1
Limits
17 lessonsWhat a function is heading toward, even where it never arrives.
1.1Limits
1.2Comparing growth rates at infinityCalculus 1
1.3Limits by algebra: factoring and rationalizingCalculus 1
1.4What a limit is: graphs and tablesPrecalculus
1.5One sided limits and when a limit existsCalculus 1
1.6The limit lawsCalculus 1
1.7Continuity
1.8Continuity at a pointCalculus 1
1.9Infinite limits and vertical asymptotesCalculus 1
1.10Limits at infinity and horizontal asymptotesCalculus 1
1.11Continuity on an intervalCalculus 1
1.12The intermediate value theoremCalculus 1
1.13Discontinuities and how to remove themCalculus 1
1.14The extreme value theoremCalculus 1
1.15Making a piecewise function continuousCalculus 1
1.16The squeeze theorem and the limit of sine x over xCalculus 1
1.17The epsilon delta definition of a limitCalculus 1
2
Derivatives
20 lessonsThe rate of change at an instant, and the rules for finding it.
2.1Derivatives
2.3Finding a derivative from the definitionCalculus 1
2.4The derivative as a functionCalculus 1
2.5The derivative at a pointCalculus 1
2.6The derivative as a rate of changeCalculus 1
2.7The power rule and basic differentiation rulesCalculus 1
2.8The product ruleCalculus 1
2.9The quotient ruleCalculus 1
2.11The chain ruleCalculus 1
2.12Combining differentiation rulesCalculus 1
2.13Derivatives of exponential functionsCalculus 1
2.14Derivatives of the trigonometric functionsCalculus 1
2.16Second and higher derivativesCalculus 1
2.17Implicit differentiationCalculus 1
2.18Derivatives of inverse functionsCalculus 1
2.20Derivatives of inverse trigonometric functionsCalculus 1
3
Applications of derivatives
16 lessonsWhat the derivative tells you about a function and the world.
3.2Tangent lines and normal linesCalculus 1
3.3Linear approximation and differentialsCalculus 1
3.4Critical points and local versus absolute extremaCalculus 1
3.5Extrema on a closed intervalCalculus 1
3.6Increasing and decreasing from the derivativeCalculus 1
3.8The mean value theoremCalculus 1
3.9Concavity and inflection pointsCalculus 1
3.10Position, velocity and accelerationCalculus 1
3.11Related ratesCalculus 1
3.12The first and second derivative testsCalculus 1
3.13Curve sketching and connecting f, f' and f''Calculus 1
3.14L'Hopital's ruleCalculus 1
3.15OptimizationCalculus 1
3.16Newton's methodCalculus 1
4
Integrals
22 lessonsAdding up infinitely many infinitely thin pieces, and the surprise that undoes the derivative.
4.1Integrals
4.2Approximating area with Riemann sumsCalculus 1
4.3Properties of definite integralsCalculus 1
4.4Signed area and integrals from geometryCalculus 1
4.6The definite integral as a limit of Riemann sumsCalculus 1
4.7Antiderivatives and indefinite integralsCalculus 1
4.8The first fundamental theorem of calculusCalculus 1
4.10The second fundamental theorem of calculusCalculus 1
4.11Improper integralsCalculus 2
4.12The trapezoidal rule and Simpson's ruleCalculus 1
4.13u-substitutionCalculus 1
4.14Antiderivatives of trig, exponential and log functionsCalculus 1
4.15Improper integrals with an infinite integrandCalculus 2
4.16Integration by partsCalculus 2
4.17Completing the square inside an integralCalculus 1
4.18Dividing before integratingCalculus 1
4.19Integration by partial fractionsCalculus 2
4.20Trigonometric substitutionCalculus 2
4.21Trigonometric integralsCalculus 2
4.22Choosing an integration techniqueCalculus 2
5
Applications of integrals
14 lessonsArea, volume, distance, and total change.
5.2Volumes with known cross sectionsCalculus 1
5.4Area between curvesCalculus 1
5.5Position, displacement and distance from velocityCalculus 1
5.6The net change theorem in contextCalculus 1
5.7Arc lengthCalculus 2
5.8Average value of a functionCalculus 1
5.9Graphing an accumulation functionCalculus 1
5.10The disk methodCalculus 1
5.11The washer methodCalculus 1
5.12The shell methodCalculus 2
5.13Work as an integralCalculus 2
5.14Surface area of a solid of revolutionCalculus 2
6
Differential equations
8 lessonsEquations about a rate, and finding the function that obeys them.
6.2Writing a differential equation from a descriptionCalculus 1
6.3Differential equations and their solutionsCalculus 1
6.4Slope fieldsCalculus 1
6.5Euler's methodCalculus 2
6.6Separable differential equationsCalculus 1
6.7Exponential growth and decay modelsCalculus 1
6.8Logistic growthCalculus 2
7
Sequences and series in calculus
20 lessonsInfinite sums: when they settle, and how functions become polynomials.
7.2Convergence of sequencesCalculus 2
7.3Infinite series and partial sumsCalculus 2
7.4The geometric series testCalculus 2
7.5The nth term test for divergenceCalculus 2
7.6Absolute and conditional convergenceCalculus 2
7.7Alternating series and their error boundCalculus 2
7.9The direct and limit comparison testsCalculus 2
7.10The integral test and p seriesCalculus 2
7.11The ratio and root testsCalculus 2
7.12Choosing a convergence testCalculus 2
7.13Power series and the interval of convergenceCalculus 2
7.14Representing functions as power seriesCalculus 2
7.16Taylor polynomialsCalculus 2
7.17The binomial seriesCalculus 2
7.18The Taylor remainder and the Lagrange error boundCalculus 2
7.19The common Maclaurin seriesCalculus 2
7.20Using series to evaluate limits and integralsCalculus 2
8
Parametric, polar and vector valued functions
6 lessonsDoing calculus on curves described by a parameter, by an angle, or by a position vector.
8.1Area in polar coordinatesCalculus 2
8.3Derivatives of parametric curvesCalculus 2
8.4Slope of a polar curveCalculus 2
8.5Arc length and speed on parametric and polar curvesCalculus 2
8.6Vector valued functions and motion in the planeCalculus 2