
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

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 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 









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