Practical class № 6.
Investigation of the function. Extremum of the function. Convexity, concavity and point of inflection. Asymptotes
Theoretical questions:
L’Hospital’s Rule;
Monotonic conditions. Extremum of function;
Convexity and concavity. Point of inflection;
Аsymptotes;
Scheme of investigation of function and charting.
Classroom assignments:
1. Find the limits using L'Hospital's rule:
1.
2.
3.
4.
5.
6.
7.
8.
9.
2.
Conduct a complete investigation of the
functions and construct their graphs: 1.
2.
3. Describe the concavity of the graph of 
for
.
4.
Find 
and
so
that the function 
has a point of inflection at
.
5. Find the domain
and all asymptotes of the following function:
.
6. Find the maximum
and minimum values of
on the interval [0, 5].
Homework:
Theoretical material: Functions of several variables and main properties. Partial derivatives and differentials.
Solve problems:
1. Find the limits using L'Hospital's rule:
1.
2.
3.
4.
2.
Conduct a complete investigation of the functions and construct their
graphs: 1.
2.
.
3.
Find the domain and all asymptotes of the following function:
.
4.
Find the domain and all asymptotes of the following function:
.
5.
Find the maximum and minimum values of
on the interval [-2, 1].
6.
Find the maximum and minimum values of
on the interval
.
Practical class № 7. Functions of several variables. Full differential
Theoretical questions:
1. Concept of functions of several variables; 2. Basic properties of functions of several variables; 3. Partial derivatives and total differential; 4. Partial derivatives and higher order differentials; 5. Tangent and surface normal; 6. Extrema of functions of several variables.
Classroom assignments:
1. Find the domain of existence of functions:
1.
2.
3.
4.
.
5.
2. Find the first order partial derivatives for the function:
1.
2.
3.
4.
3. Find the second order partial derivatives:
1.
2.
3.
4.
5.
4. Find derivatives of functions defined implicitly:
1.
2.
3.
4.
5.
5.
Find
, if
.
6.
Find
, if
7. Find the third order partial derivatives:
8. Find the third order partial derivatives:
1.
2.
3.
9.
Write the equation of the tangent plane
and normal to the curve
,
given by the equation
in point
:
1.
2.
10. Find the extremum of functions:
1.
2.
3.
4.
5.
(x>0, y>0).
Homework:
Theoretical material: Antiderivative and indefinite integral.
Solve problems:
1. Find the second order partial derivatives:
1. 2. 3. 4. 5. .
2. Find the third order partial derivatives:
1. 2.
3. 4.
3. Find the extremum of functions:
1.
. 2.
Practical class № 8. Antiderivative. Indefinite integral and its properties. Table of integrals. Main methods of integration
Theoretical questions:
Concept of antiderivative and indefinite integral. 2. Main properties of indefinite integral. 3. Main methods of integration. 4. Integration of rational fractions. 5. Integration of expressions containing trigonometric functions. 6. Integration of irrational functions.
Classroom assignments:
1. Calculate integrals using table of integrals:
1.
2.
3.
4.
5.
2. Find integrals using appropriate substitution:
1.
2.
3.
4.
5.
3. Find integrals using method of integration by parts:
1.
.
2.
.
3.
.
4.
5.
.
4. Calculate integrals:
1.
2.
3.
4.
5.
6.
7.
8.
5. Calculate integrals:
1.
2.
3.
4.
5.
6. Find integrals using appropriate substitution:
1.
2.
3.
4.
5.
7. Calculate integrals:
1.
2.
3.
4.
5.
6.
7.
8.
Homework:
Theoretical material: The definite integral. Problems leading to the definite integral. The Newton-Leibniz formula.
Solve problems:
Calculate
integrals: 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
PRACTICAL CLASS № 9-10.
Properties of definite integrals. The Newton-Leibniz formula. Applications of definite integrals in economics. Calculation the arc length, the amount of body rotation. The improper integral
Theoretical questions:
1. Newton – Leibniz formula. 2. Basic methods of integration. 3. Geometric applications of the definite integral.
Classroom assignments:
1. Find integrals using Newton – Leibniz formula
1.
2.
3.
4.
5.
2.
Calculate integrals: 1.
2.
3.
4.
5.
3.
Find the area bounded by the curve
,
the x-axis, and the lines
and
.
4.
Find the area of the ellipse
.
5.
Find the area bounded by the curve
,
for
.
6.
Find the area of the lemniscate
.
7.
Find the length of the cissoid
from
to
.
Homework:
Theoretical material: First order differential equations. Differential equations of higher orders.
Solve problems:
1.
Find the area bounded by the parabola
and the straight line
.
2
Find the area bounded by the ellipse
.
3.
Find the area of the ellipse whose parametric equations are
and
.
4.
Find the parabola
from (0,0) to (-4,4).