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Covering the essential topics of calculus 2.

Videos on a third course in calculus (Multivariable Calculus).

Covering the basics of Calculus III

This is a variation on 18.02 Multivariable Calculus. It covers the same topics as in 18.02, but with more focus on mathematical concepts.

Acknowledgement

Prof. McKernan would like to acknowledge the contributions of Lars Hesselholt to the development of this course.

Calculus One is a first introduction to differential and integral calculus, emphasizing engaging examples from everyday life.

Calculus One is a first introduction to differential and integral calculus, emphasizing engaging examples from everyday life.

Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

This is an undergraduate course on differential calculus in one and several dimensions. It is intended as a one and a half term course in calculus for students who have studied calculus in high school. The format allows it to be entirely self contained, so that it is possible to follow it without any background in calculus.

18.014, Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus.

Sample questions from the A.P. Calculus AB and BC exams (both multiple choice and free answer). 2011 Calculus AB Free Response #1a. 2011 Calculus AB Free Response #1 parts b c d. 2011 Calculus AB Free Response #2 (a & b). 2011 Calculus AB Free Response #2 (c & d). 2011 Calculus AB Free Response #3 (a & b). 2011 Calculus AB Free Response #3 (c). 2011 Calculus AB Free Response #4a. 2011 Calculus AB Free Response #4b. 2011 Calculus AB Free Response #4c. 2011 Calculus AB Free Response #4d. 2011 Calculus AB Free Response #5a. 2011 Calculus AB Free Response #5b. 2011 Calculus AB Free Response #5c.. 2011 Calculus AB Free Response #6a. 2011 Calculus AB Free Response #6b. 2011 Calculus AB Free Response #6c. AP Calculus BC Exams: 2008 1 a. AP Calculus BC Exams: 2008 1 b&c. AP Calculus BC Exams: 2008 1 c&d. AP Calculus BC Exams: 2008 1 d. Calculus BC 2008 2 a. Calculus BC 2008 2 b &c. Calculus BC 2008 2d. 2011 Calculus BC Free Response #1a. 2011 Calculus BC Free Response #1 (b & c). 2011 Calculus BC Free Response #1d. 2011 Calculus BC Free Response #3a. 2011 Calculus BC Free Response #3 (b & c). 2011 Calculus BC Free Response #6a. 2011 Calculus BC Free Response #6b. 2011 Calculus BC Free Response #6c. 2011 Calculus BC Free Response #6d. 2011 Calculus AB Free Response #1a. 2011 Calculus AB Free Response #1 parts b c d. 2011 Calculus AB Free Response #2 (a & b). 2011 Calculus AB Free Response #2 (c & d). 2011 Calculus AB Free Response #3 (a & b). 2011 Calculus AB Free Response #3 (c). 2011 Calculus AB Free Response #4a. 2011 Calculus AB Free Response #4b. 2011 Calculus AB Free Response #4c. 2011 Calculus AB Free Response #4d. 2011 Calculus AB Free Response #5a. 2011 Calculus AB Free Response #5b. 2011 Calculus AB Free Response #5c.. 2011 Calculus AB Free Response #6a. 2011 Calculus AB Free Response #6b. 2011 Calculus AB Free Response #6c. AP Calculus BC Exams: 2008 1 a. AP Calculus BC Exams: 2008 1 b&c. AP Calculus BC Exams: 2008 1 c&d. AP Calculus BC Exams: 2008 1 d. Calculus BC 2008 2 a. Calculus BC 2008 2 b &c. Calculus BC 2008 2d. 2011 Calculus BC Free Response #1a. 2011 Calculus BC Free Response #1 (b & c). 2011 Calculus BC Free Response #1d. 2011 Calculus BC Free Response #3a. 2011 Calculus BC Free Response #3 (b & c). 2011 Calculus BC Free Response #6a. 2011 Calculus BC Free Response #6b. 2011 Calculus BC Free Response #6c. 2011 Calculus BC Free Response #6d.

Minima, maxima, and critical points. Rates of change. Optimization. Rates of change. L'Hopital's rule. Mean value theorem. Minima, maxima and critical points. Testing critical points for local extrema. Identifying minima and maxima for x^3 - 12x - 5. Concavity, concave upwards and concave downwards intervals. Recognizing concavity exercise. Recognizing concavity. Inflection points. Graphing using derivatives. Another example graphing with derivatives. Minimizing sum of squares. Optimizing box volume graphically. Optimizing box volume analytically. Optimizing profit at a shoe factory. Minimizing the cost of a storage container. Expression for combined area of triangle and square. Minimizing combined area. Rates of change between radius and area of circle. Rate of change of balloon height. Related rates of water pouring into cone. Falling ladder related rates. Rate of change of distance between approaching cars. Speed of shadow of diving bird. Mean Value Theorem. Introduction to L'H

Divergence theorem intuition. Divergence theorem examples and proofs. Types of regions in 3D. 3-D Divergence Theorem Intuition. Divergence Theorem Example 1. Why we got zero flux in Divergence Theorem Example 1. Type I Regions in Three Dimensions. Type II Regions in Three Dimensions. Type III Regions in Three Dimensions. Divergence Theorem Proof (part 1). Divergence Theorem Proof (part 2). Divergence Theorem Proof (part 3). Divergence Theorem Proof (part 4). Divergence Theorem Proof (part 5). 3-D Divergence Theorem Intuition. Divergence Theorem Example 1. Why we got zero flux in Divergence Theorem Example 1. Type I Regions in Three Dimensions. Type II Regions in Three Dimensions. Type III Regions in Three Dimensions. Divergence Theorem Proof (part 1). Divergence Theorem Proof (part 2). Divergence Theorem Proof (part 3). Divergence Theorem Proof (part 4). Divergence Theorem Proof (part 5).

Volume under a surface with double integrals. Triple integrals as well. Double Integral 1. Double Integrals 2. Double Integrals 3. Double Integrals 4. Double Integrals 5. Double Integrals 6. Triple Integrals 1. Triple Integrals 2. Triple Integrals 3. Double Integral 1. Double Integrals 2. Double Integrals 3. Double Integrals 4. Double Integrals 5. Double Integrals 6. Triple Integrals 1. Triple Integrals 2. Triple Integrals 3.

Indefinite integral as anti-derivative. Definite integral as area under a curve. Integration by parts. U-substitution. Trig substitution. Antiderivatives and indefinite integrals. Indefinite integrals of x raised to a power. Antiderivative of hairier expression. Basic trig and exponential antiderivatives. Antiderivative of x^-1. Simple Riemann approximation using rectangles. Generalizing a left Riemann sum with equally spaced rectangles. Rectangular and trapezoidal Riemann approximations. Trapezoidal approximation of area under curve. Riemann sums and integrals. Deriving integration by parts formula. Antiderivative of xcosx using integration by parts. Integral of ln x. Integration by parts twice for antiderivative of (x^2)(e^x). Integration by parts of (e^x)(cos x). U-substitution. U-substitution example 2. U-substitution Example 3. U-substitution with ln(x). Doing u-substitution twice (second time with w). U-substitution and back substitution. U-substitution with definite integral. (2^ln x)/x Antiderivative Example. Another u-substitution example. Riemann sums and integrals. Intuition for Second Fundamental Theorem of Calculus. Evaluating simple definite integral. Definite integrals and negative area. Area between curves. Area between curves with multiple boundaries. Challenging definite integration. Introduction to definite integrals. Definite integrals (part II). Definite Integrals (area under a curve) (part III). Definite Integrals (part 4). Definite Integrals (part 5). Definite integral with substitution. Introduction to trig substitution. Another substitution with x=sin (theta). Integrals: Trig Substitution 1. Trig and U substitution together (part 1). Trig and U substitution together (part 2). Trig substitution with tangent. Integrals: Trig Substitution 2. Integrals: Trig Substitution 3 (long problem). Fundamental theorem of calculus. Applying the fundamental theorem of calculus. Swapping the bounds for definite integral. Both bounds being a function of x. Proof of Fundamental Theorem of Calculus. Connecting the first and second fundamental theorems of calculus. Introduction to improper integrals. Improper integral with two infinite bounds. Divergent improper integral. Antiderivatives and indefinite integrals. Indefinite integrals of x raised to a power. Antiderivative of hairier expression. Basic trig and exponential antiderivatives. Antiderivative of x^-1. Simple Riemann approximation using rectangles. Generalizing a left Riemann sum with equally spaced rectangles. Rectangular and trapezoidal Riemann approximations. Trapezoidal approximation of area under curve. Riemann sums and integrals. Deriving integration by parts formula. Antiderivative of xcosx using integration by parts. Integral of ln x. Integration by parts twice for antiderivative of (x^2)(e^x). Integration by parts of (e^x)(cos x). U-substitution. U-substitution example 2. U-substitution Example 3. U-substitution with ln(x). Doing u-substitution twice (second time with w). U-substitution and back substitution. U-substitution with definite integral. (2^ln x)/x Antiderivative Example. Another u-substitution example. Riemann sums and integrals. Intuition for Second Fundamental Theorem of Calculus. Evaluating simple definite integral. Definite integrals and negative area. Area between curves. Area between curves with multiple boundaries. Challenging definite integration. Introduction to definite integrals. Definite integrals (part II). Definite Integrals (area under a curve) (part III). Definite Integrals (part 4). Definite Integrals (part 5). Definite integral with substitution. Introduction to trig substitution. Another substitution with x=sin (theta). Integrals: Trig Substitution 1. Trig and U substitution together (part 1). Trig and U substitution together (part 2). Trig substitution with tangent. Integrals: Trig Substitution 2. Integrals: Trig Substitution 3 (long problem). Fundamental theorem of calculus. Applying the fundamental theorem of calculus. Swapping the bounds for definite integral. Both bounds being a function of x. Proof of Fundamental Theorem of Calculus. Connecting the first and second fundamental theorems of calculus. Introduction to improper integrals. Improper integral with two infinite bounds. Divergent improper integral.

Limit introduction, squeeze theorem, and epsilon-delta definition of limits. Introduction to limits. Limit at a point of discontinuity. Determining which limit statements are true. Limit properties. Limit example 1. Limits 1. One-sided limits from graphs. One-sided limits from graphs. Introduction to Limits. Limit Examples (part 1). Limit Examples (part 2). Limit Examples (part 3). Limit Examples w/ brain malfunction on first prob (part 4). More Limits. Limits 1. Limits and infinity. Limits at positive and negative infinity. More limits at infinity. Limits with two horizontal asymptotes. Limits 2. Squeeze Theorem. Proof: lim (sin x)/x. Limit intuition review. Building the idea of epsilon-delta definition. Epsilon-delta definition of limits. Proving a limit using epsilon-delta definition. Limits to define continuity. Continuity. Epsilon Delta Limit Definition 1. Epsilon Delta Limit Definition 2. Introduction to limits. Limit at a point of discontinuity. Determining which limit statements are true. Limit properties. Limit example 1. Limits 1. One-sided limits from graphs. One-sided limits from graphs. Introduction to Limits. Limit Examples (part 1). Limit Examples (part 2). Limit Examples (part 3). Limit Examples w/ brain malfunction on first prob (part 4). More Limits. Limits 1. Limits and infinity. Limits at positive and negative infinity. More limits at infinity. Limits with two horizontal asymptotes. Limits 2. Squeeze Theorem. Proof: lim (sin x)/x. Limit intuition review. Building the idea of epsilon-delta definition. Epsilon-delta definition of limits. Proving a limit using epsilon-delta definition. Limits to define continuity. Continuity. Epsilon Delta Limit Definition 1. Epsilon Delta Limit Definition 2.

Line integral of scalar and vector-valued functions. Green's theorem and 2-D divergence theorem. Introduction to the Line Integral. Line Integral Example 1. Line Integral Example 2 (part 1). Line Integral Example 2 (part 2). Position Vector Valued Functions. Derivative of a position vector valued function. Differential of a vector valued function. Vector valued function derivative example. Line Integrals and Vector Fields. Using a line integral to find the work done by a vector field example. Parametrization of a Reverse Path. Scalar Field Line Integral Independent of Path Direction. Vector Field Line Integrals Dependent on Path Direction. Path Independence for Line Integrals. Closed Curve Line Integrals of Conservative Vector Fields. Example of Closed Line Integral of Conservative Field. Second Example of Line Integral of Conservative Vector Field. Green's Theorem Proof Part 1. Green's Theorem Proof (part 2). Green's Theorem Example 1. Green's Theorem Example 2. Constructing a unit normal vector to a curve. 2 D Divergence Theorem. Conceptual clarification for 2-D Divergence Theorem. Introduction to the Line Integral. Line Integral Example 1. Line Integral Example 2 (part 1). Line Integral Example 2 (part 2). Position Vector Valued Functions. Derivative of a position vector valued function. Differential of a vector valued function. Vector valued function derivative example. Line Integrals and Vector Fields. Using a line integral to find the work done by a vector field example. Parametrization of a Reverse Path. Scalar Field Line Integral Independent of Path Direction. Vector Field Line Integrals Dependent on Path Direction. Path Independence for Line Integrals. Closed Curve Line Integrals of Conservative Vector Fields. Example of Closed Line Integral of Conservative Field. Second Example of Line Integral of Conservative Vector Field. Green's Theorem Proof Part 1. Green's Theorem Proof (part 2). Green's Theorem Example 1. Green's Theorem Example 2. Constructing a unit normal vector to a curve. 2 D Divergence Theorem. Conceptual clarification for 2-D Divergence Theorem.

Thinking about forms of derivatives in multi-dimensions and for vector-valued functions: partial derivatives, gradient, divergence and curl. Partial Derivatives. Partial Derivatives 2. Gradient 1. Gradient of a scalar field. Divergence 1. Divergence 2. Divergence 3. Curl 1. Curl 2. Curl 3. Partial Derivatives. Partial Derivatives 2. Gradient 1. Gradient of a scalar field. Divergence 1. Divergence 2. Divergence 3. Curl 1. Curl 2. Curl 3.

Sequences, series and approximating functions. Maclaurin and Taylor series. Sequences and Series (part 1). Sequences and series (part 2). Maclaurin and Taylor Series Intuition. Cosine Taylor Series at 0 (Maclaurin). Sine Taylor Series at 0 (Maclaurin). Taylor Series at 0 (Maclaurin) for e to the x. Euler's Formula and Euler's Identity. Visualizing Taylor Series Approximations. Generalized Taylor Series Approximation. Visualizing Taylor Series for e^x. Error or Remainder of a Taylor Polynomial Approximation. Proof: Bounding the Error or Remainder of a Taylor Polynomial Approximation. Polynomial approximation of functions (part 1). Polynomial approximation of functions (part 2). Approximating functions with polynomials (part 3). Polynomial approximation of functions (part 4). Polynomial approximations of functions (part 5). Polynomial approximation of functions (part 6). Polynomial approximation of functions (part 7). Taylor Polynomials. Sequences and Series (part 1). Sequences and series (part 2). Maclaurin and Taylor Series Intuition. Cosine Taylor Series at 0 (Maclaurin). Sine Taylor Series at 0 (Maclaurin). Taylor Series at 0 (Maclaurin) for e to the x. Euler's Formula and Euler's Identity. Visualizing Taylor Series Approximations. Generalized Taylor Series Approximation. Visualizing Taylor Series for e^x. Error or Remainder of a Taylor Polynomial Approximation. Proof: Bounding the Error or Remainder of a Taylor Polynomial Approximation. Polynomial approximation of functions (part 1). Polynomial approximation of functions (part 2). Approximating functions with polynomials (part 3). Polynomial approximation of functions (part 4). Polynomial approximations of functions (part 5). Polynomial approximation of functions (part 6). Polynomial approximation of functions (part 7). Taylor Polynomials.

This course provides a brisk, challenging, and dynamic treatment of differential and integral calculus, with an emphasis on conceptual understanding and applications to the engineering, physical, and social sciences.

Using definite integrals with the shell and disc methods to find volumes of solids of revolution. Disk method around x-axis. Generalizing disc method around x-axis. Disc method around y-axis. Disc method (washer method) for rotation around x-axis. Generalizing the washer method. Disc method rotation around horizontal line. Washer method rotating around non-axis. Part 2 of washer for non axis rotation. Disc method rotating around vertical line. Calculating integral disc method around vertical line. Washer or ring method for vertical line rotation. Evaluating integral for washer method around vertical line. Shell method for rotating around vertical line. Evaluating integral for shell method example. Shell method for rotating around horizontal line. Shell method with two functions of x. Calculating integral with shell method. Shell method with two functions of y. Part 2 of shell method with 2 functions of y. Disc method: function rotated about x-axis. Disc method (rotating f(x) about x axis). Volume of a sphere. Disc method with outer and inner function boundaries. Shell method to rotate around y-axis. Disk method: rotating x=f(y) around the y-axis. Shell method around a non-axis line. Shell method around a non-axis line 2. Disk method around x-axis. Generalizing disc method around x-axis. Disc method around y-axis. Disc method (washer method) for rotation around x-axis. Generalizing the washer method. Disc method rotation around horizontal line. Washer method rotating around non-axis. Part 2 of washer for non axis rotation. Disc method rotating around vertical line. Calculating integral disc method around vertical line. Washer or ring method for vertical line rotation. Evaluating integral for washer method around vertical line. Shell method for rotating around vertical line. Evaluating integral for shell method example. Shell method for rotating around horizontal line. Shell method with two functions of x. Calculating integral with shell method. Shell method with two functions of y. Part 2 of shell method with 2 functions of y. Disc method: function rotated about x-axis. Disc method (rotating f(x) about x axis). Volume of a sphere. Disc method with outer and inner function boundaries. Shell method to rotate around y-axis. Disk method: rotating x=f(y) around the y-axis. Shell method around a non-axis line. Shell method around a non-axis line 2.