Calculus Revisited: Single Variable Calculus

Video Lectures

Displaying all 37 video lectures.
I. Course Introduction
Lecture 1
Preface
Play Video		 Preface
In this video lecture, Professor Gross introduces the course consisting of:

Lecture Videos: 
Approximately 30-45 minutes per video

Supplementary Notes:
Prerequisite materials, detailed proofs, and deeper treatments of selected topics

Study Guides: Exercises with solutions, including a pre-test for each topic

Blackboard Photos: Photographs of every chalkboard used in the videotapes, for lecture preview or review

*Note: The course makes reference to the out-of-print textbook cited below, but any newer textbook will suffice to expand on topics covered in the video lectures.

Thomas, George B. Calculus and Analytic Geometry. Reading, Mass: Addison-Wesley, 1972. ISBN: 9780201075250
II. Sets, Functions, and Limits
Lecture 2
Analytic Geometry
Play Video		 Analytic Geometry
In this video lecture, Prof. Herbert Gross discusses Cartesian coordinates, curves as sets of points, graphs of functions, equations of straight lines and simultaneous linear equations.
Lecture 3
Functions
Play Video		 Functions
In this video lecture, Prof. Herbert Gross discusses notations, concepts of onto and one-to-one, the arithmetic of functions of a real variable, intervals and deleted neighborhoods, absolute values and composition of functions. 
Lecture 4
Inverse Functions
Play Video		 Inverse Functions
In this video lecture, Prof. Herbert Gross discusses the concept of an inverse function, graphical interpretation, single-valued and multivalued functions and branches of functions.
Lecture 5
Derivatives and Limits
Play Video		 Derivatives and Limits
In this video lecture, Prof. Herbert Gross discusses instantaneous speed as an outgrowth of average speed, the definition of a limit, instantaneous speed as a limit, the formal definition of limit and some consequences.
Lecture 6
A More Rigorous Approach to Limits
Play Video		 A More Rigorous Approach to Limits
In this video lecture, Prof. Herbert Gross continues on the previous lecture. Important limit properties are developed as theorems from the formal definition of limit.
Lecture 7
Mathematical Induction
Play Video		 Mathematical Induction
In this video lecture, Prof. Herbert Gross discusses the meaning of mathematical induction. He gives some examples of what mathematical induction is and isn't, as well as applications to limit theorems.
III. Differentiation
Lecture 8
Derivatives of Some Simple Functions
Play Video		 Derivatives of Some Simple Functions
In this video lecture, Prof. Herbert Gross discusses the definition of a derivative. He teaches about the derivative of x to the n where n is an integer, as well as the derivatives of sums, differences, products, and quotients.
Lecture 9
Approximations and Infinitesimals
Play Video		 Approximations and Infinitesimals
In this video lecture, Prof. Herbert Gross provides:
- An approximation of delta y by f(x) delta x
- A discussion of that difference between delta y and f'(x) delta x
- An introduction to the chain rule
Lecture 10
Composite Functions and the Chain Rule
Play Video		 Composite Functions and the Chain Rule
In this video lecture, Prof. Herbert Gross talks about the composition of functions, graphical interpretation, applications to parametric equations and using the chain rule to extend the concept of finding derivatives.
Lecture 11
Differentiation of Inverse Functions
Play Video		 Differentiation of Inverse Functions
In this video lecture, Prof. Herbert Gross discusses the concept of an inverse function, differentiation of an inverse function and when a function is invertible.
Lecture 13
Continuity
Play Video		 Continuity
In this video lecture, Prof. Herbert Gross discusses:
- Physical interpretation of continuity
- The definition of continuity in terms of limits
- A geometric interpretation of continuity
- Analytic consequences
Lecture 14
Curve Plotting
Play Video		 Curve Plotting
In this video lecture, Prof. Herbert Gross provides:
- A basic pre-calculus review
- A lecture on even and odd functions and other symmetries
- A discussion on the role of the first and second derivatives in curve plotting, stationary points and inflections
Lecture 15
Maxima and Minima
Play Video		 Maxima and Minima
In this video lecture, Prof. Herbert Gross discusses high and low points of a curve, techniques for finding them, applications to finding maxima and minima of functions and physical applications.
Lecture 16
Rolle's Theorem and its Consequences
Play Video		 Rolle's Theorem and its Consequences
Statement of Rolle's Theorem; a geometric interpretation; some cautions; the Mean Value Theorem; consequences of the Mean Value Theorem.
Lecture 17
Inverse Differentiation
Play Video		 Inverse Differentiation
In this video lecture, Prof. Herbert Gross talks about the "opposite" of differentiation: Inverse Differentiation. He teaches how to find f(x) knowing f'(x), and gives some examples, formulas and correct notations.
Lecture 18
The
Play Video		 The "Definite" Indefinite Integral
In this video lecture, Prof. Herbert Gross discusses the meaning of the definite integral as g(b) - g(a) where g'(x) = f(x) and provides some applications.
IV. The Circular Functions
Lecture 19
Circular Functions
Play Video		 Circular Functions
In this video lecture, Prof. Herbert Gross discusses trigonometric functions without angles, the logic of radian measure, definition of circular functions, and derivatives of sin x and cos x.
Lecture 20
Inverse Circular Functions
Play Video		 Inverse Circular Functions
In this video lecture, Prof. Herbert Gross discusses the meaning of arc sin x in terms of the sine function, the derivative of arc sin x in terms of the derivative of sin x, and some applications.
V. The Definite Integral
Lecture 21
The Definite Integral
Play Video		 The Definite Integral
In this video lecture, Prof. Herbert Gross discusses:
- Axiomatic approach to area
- Area approximations by upper and lower bounds
- The method of exhaustion
- Using limits to find areas of non-rectilinear regions
- Piecewise continuity
- Trapezoidal approximations
Lecture 22
Marriage of Differential and Integral Calculus
Play Video		 Marriage of Differential and Integral Calculus
In this video lecture, Prof. Herbert Gross discusses:
- First Fundamental Theorem of Integral Calculus; some applications
- Second Fundamental Theorem of Integral Calculus; some applications;
- Significance of the two theorems
Lecture 23
Three-Dimensional Area
Play Video		 Three-Dimensional Area
In this video lecture, Prof. Herbert Gross discusses:
- Extending the axioms of area to volume; some applications
- The method of cylindrical shells
Lecture 24
One-Dimensional Area
Play Video		 One-Dimensional Area
In this video lecture, Prof. Herbert Gross discusses:
- The main difference between arc-length and either area or volume
- The limit definition of arc-length
- Approximating errors and their magnitude when we use infinite sums
VI. Transcendental Functions
Lecture 25
Logarithms without Exponents
Play Video		 Logarithms without Exponents
In this video lecture, Prof. Herbert Gross discusses:
- The concept of the natural logarithm
- The notion of the rate of change being proportional to the amount present
- The general concept of a logarithmic function
- ln x in terms of differential and integral calculus
- The meaning of the number e as the base of the natural logarithms
Lecture 26
Inverse Logarithms
Play Video		 Inverse Logarithms
In this video lecture, Prof. Herbert Gross discusses:
- The invertibility of the logarithmic function
- e to the x as the inverse of ln x
- A discussion of exponential functions and some applications
Lecture 27
What a Difference a Sign Makes
Play Video		 What a Difference a Sign Makes
In this video lecture, Prof. Herbert Gross discusses:
- Hyperbolic functions
- Comparisons with circular functions
- Relationship between hyperbolic functions and exponential functions
- Applications of calculus to hyperbolic functions
Lecture 28
Inverse Hyperbolic Functions
Play Video		 Inverse Hyperbolic Functions
In this video lecture, Prof. Herbert Gross discusses:
- The theory of inverse functions applied to the hyperbolic functions
- Some formulas for differentiation and integration; some applications
VII. More Integration Techniques
Lecture 29
Some Basic Recipes
Play Video		 Some Basic Recipes
In this video lecture, Prof. Herbert Gross discusses:
- A review and extension of previous results for finding f(x) knowing f'(x)
- Particular emphasis on the case where f'(x) involves the sum and/or difference of two squares
- Completing the square
Lecture 30
Partial Fractions
Play Video		 Partial Fractions
In this video lecture, Prof. Herbert Gross discusses:
- The concept of partial fractions
- Finding f(x) when f'(x) is the quotient of two polynomials
- Some notes about identities
- Application of partial fractions to the case where f is of the form f(sinx, cos x)
Lecture 31
Integration by Parts
Play Video		 Integration by Parts
In this video lecture, Prof. Herbert Gross discusses:
- Using the identity d(uv) = udv + vdu to find the integral of udv knowing the integral of vdu
- Using the technique to evaluate certain integrals
- Reduction formulas
- Some applications
Lecture 32
Improper Integrals
Play Video		 Improper Integrals
In this video lecture, Prof. Herbert Gross discusses:
- The problem of trying to study the integral of f(x)dx when f(x) is not continuous on the interval [a,b]
- What happens if the limits of integration are not finite
- Importance of improper integrals
VIII. Infinite Series
Lecture 33
Many Versus Infinite
Play Video		 Many Versus Infinite
In this video lecture, Prof. Herbert Gross discusses:
- Discussion of how infinity differs from "very large"
- Some sublte and not-so-subtle consequences of the difference
- The case against intuition
- Motivating infinite series in terms of finding area as a limit
Lecture 34
Positive Series
Play Video		 Positive Series
In this video lecture, Prof. Herbert Gross discusses:
- The special case wherein each term in the series is non-negative
- The concept of convergence
- The comparison test
- The ratio test
- The integral test
Lecture 35
Absolute Convergence
Play Video		 Absolute Convergence
In this video lecture, Prof. Herbert Gross discusses:
- Non-absolute convergence
- Conditional and absolute convergence
- A series converging when each of its negative terms is replaced by the absolute value of that term
- Geometric interpretation
Lecture 36
Polynomial Approximations
Play Video		 Polynomial Approximations
In this video lecture, Prof. Herbert Gross discusses:
- Using an nth degree polynomial to approximate a function f(x)
- How to choose the coefficients
- Power series
- Taylor's Remainder Theorem
- Expressing functions in terms of power series
Lecture 37
Uniform Convergence
Play Video		 Uniform Convergence
In this video lecture, Prof. Herbert Gross discusses:
- Point-wise convergence versus uniform convergence
- Some important consequences of uniform convergence
- Applications of uniform convergence to the study of power series
Lecture 38
Uniform Convergence of Power Series
Play Video		 Uniform Convergence of Power Series
In this video lecture, Prof. Herbert Gross discusses:
- Weirstrass M-test
- Using power series to evaluate definite integrals when we do not know the anti-derivative of the integrand