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| Solved
Problems
Contents - A |
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| The
limit of a function |
Continuous
function |
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Limits
at infinity (or limits of functions as x approaches
positive or negative infinity) |
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Vertical
asymptote |
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Horizontal
asymptote |
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Slant
or oblique asymptote |
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Limits of
functions
properties |
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Limits of
functions
properties use |
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Limits of
rational
functions |
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Evaluating
the limit of a rational function at infinity |
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Evaluating
the limit of a rational function at a point |
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The limit of a rational function that is defined at the given point |
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The limit of a rational function that is not defined at the
given point |
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The limit of a rational function at infinity
containing roots (irrational expressions) |
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The limit of a rational function at a point containing
irrational expressions, use of substitution |
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Evaluating
trigonometric
limits |
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Evaluating
limits of functions
based on the definition of the natural number e |
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Use of the composition
rule to evaluate limits of functions |
| Differential
calculus, derivatives and differentials
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The derivative of a
function |
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Definition of the derivative of a function |
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The
equation of the line tangent to the given curve at the given point |
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Determining the derivative of a function as the limit of the
difference quotient |
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Determining the
lines tangent to the graph of a function from a point outside the
function |
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Derivatives of
functions |
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The
quotient rule |
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Differentiation
using the chain rule |
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Table
of derivatives of elementary functions |
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Differentiation
rules
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The chain rule
applications |
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Implicit
differentiation |
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Generalized
power rule |
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Logarithmic differentiation |
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Derivative
of a composite exponential function |
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Use of the logarithmic differentiation |
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Derivative of the inverse function |
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Derivative of
parametric functions, parametric derivatives |
| Applications
of the derivative |
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Tangent, normal
subtangent and subnormal |
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Properties
of the parabola |
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Property
of power functions |
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Property
of exponential functions |
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Angle
between two curves |
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Differential
of a function |
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Use
of differential to approximate the value of a function |
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Rules
for differentials |
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Differentials
of some basic functions |
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Higher order derivatives |
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Higher derivatives of composite functions
examples |
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Higher derivatives of
implicit
functions examples |
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Higher derivatives of parametric functions |
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Higher order differentials |
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L'Hospital's
rule - evaluation of limits of indeterminate forms |
| Applications
of differentiation - the graph of a function and its derivative |
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Definition
of increasing and decreasing |
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Rolle's
theorem |
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The
mean value theorem |
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Generalization
of the mean value theorem |
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Concavity of the graph of a
function |
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Points
of inflection |
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Approximate
solution to an equation, Newton's method (or Newton-Raphson
method) |
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Representing
polynomial using Maclaurin's and Taylor's formula |
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The
approximation of the sine function
by polynomial using Taylor's or Maclaurin's formula |
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Extreme
points, local (or relative) maximum and local minimum |
| The
first derivative test |
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Finding and classifying
critical (or stationary) points, examples |
| Integral
calculus - the
definite integral |
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The area between the graph
of a function and the x-axis over a closed interval |
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Calculating
a definite integral from the definition |
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Physical
applications of the definite integral |
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Describing
motion of the objects using velocity - time graphs |
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Evaluating
the area under the graph of a function using the definition of the
definite integral, examples |
| The
definite and indefinite integrals |
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The area between the graph
of a function and the x-axis over a closed interval |
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Geometric
interpretation of the definite integral |
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Integration
- inverse of differentiation |
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Evaluating
the indefinite integral |
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The fundamental theorem of differential calculus |
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The fundamental theorem of integral calculus |
| Cavalieri
- Gregory formula for quadrature of the parabola |
| The
indefinite integral |
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Basic
rules of integration
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Table
of indefinite integrals
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Evaluation of indefinite integrals using some basic integration rules
and formulas, examples |
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Substitution rule |
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Evaluating
indefinite integrals using substitution rule, examples
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Integration by parts rule |
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Evaluating the indefinite integrals using the integration by parts
formula, examples
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The indefinite integrals
containing quadratic polynomial (trinomial) |
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The indefinite integrals
containing quadratic polynomial, examples
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Recursion
formulas - use
of integration by parts formula |
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Integrating rational functions |
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Use
of the partial fraction decomposition to integrate a proper
rational function, examples
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The
Ostrogradski method of the integration of a proper rational
functions
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The
Ostrogradski method of the integration of a proper rational
functions, examples
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Integrating
irrational functions |
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Solving
irrational functions integrals, examples |
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Integrating
irrational functions using Euler's
substitutions, examples
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Binomial
integral
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Solving
binomial integrals, examples
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Trigonometric
integrals |
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Solving
trigonometric integrals, examples |
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Integrals
of rational functions containing sine and cosine
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Integrals
of the hyperbolic functions
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Substitution
and definite integration |
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Substitution
and definite integration example
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Integrations
by parts and the definite integral |
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The
improper integrals |
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Differentiation
and integration of infinite series
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Differentiation
of power series |
| Applications
of the definite integral
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The
area of a region in the plane |
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The
area between the graph of a curve and the coordinate axis
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The
area bounded by a parametric curve
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The
area in polar coordinates
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The
area of the sector of a parametric curve
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The
area between two curves |
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Length
of plane curve, arc length |
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Arc length of a parametric curve
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Arc length of a curve in polar coordinates
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The
volume
of a solid of revolution
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The
volume
of a sphere
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The
volume
of a spherical segment
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The
volume
of a cone
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The
surface area of a solid of revolution |
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The
lateral surface area of a cone
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The surface area of a spherical
cap
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The surface area of an
ellipsoid
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| TRIGONOMETRY
- solved problems |
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Trigonometry |
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Degrees
to radians and radians to degrees conversion |
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Signs of trigonometric
functions
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Trigonometric
functions of arcs from 0
to ±
2p
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Trigonometric
functions of negative arcs or angles
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Trigonometric
functions of complementary angles
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Trigonometric
functions of supplementary angles
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Trigonometric
functions of arcs that differ on p/2
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Trigonometric
functions of arcs that differ on p
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Trigonometric
functions of arcs whose
sum is
2p |
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The values of the trigonometric functions of arcs that are multipliers of
30°
(p/6)
and 45°
(p/4)
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Calculation of values of trigonometric functions of an arbitrary angle
x |
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Basic relationships
between trigonometric functions of the same angle, examples |
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Trigonometric identities,
examples |
| Trigonometric
functions and equations |
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Trigonometric functions |
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Trigonometric
equations |
| Applications of trigonometry |
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Solving right triangles examples |
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Solving oblique or scalene triangles examples |
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Sections of solids |
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Applications
of trigonometry in geodesy (or plane surveying) and physics |
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An example of using trigonometry in
optics |
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| VECTORS
- solved problems |
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| Vectors
in a plane |
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Addition, subtraction
and scalar multiplication of vectors |
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Vectors
and a coordinate system, Cartesian vectors |
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Vectors in
a coordinate plane (a two-dimensional system of coordinates), Cartesian
vectors, examples
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| Vectors in
three-dimensional space in terms of Cartesian coordinates |
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Vectors in
three-dimensional space in terms of Cartesian coordinates, examples
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Scalar product or dot
product or inner product |
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Vector
product or cross product |
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An example for the vector product in physics
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The vector
product in component form
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The
mixed product or scalar triple product expressed in terms of components |
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Vector
product and mixed product, examples |
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| COORDINATE
GEOMETRY - solved problems |
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| Line in a coordinate plane
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The parametric equations of a line |
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The parametric equations of a line
examples
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Hessian normal form of the equation of a line |
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Distance between a point and a line |
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Condition that three points lie on the same line |
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| Points,
lines and planes in three-dimensional ( 3D ) coordinate system
represented by vectors |
| Line in three-dimensional
coordinate
system |
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The parametric equations of a line
in 3D
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Equation of a line defined by
direction vector and a point - Symmetric equation of a line |
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Equation of a line defined by
direction vector and a point examples |
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The angles that a line
forms with coordinate axes
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Angle between lines
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Condition for intersection of two
lines in a 3D space
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| The equation of a
plane in a
3D coordinate system |
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The equation of
the plane through three points
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The distance of a point
and a plane -
plane given in general form
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Angle (dihedral angle) between two
planes
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| Line
and plane
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The line of intersection of two planes
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Projection of a line onto coordinate planes
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How determine two planes of which, a given line is their
intersection line
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Intersection point of a line and a plane
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Sheaf or
pencil of planes
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| Points, lines and planes relations
in 3D space, examples
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The angle between line and plane
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Point,
line and plane - orthogonal projections, distances,
perpendicularity of line and plane
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Through a given point pass a line perpendicular to a given plane
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Given a line and a point, through
the point lay a plane perpendicular to the line
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Projection of a point onto a plane
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Projection of a point onto a line
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Through a given point lay a line perpendicular to a given line
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Projection of a line onto a plane
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Plane laid through a given point, such that be parallel with two skew lines
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Plane
laid through a given point, such that be parallel with two parallel
lines
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Distance between point and line
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Distance between point and plane
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Distance between parallel lines
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Distance between two skew lines
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