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
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