of the elliptic paraboloid.

Оy

11.9. Hyperbolic Paraboloid

A surface is called a hyperbolic paraboloid (Fig. 2.35) if its standard equation is

3x2 +4 y2 −8z2 −18x +8y +32z −1 =

3(x2 −6x +9) +4( y2 +2y +1)−8(z2 −4z +4) −1−27 −4 +32 = = 3(x −3)2 +4( y +1)2 −8(z −2)2 .

So the given equation looks as follows:

3(x −3)2 + 4( y +1)2 −8(z −2)2 = 0 ,

or

that is a standard equation of a cone whose vertex is a point P(3; –1; 2) (in the system Оxyz).

Micromodule 11

CLASS AND HOME ASSINMENTS

1. Determine what surface is defined by the following equation:

2. Determine what surface is defined by the following equation:

a)x2 + y2 + z2 −2x +4y +6z −2 = 0 ;

b)5x2 +4 y2 +3z2 +10x −8y −6z = 0 ;

c)y2 +2x −4y +16 = 0 .

4. Form an equatin of a sphere whose points А(5; 0; –2) and В(1; 4; 0) are endpoints of its diameter.

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5. Form an equation of a revolution paraboloid obtained by rotation of the

2. а)a sphere; b)an ellipsoid; c)a parabolic cylinder. 3. (4;–3; 2). 4. (x − 3)2 + + ( y − 2)2 + (z + 1)2 = 9 . 5. x2 + y2 = 12z.

Micromodule 11

SELF-TEST ASSINMENTS

11.1. General equation of a second order surface is given. Determine what surface is defined by that equation and reduce it to standard form.

11.1.1.36x2 + 4 y2 − 8y + 9z2 − 32 = 0.

11.1.2.x2 + 2 y2 − 4x + 4 y − 4z + 6 = 0.

11.1.3.x2 + 16 y2 − 4z2 − 4x + 8z = 0.

11.1.4.3x2 + y2 − 6x + 4 y − 2z + 7 = 0.

11.1.5.x2 + 4 y2 − 4x − 24y − 4z + 40 = 0.

11.1.6.x2 + y2 − z2 − 2x + 4y + 6z − 4 = 0.

11.1.7.x2 + y2 − z2 + 4x + 2 y + 4z + 1 = 0.

11.1.8.x2 + y2 − z2 − 6x − 2y − 2z + 9 = 0.

11.1.9.x2 + y2 − z2 − 4x − 4 y − 4z + 4 = 0.

11.1.10.4x2 + y2 − 2z2 − 8x + 4z + 2 = 0.

11.1.11.12x2 + 3y2 + 4z2 − 24x + 12 y + 8z + 16 = 0.

11.1.12.4x2 + 2 y2 + z2 − 8x − 4 y − 4z + 6 = 0.

11.1.13.4x2 + 9y2 + 36z2 − 36y − 72z + 36 = 0.

11.1.14.3x2 + y2 + 9z2 + 12x − 2 y + 4 = 0.

11.1.15.x2 + 5y2 + 5z2 − 4x − 20z − 1 = 0.

11.1.16.3x2 + 4y2 − 4z2 − 8y + 8z − 12 = 0.

11.1.17.3x2 + 2y2 − 6z2 − 6x + 4y − 1 = 0.

11.1.18.x2 + 3y2 − z2 − 4x + 4z + 1 = 0.

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11.1.19.4x2 + 9y2 − 36z2 + 8x − 18y − 23 = 0.

11.1.20.2x2 + 5y2 − 10z2 + 8x + 20z − 12 = 0.

11.1.21.4x2 + y2 − 4z2 − 16x − 2 y + 21 = 0.

11.1.22.9x2 + y2 − 9z2 + 18x + 2 y + 19 = 0.

11.1.23.2x2 + y2 − 2z2 − 12x − 4 y + 22 = 0.

11.1.24.4x2 + 4 y2 − z2 + 16x + 8y + 2z + 23 = 0.

11.1.25.4x2 + 2 y2 − z2 − 8y − 2z + 11 = 0.

11.1.26.2x2 − y2 − 4x − 2y − 6z + 1 = 0.

11.1.27.5x2 − 4y2 + 10x + 8y − 20z + 1 = 0.

11.1.28.9x2 − 4 y2 − 16 y − 36z − 52 = 0.

11.1.29.4x2 − y2 + 4y − 4z + 4 = 0.

11.1.30.4x2 − y2 − 16x − 2y − 4z + 7 = 0.

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на розривність, infinitesimals — нескінченно малі, infinitely large — нескін-

ченно велика, indeterminate form — невизначений вираз, indeterminacy —

невизначеність, domain of definition — область визначення, a range — множина значень, inverse function — обернена функція, еven function —

парна функція, odd function — непарна функція, periodic function —

періодична функція, honorable limit — важлива границя, derivative — похідна, differential — диференціал, relative extremum — відносний екстре-

мум, asymptote — асимптота, inflection point — точка перегину, convexity —

опуклість, concavity –вгнутість.

Main tasks: 1. Evaluation of limits. 2. Investigation of continuity. 3. Evaluation of the first and higher order derivatives. 4. Evaluation of the derivatives of the implicit functions and the parametric functions. Logarithmic differentiation. 5. Evaluation of the differentials of functions. 6. The usage of derivative. 7. The usage of the differentials. 8. Functions investigation. 9. Graph construction.

THE STUDENT MUST BE READY TO DO

THE FOLLOWING ASSIGNMENTS

1.Concepts, definitions, formulations:

1.Sets. Classification of numerical sets. Operations on sets. The modules of a real number.

2.A sequence.

3.A function. Classification of functions. The elementary functions. An inverse function. A composite function.

4.The Limit of a numerical sequence. The Limit of a function. Infinitesi-

mals.

5.Continuity. Continuity of a function at a point and on an interval. Properties of continuous functions. Points of discontinuity and its classification.

6.Definition of a derivative. Geometrical and physical interpretation.

7.A table of derivatives.

8.Rules of differentiation.

9.A connection between continuity and differentiability.

10.A differential. Geometrical interpretation of a differential.

11.The usage of the differentials.

12.Evaluation of the first and higher order derivatives.

13.Leibniz’s formula.

14.Lagrange’s formula.

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18.Investigation for function increase and decrease on the given interval.

19.Investigation of a function for extremum.

20.Minimum and maximum values on the interval.

21.Concavity intervals. Inflection points.

22.Asymptotes.

23.Plan of graph construction.

2.Proofs and conclusions

1.Theorems about limits.

2.The first and the second honorable limits.

3.Theorems about equivalent infinitesimals.

4.The derivatives of elementary functions.

5.The first order and higher order derivatives of the parametric functions.

6.Theorem about continuity of differentiable functions.

7.Geometrical interpretation of the first order differential.

8.Equation of a tangent line and a normal to the curve.

9.Lagrange’s and Fermat’s theorems.

11.The necessary monotony conditions.

12.The necessary and sufficient extremum conditions.

13.Curve asymptotes seeking rule.

3.Assignments

1.Find the derivatives of functions.

2.Find the derivatives of composite functions, implicit functions and parametric functions.

3.Find the differentials of functions.

4.Find the derivatives and the differentials of higher order.

5.Solve tasks for geometrical and physical interpretation of a derivative.

6.Investigate elementary functions.

7.Sketch the graphs of elementary functions.

8.Evaluate the limits.

9.Evaluate the limits using the equivalent infinitesimals.

10.Investigate functions for continuity.

11.Find different limits with the help of L’Hospital’s rule.

12.Find intervals of function increase and decrease, local extremum.

13.Find concavity intervals.

14.Find graph asymptotes.

15.Construct the graph.

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