
Solved
Problems
Contents  A 





The
limit of a function 
Continuous
function 
Limits
at infinity (or limits of functions as x approaches
positive or negative infinity) 
Vertical
asymptote 
Horizontal
asymptote 
Slant
or oblique asymptote 
Limits of
functions
properties 
Limits of
functions
properties use 
Limits of
rational
functions 
Evaluating
the limit of a rational function at infinity 
Evaluating
the limit of a rational function at a point 
The limit of a rational function that is defined at the given point 
The limit of a rational function that is not defined at the
given point 
The limit of a rational function at infinity
containing roots (irrational expressions) 
The limit of a rational function at a point containing
irrational expressions, use of substitution 
Evaluating
trigonometric
limits 
Evaluating
limits of functions
based on the definition of the natural number e 
Use of the composition
rule to evaluate limits of functions 
Differential
calculus, derivatives and differentials

The derivative of a
function 
Definition of the derivative of a function 
The
equation of the line tangent to the given curve at the given point 
Determining the derivative of a function as the limit of the
difference quotient 
Determining the
lines tangent to the graph of a function from a point outside the
function 
Derivatives of
functions 
The
quotient rule 
Differentiation
using the chain rule 
Table
of derivatives of elementary functions 
Differentiation
rules

The chain rule
applications 
Implicit
differentiation 
Generalized
power rule 
Logarithmic differentiation 
Derivative
of a composite exponential function 
Use of the logarithmic differentiation 
Derivative of the inverse function 
Derivative of
parametric functions, parametric derivatives 
Applications
of the derivative 
Tangent, normal
subtangent and subnormal 
Properties
of the parabola 
Property
of power functions 
Property
of exponential functions 
Angle
between two curves 
Differential
of a function 
Use
of differential to approximate the value of a function 
Rules
for differentials 
Differentials
of some basic functions 
Higher order derivatives 
Higher derivatives of composite functions
examples 
Higher derivatives of
implicit
functions examples 
Higher derivatives of parametric functions 
Higher order differentials 
L'Hospital's
rule  evaluation of limits of indeterminate forms 
Applications
of differentiation  the graph of a function and its derivative 
Definition
of increasing and decreasing 
Rolle's
theorem 
The
mean value theorem 
Generalization
of the mean value theorem 
Concavity of the graph of a
function 
Points
of inflection 
Approximate
solution to an equation, Newton's method (or NewtonRaphson
method) 
Representing
polynomial using Maclaurin's and Taylor's formula 
The
approximation of the sine function
by polynomial using Taylor's or Maclaurin's formula 
Extreme
points, local (or relative) maximum and local minimum 
The
first derivative test 
Finding and classifying
critical (or stationary) points, examples 
Integral
calculus  the
definite integral 
The area between the graph
of a function and the xaxis over a closed interval 
Calculating
a definite integral from the definition 
Physical
applications of the definite integral 
Describing
motion of the objects using velocity  time graphs 
Evaluating
the area under the graph of a function using the definition of the
definite integral, examples 
The
definite and indefinite integrals 
The area between the graph
of a function and the xaxis over a closed interval 
Geometric
interpretation of the definite integral 
Integration
 inverse of differentiation 
Evaluating
the indefinite integral 
The fundamental theorem of differential calculus 
The fundamental theorem of integral calculus 
Cavalieri
 Gregory formula for quadrature of the parabola 
The
indefinite integral 
Basic
rules of integration

Table
of indefinite integrals

Evaluation of indefinite integrals using some basic integration rules
and formulas, examples 
Substitution rule 
Evaluating
indefinite integrals using substitution rule, examples

Integration by parts rule 
Evaluating the indefinite integrals using the integration by parts
formula, examples

The indefinite integrals
containing quadratic polynomial (trinomial) 
The indefinite integrals
containing quadratic polynomial, examples

Recursion
formulas  use
of integration by parts formula 
Integrating rational functions 
Use
of the partial fraction decomposition to integrate a proper
rational function, examples

The
Ostrogradski method of the integration of a proper rational
functions

The
Ostrogradski method of the integration of a proper rational
functions, examples

Integrating
irrational functions 
Solving
irrational functions integrals, examples 
Integrating
irrational functions using Euler's
substitutions, examples

Binomial
integral

Solving
binomial integrals, examples

Trigonometric
integrals 
Solving
trigonometric integrals, examples 
Integrals
of rational functions containing sine and cosine

Integrals
of the hyperbolic functions

Substitution
and definite integration 
Substitution
and definite integration example

Integrations
by parts and the definite integral 
The
improper integrals 
Differentiation
and integration of infinite series

Differentiation
of power series 
Applications
of the definite integral

The
area of a region in the plane 
The
area between the graph of a curve and the coordinate axis

The
area bounded by a parametric curve

The
area in polar coordinates

The
area of the sector of a parametric curve

The
area between two curves 
Length
of plane curve, arc length 
Arc length of a parametric curve

Arc length of a curve in polar coordinates

The
volume
of a solid of revolution

The
volume
of a sphere

The
volume
of a spherical segment

The
volume
of a cone

The
surface area of a solid of revolution 
The
lateral surface area of a cone

The surface area of a spherical
cap

The surface area of an
ellipsoid


TRIGONOMETRY
 solved problems 

Trigonometry 
Degrees
to radians and radians to degrees conversion 
Signs of trigonometric
functions

Trigonometric
functions of arcs from 0
to ±
2p

Trigonometric
functions of negative arcs or angles

Trigonometric
functions of complementary angles

Trigonometric
functions of supplementary angles

Trigonometric
functions of arcs that differ on p/2

Trigonometric
functions of arcs that differ on p

Trigonometric
functions of arcs whose
sum is
2p 
The values of the trigonometric functions of arcs that are multipliers of
30°
(p/6)
and 45°
(p/4)

Calculation of values of trigonometric functions of an arbitrary angle
x 
Basic relationships
between trigonometric functions of the same angle, examples 
Trigonometric identities,
examples 
Trigonometric
functions and equations 
Trigonometric functions 
Trigonometric
equations 
Applications of trigonometry 
Solving right triangles examples 
Solving oblique or scalene triangles examples 
Sections of solids 
Applications
of trigonometry in geodesy (or plane surveying) and physics 
An example of using trigonometry in
optics 

VECTORS
 solved problems 

Vectors
in a plane 
Addition, subtraction
and scalar multiplication of vectors 
Vectors
and a coordinate system, Cartesian vectors 
Vectors in
a coordinate plane (a twodimensional system of coordinates), Cartesian
vectors, examples

Vectors in
threedimensional space in terms of Cartesian coordinates 
Vectors in
threedimensional space in terms of Cartesian coordinates, examples

Scalar product or dot
product or inner product 
Vector
product or cross product 
An example for the vector product in physics

The vector
product in component form

The
mixed product or scalar triple product expressed in terms of components 
Vector
product and mixed product, examples 

COORDINATE
GEOMETRY  solved problems 

Line in a coordinate plane

The parametric equations of a line 
The parametric equations of a line
examples

Hessian normal form of the equation of a line 
Distance between a point and a line 
Condition that three points lie on the same line 

Points,
lines and planes in threedimensional ( 3D ) coordinate system
represented by vectors 
Line in threedimensional
coordinate
system 
The parametric equations of a line
in 3D

Equation of a line defined by
direction vector and a point  Symmetric equation of a line 
Equation of a line defined by
direction vector and a point examples 
The angles that a line
forms with coordinate axes

Angle between lines

Condition for intersection of two
lines in a 3D space

The equation of a
plane in a
3D coordinate system 
The equation of
the plane through three points

The distance of a point
and a plane 
plane given in general form

Angle (dihedral angle) between two
planes

Line
and plane

The line of intersection of two planes

Projection of a line onto coordinate planes

How determine two planes of which, a given line is their
intersection line

Intersection point of a line and a plane

Sheaf or
pencil of planes

Points, lines and planes relations
in 3D space, examples

The angle between line and plane

Point,
line and plane  orthogonal projections, distances,
perpendicularity of line and plane

Through a given point pass a line perpendicular to a given plane

Given a line and a point, through
the point lay a plane perpendicular to the line

Projection of a point onto a plane

Projection of a point onto a line

Through a given point lay a line perpendicular to a given line

Projection of a line onto a plane

Plane laid through a given point, such that be parallel with two skew lines

Plane
laid through a given point, such that be parallel with two parallel
lines

Distance between point and line

Distance between point and plane

Distance between parallel lines

Distance between two skew lines










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