|
| 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 Newton-Raphson
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 x-axis 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 x-axis 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
|
|
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
|
|
| GEOMETRY
- solved problems |
|
| Triangles |
|
Congruence of triangles
|
|
Similarity of triangles,
division of a line segment in a given ratio |
|
Solving
the right triangle |
|
Triangle
formulas
|
|
The
sine law (or the sine rule) and the cosine law |
|
The sine law |
|
The cosine law |
|
Calculating angles
of an oblique triangle |
|
Solving
the oblique triangle - use of the sine law and the cosine law |
|
Triangle
solved problems examples |
| Plane
figures: quadrilaterals, polygons - regular polygons and
circle |
|
Plane figures, solved
problems examples |
| Solid figures:
prisms, pyramids, regular polyhedrons and solids of revolution |
|
Solid
figures, solved problems examples |
| Geometry - 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 |
|
| 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 |
|
| ANALYTIC
GEOMETRY
- Conic sections - solved problems |
|
| Circle |
|
Circle through three points
|
|
Circle
and line
|
|
Line circle intersection
|
|
Equation of a tangent at a point
of a circle with the center at the origin
|
|
Equation of a tangent at a point
of a translated circle |
|
Condition of tangency - Condition for a line to be the tangent to a circle
|
|
Tangents to a circle from a point outside the circle - use of the
tangency condition
|
|
Angle between a line and a circle
|
|
Angle between two circles
|
| Ellipse
|
|
Equation of the ellipse, standard
equation of the ellipse
|
|
Ellipse
and line |
|
Condition for a line to be the tangent to an
ellipse
|
|
Tangents to an ellipse from a point outside the ellipse
|
|
Angle between
the circle and the ellipse
|
| Hyperbola
|
|
Equation of the hyperbola
|
|
Properties of the hyperbola
|
|
Translated hyperbola
|
|
The vertex form of the equation of the hyperbola
|
|
Hyperbola
and line
|
|
Condition for a line to be the tangent to the hyperbola -
tangency condition
|
|
Tangents
to the hyperbola from a point outside the hyperbola |
|
Angle
between a curve and the hyperbola
|
| Parabola
|
|
Vertex form of the equation of a parabola
|
|
Equation of a translated parabola - the standard form
|
|
Parabola
and Line
|
|
Condition for a line to be the tangent to the parabola
- tangency condition
|
|
| 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 two-dimensional system of coordinates), Cartesian
vectors, examples
|
| Vectors in
three-dimensional space in terms of Cartesian coordinates |
|
Vectors in
three-dimensional 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 |
|
|
|
|
|
|
|
 |
|
|
| Back
to beginning solved
problems contents |
|
 |
|
| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |