Calculus I
Functions, limits, continuity, and the basics of derivatives and integrals.
Textbook: Robert Adams, Christopher Essex, Calculus, Eighth Edition, Pearson
- Week 1Functions, New Functions From Old Ones, Exponential Functions
- Functions, transformations and exponential functionsDomain, composition, even/odd functions, graph shifts and exponential equations. Every question for the rest of the semester builds on this language.
- Week 2Inverse Functions, Logarithmic Functions, Trigonometric Functions
- Inverse functions, logarithms and trigonometric functionsFinding inverse functions, logarithm rules, log/exponential equations, unit circle values and inverse trigonometric functions. The alphabet of the derivative weeks.
- Week 3Limits, Infinite Limits
- Limits and infinite limitsThe meaning of a limit, one-sided limits, techniques for resolving the 0/0 indeterminate form, sin x / x, the Squeeze Theorem and vertical asymptotes. The first exam question almost always comes from here.
- Week 4Limits at Infinity, Continuity
- Limits at infinity and continuityLimits as x goes to infinity, horizontal and oblique asymptotes, the three conditions of continuity, types of discontinuity and the Intermediate Value Theorem. Asymptote questions and 'what should k be' questions come from here.
- Week 5Introduction to the Derivative, the Product Rule and the Quotient Rule
- The definition of the derivative, product and quotient rulesComputing the derivative from the limit definition, the tangent line, differentiability, the power rule, and the product and quotient rules. A 'Use the definition' question is almost certain on the midterm.
- Week 6The Derivative of Trigonometric Functions, the Chain Rule and Implicit Differentiation
- Trigonometric derivatives, the chain rule and implicit differentiationDerivatives of the six trigonometric functions, the chain rule for nested functions, and implicit differentiation for curves where y cannot be written in terms of x. The longest derivative questions on the midterm come from here.
- Week 7The Derivative of Logarithmic, Exponential and Inverse Trigonometric Functions
- Derivatives of logarithmic, exponential and inverse trigonometric functionsDerivatives of e^x, a^x, ln x, log_a x and arcsin, arccos, arctan, logarithmic differentiation and the derivative of an inverse function. How to solve questions like x^x that seem to have 'no rule'.
- Week 8Related Rates Problem, Extreme Values
- Related rates and extreme valuesThe rate relationship between quantities that change with time (related rates), and finding the absolute maximum/minimum on a closed interval. Two classic word-problem types, worth many points.
- Week 9The Mean Value Theorem and Curve Sketching
- The Mean Value Theorem and curve sketchingRolle's and the Mean Value Theorem, the increasing/decreasing test, first and second derivative tests, concavity, inflection points and step-by-step graph sketching. The full template for a 'Sketch the graph' question.
- Week 10Optimization Problems, Linear Approximation and Differentials
- Optimization, linear approximation and differentialsTurning a word problem into a single-variable function and finding its maximum/minimum, approximating with the tangent line, and estimating error with differentials. The classic word problem of the final.
- Week 11Areas as Limits of Sums, the Definite Integral, Antiderivatives
- Area as a limit of sums, the definite integral and the antiderivativeSigma notation, left/right/midpoint Riemann sums, the definition of area as a limit, properties of the definite integral and the antiderivative (indefinite integral) table. The foundation of the integral weeks.
- Week 12The Fundamental Theorem of Calculus, The Substitution Method
- The Fundamental Theorem of Calculus and substitutionThe theorem that links derivative and integral: the derivative of an integral function and evaluating a definite integral with an antiderivative. Then the chain rule for integrals: u-substitution.
- Week 13Integration by Parts, Integration by Partial Fractions
- Integration by parts and partial fractionsIntegration by parts for products (LIATE, the tabular method, cyclic integrals) and partial fractions for integrating rational functions. The long integral questions of the final.
- Week 14Inverse Substitution and Review
- Inverse substitution: trigonometric substitutionThe substitutions x = a sin θ, a tan θ, a sec θ when you see a² - x², a² + x², x² - a² inside roots, returning with a triangle, completing the square and algebraic inverse substitutions.
- Final review: limits, derivatives, applications and integralsA one-page summary of the whole semester: the main formula of each topic, decision charts and 12 mixed questions like the final.