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) 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.