Calculus I — Differential & Integral Calculus

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• The tangent line problem and rates of change ~12 min
• Introduction to limits ~12 min
• Estimating limits numerically and graphically ~10 min
• One-sided limits ~12 min
• Limit laws ~12 min
• Algebraic techniques for limits ~14 min
• The Squeeze Theorem ~10 min
• The limit of sin(x)/x ~10 min
• Infinite limits and vertical asymptotes ~12 min
• Limits at infinity and horizontal asymptotes ~12 min
• Continuity at a point ~10 min
• Types of discontinuity ~12 min
• Continuity on an interval ~10 min
• The Intermediate Value Theorem ~10 min
• The precise (ε-δ) definition of a limit ~30 min
• Writing an ε-δ proof ~35 min
• ε-δ proofs beyond straight lines ~40 min

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• Average vs. instantaneous rate of change ~12 min
• The definition of the derivative ~14 min
• The derivative as a function ~12 min
• Reading f' from the graph of f ~12 min
• Differentiability and continuity ~12 min
• Constant, power, and constant-multiple rules ~12 min
• Sum and difference rules ~10 min
• The product rule ~14 min
• The quotient rule ~12 min
• Derivatives of trigonometric functions ~12 min
• Higher-order derivatives ~10 min
• Velocity and acceleration ~12 min
• Rates of change in the sciences ~10 min

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• The chain rule ~15 min
• Recognizing composite functions before differentiating ~12 min
• Chain rule with power functions ~12 min
• Chain rule with trigonometric functions ~12 min
• Chain rule with exponential and logarithmic functions ~14 min
• Combining the chain rule with the product and quotient rules ~15 min
• Implicit differentiation ~15 min
• Tangent lines to implicit curves ~13 min
• Derivatives of inverse functions ~14 min
• Derivatives of inverse trigonometric functions ~14 min

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• Related rates -- the general method ~18 min
• Related rates -- geometric scenes ~18 min
• Related rates -- volume and rate scenes ~18 min
• Linear approximation and differentials ~18 min
• Newton's method ~18 min

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• Absolute extrema and the Extreme Value Theorem ~16 min
• Local extrema and Fermat's theorem ~16 min
• Finding absolute extrema on a closed interval ~16 min
• Rolle's Theorem ~16 min
• The Mean Value Theorem ~18 min
• Consequences of the Mean Value Theorem ~15 min
• Increasing/decreasing functions and the First Derivative Test ~17 min
• Classifying local extrema with the First Derivative Test ~16 min
• Concavity and inflection points ~16 min
• The Second Derivative Test ~16 min
• Curve sketching -- putting it all together ~20 min
• Curve sketching -- worked examples on harder functions ~20 min
• Limits at infinity revisited / end behavior for curve sketching ~16 min
• Indeterminate forms and L'Hopital's Rule ~18 min
• L'Hopital's Rule -- other indeterminate forms ~18 min
• Applying derivatives to graphs -- reading f' and f'' to reconstruct f ~18 min

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• Optimization: the general method ~20 min
• Optimization: geometric problems ~20 min
• Optimization: applied problems ~20 min
• Optimization: domain constraints and endpoint candidates ~20 min

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• Antiderivatives: the reverse of differentiation ~14 min
• Basic antiderivative rules ~16 min
• Antiderivatives of trigonometric and exponential functions ~15 min
• Initial value problems ~16 min
• Approximating area under a curve -- Riemann sums ~18 min
• Sigma notation and the limit definition of the definite integral ~18 min
• The definite integral: properties ~16 min
• The definite integral and signed area ~17 min
• The Trapezoidal Rule and comparing approximation methods ~17 min
• Applications of Riemann sums ~16 min

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• FTC Part 1: the accumulation function ~19 min
• FTC Part 1: the chain rule twist ~17 min
• FTC Part 2: evaluating definite integrals ~18 min
• Indefinite integrals and the Net Change Theorem ~16 min
• The Fundamental Theorem -- worked applications ~18 min
• Why FTC matters: derivatives and integrals as inverse operations ~16 min

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• U-substitution: reversing the chain rule ~18 min
• Choosing u: recognizing a substitution ~18 min
• U-substitution with definite integrals ~17 min
• Substitution with trig and exponential integrands ~17 min
• Area between two curves ~17 min
• Volumes by disks and washers ~19 min
• Average value and the Mean Value Theorem for Integrals ~18 min