**Calculus Video Series**

Calculus video series on DVD consists of 56 DVDs. Each DVD is approximately 25 to 30 minutes in length. The DVDs cover the material in a standard college freshman year calculus course, and suitable for science and engineering majors. Topics covered include limits, continuity, differentiation, applications of differential calculus to graphing and optimizing functions, transcendental functions and their derivatives, integral calculus and applications to areas and volumes, L'Hopital's Rule, sequences and series, elementary vector algebra with dot products and cross-products.

The first 32 segments can also be used to supplement the typical one-semester elementary or basic calculus course, suitable for business majors and students of the liberal arts.

**6001 - Rectangular Coordinates and Graphing. **This segment covers the rectangular coordinate system and representation of ordered pairs of real numbers as points in the plane as well as the representation of points in the plane by ordered pairs of real numbers.

**6002 - Functions and Their Graphs. **This segment covers the mathematical definition of the word "function", the function notation, graphing of functions, and some simple examples of functions and their graphs.

**6003 - Average Rate of Change and Slope of Lines. **This segment covers average rate of change for a function between two points, slope of lines, the relation between slope, average rate of change, and velocity.

**6004 - Formulas for Lines. **This segment covers the point-slope and slope-intercept forms for equations of lines as well as problems involving equations for parallel and perpendicular lines.

**6005 - Limits and Continuity. **This segment covers computations of limits using limit rules, the definition of continuity, and the use of continuity in computations of limits.

**6006 - The Delta-Process and Instantaneous Rates of Change. **This segment covers the computation of derivatives or instantaneous rates of change as limits of difference quotients or limits of average rates of change.

**6007 - Tangent Lines. **This segment covers the computation of equations of tangent lines to graphs of functions using the differentiation rules and point-slope form for equations of lines.

**6008 - Differentiation Rules (Powers and Sums). **This segment covers the power, sum, difference, and scalar multiplication rules for differentiation.

**6009 - The Product and Quotient Rules. **This segment covers the product and quotient rules for differentiation including examples showing the power rule for positive integer powers as a consequence of the product rule.

**6010 - Composite Functions and the Chain Rule. **This segment covers the definition of the composition operation for functions as well as examples of computations with composite functions. The chain rule for differentiation of composite functions is covered and examples are covered illustrating how and when to apply the chain rule.

**6011 - Optimization Using Differentiation: Critical Points. **This segment covers the technique of optimization of a function by finding all zeroes of the derivative.

**6012 - Second Derivative, Inflection Points and Concavity. **This segment covers higher derivatives, inflection points, concavity, and examples illustrating the use of these concepts in graphing and in optimization.

**6013 - Implicit Differentiation. **This segment covers implicit differentiation and its use in finding slopes of tangent lines to curves at specific points when the curves are defined implicitly by equations.

**6014 - Inverse Functions and Their Derivatives. **This segment covers inverse functions and their graphical relationships as well as methods for finding inverses of invertible functions and for finding derivatives of the inverses.

**6015 - Exp, Log and Differentiation.**

This segment covers the exponential and logarithmic functions as well as their derivatives, and techniques for differentiation of functions involving the exponential and logarithmic functions using differentiation rules.

**6016 - Logarithmic Differentiation. **This segment covers logarithmic differentiation and its use in differentiating products with many factors as well as complicated exponential expressions.

**6017 - Applications to Growth and Decay. **This segment covers the use of exponential and logarithmic functions in solving problems where rate of change of a quantity is proportional to the amount of that quantity. Applications include population growth, radioactive decay, and continuously compounded interest.

**6018 - Trig Functions. **This segment reviews the trig functions and covers their derivatives and the use of differentiation rules to differentiate functions involving trig functions.

**6019 - Related Rates. **This segment covers applications involving related rates using differentiation rules to differentiate equations relating various quantities to obtain equations relating rates of change.

**6020 - Differentiation and Approximation. **This segment covers the use of differentiation to obtain linear approximations to function values at points near points where values and derivatives are computable.

**6021 - Taylor's Formula. **This segment covers Taylor's formula and its use in approximating function values as well as problems of finding Taylor polynomials for functions.

**6022 - Areas, Antidifferentiation and the Fundamental Theorem of Calculus. **This segment is an introduction to the ideas of integral calculus and the use of antidifferentiation and the fundamental theorem of calculus in the computation of area.

**6023 - Integration Formulas. **This segment covers the power, sum, difference, and scalar multiplication rules for integration and techniques for reducing antidifferentiation of certain types of functions to application of these rules.

**6024 - Substitution. **This segment covers the power, sum, difference, and scalar multiplication rules for integration and techniques for reducing antidifferentiation of certain types of functions to application of these rules.

**6025 - Integration by Parts. **This segment covers integration by parts and techniques for using integration by parts to antidifferentiate certain classes of functions.

**6026 - Definite Integrals and Areas. **This segment covers the computation of areas for regions bounded by curves using the definite integral.

**6027 - Definite Integrals, Substitution, and Integration by Parts. **This segment covers definite integrals which can be computed by substitution and/or integration by parts.

**6028 - Advanced Area Problems.**

This segment covers more difficult examples of area where boundaries may involve several curves and computations involve more than one definite integral.

**6029 - Volume Problems. **This segment covers volumes for solids of revolution as well as Cavallieri's principle for finding volumes from cross-sectional area functions by integration.

**6030 - Advanced Volume Problems. **This segment covers problems of finding volumes of more complicated geometric solids including the torus, the ball with a hole drilling through, and the intersection of two solid cylinders.

**6031 - Applications to Physics.**

This segment covers applications to calculus to Newtonian mechanics, the laws of motion for an object with one degree of freedom of movement, the concepts of potential and kinetic energy, conservation of energy and gravitation.

**6032 - Applications to Business. **This segment covers applications of calculus to business problems, the calculus interpretation of the word "marginal" as used in business, marginal cost, marginal profit, marginal revenue, and optimization problems arising in business.

**6033 - Special Trigonometric Limits. **The special trigonometric limits of sin(x) over x and (1-cos(x)) over x as x approaches zero are reviewed and examples are worked involving limits of algebraic expressions involving trigonometric functions which can be evaluated by reduction to one of the former cases.

**6034 - General Limits. **The basic theorems on limits are reviewed including the composition or substitution theorem, the squeeze theorem, and its corollary, the fact that a product of a bounded function by a function with limit zero must also have limit zero. Examples of limits of a more advanced nature are worked which illustrate the use of these theorems.

**6035 - One-Sided Limits. **The definition of one-sided limit is given and visually illustrated. The theorem on the relation between one-sided limits and two-sided (ordinary) limits is reviewed, and examples worked both for the computation of one-sided limits and the use of one-sided limits to show non-existence of certain two-sided limits.

**6036 - Limits Using Continuity. **The elementary examples of limits are worked out using the idea of extending a continuous function to include the limit point in the domain as in the removal of a singularity.

**6037 - Hyperbolic Functions.**

The hyperbolic trigonometric functions are defined, their basic identities are reviewed as well as their derivatives. Examples of differentiation involving hyperbolic functions are worked.

**6038 - L'Hopital's rule.**

L'Hopital's rule for computation of limits of indeterminate form is reviewed and examples of limit problems requiring L'Hopital's rule are worked.

**6039 - Trigonometric Integrals. **Techniques and reduction formulas for integrating products of trigonometric functions are reviewed. Examples are worked illustrating the various cases.

**6040 - Trigonometric Substitutions. **Techniques of using trigonometric substitution to simplify integrands are reviewed and examples are worked showing how to integrate functions containing quadratic expressions by trigonometric substitution.

**6041 - Partial Fractions. **The technique of integrating a rational function by expressing it as a sum of partial fractions is reviewed and illustrated in worked examples.

**6042 - Improper Integrals. **The definition of an improper integral as a limit of proper integrals is reviewed and examples of improper integrals are worked.

**6043 - Area in Polar Coordinates. **The technique and formula for area in polar coordinates is reviewed. Examples are worked using the integration formula for area in polar coordinates to compute areas of regions bounded by curves expressed in polar coordinates.

**6044 - Sequences and Convergence. **The basic definitions of sequences, convergence, and divergence are reviewed. The theorem on use of computing limits of sequences in terms of limits of continuous functions is discussed, and examples are worked for illustration.

**6045 - Summation Notation. **The sigma notation for summation is reviewed, examples are worked showing how to compute sums expressed in sigma notation. The concept of dummy index is discussed and examples are worked showing how to change indices in the sigma notation via substitution.

**6046 - Infinite Series. **The basic definitions of infinite series and partial sums are reviewed, the definitions of convergence and divergence for infinite series are reviewed and the nth term test for divergence of an infinite series is reviewed. Examples of telescoping series and geometric series are worked as well as examples showing the use of the nth term test to prove divergence of certain series.

**6047 - Comparison Test. **The comparison test and limit comparison tests are reviewed and discussed for infinite series and examples are worked illustrating their use in determining convergence or divergence of certain infinite series.

**6048 - Integral Test. **The comparison test and limit comparison tests are reviewed and discussed for infinite series and examples are worked illustrating their use in determining convergence or divergence of certain infinite series.

**6049 - Absolute Convergence and Alternating Series. **Absolute convergence is reviewed as well as forms of the comparison test and limit comparison for series with negative as well as positive terms in the determination of absolute convergence. Conditional convergence is reviewed. Alternating series are defined and the nth term test for convergence of an alternating series is reviewed. Examples illustrating the concepts are worked as well as examples using the nth term to estimate the error in a partial sum and examples of finding the proper partial sum for estimating an alternating series sum to within predetermined error tolerance.

**6050 - Ratio Test and Root Test. **The ratio test and the root test for determining convergence or divergence of infinite series are reviewed and discussed. Examples are worked illustrating both tests as well as how to choose between the two n tests from the form of the nth term.

**6051 - Power Series.**

Power series, radius of convergence, and interval of convergence are defined and discussed as well as the theorems on termwise differentiation and integration of power series. The ratio and root test forms for determining radius of convergence are reviewed and examples are worked illustrating their use.

**6052 - Taylor Series. **The Taylor series of a function is defined and the Lagrange form of the Taylor remainder is reviewed and used to show certain functions equal their Taylor series. Examples are also worked illustrating techniques of algebra combined with termwise differentiation and integration to obtain Taylor series of certain functions from the formula for the sum of the geometric series.

**6053 - Vectors. **Vectors are defined as arrows in space and the basic rules of vector addition and scalar multiplication are discussed visually. The commutative and associative laws are visually demonstrated for vector addition and the distributive law for scalar multiplication is demonstrated visually. The notion of a space of vectors is discussed and the definition of a frame of vectors is given in cases of all vectors in a line, a plane, or 3-dimensional space. The formulas for computing the addition and scalar multiplication in coordinates relative to a frame are demonstrated in one and two dimensions and reviewed for three dimensions.

**6054 - Dot Product and Length.**

The geometric definition of dot product for vectors as arrows in space is given and the commutative and distributive laws are demonstrated visually. The formulas for computing the dot product in coordinates relative to a frame are demonstrated as well as the utility of an orthonormal frame for simplifying the formulas. The relation of dot product to length is reviewed and demonstrated. The importance of understanding the geometry of the vector operations is emphasized throughout in order to facilitate the use of vector techniques in applications.

**6055 - Vector Component Computations. **The formulas for computing vector addition, scalar multiplication, and dot product are reviewed and used together with their geometric properties to derive geometric formulas and equations. The coordinate formulas for the distance between a pair of points in space, the normalization of a vector, the equation of a sphere, and the distance from a point to a plane are demonstrated using vectors.

**6056 - Vector Cross Product. **The geometric definition of the cross product and the right hand rule are given and demonstrated visually. the relation between lengths of the cross product are reviewed. The anticommutative and distributive laws for the cross product are geometrically and visually demonstrated and consequences examined. The coordinate formula for the cross product using coordinates relative to the standard right hand coordinate system is derived and the technique for each calculation using two by two determinants are demonstrated and illustrated by example.

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