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Pathway 02 · University

University Calculus

Single-variable and multivariable calculus as taught across Calculus I–III: limits, derivatives, integrals, series, partial derivatives and multiple integrals. The emphasis is on knowing why each technique applies, not only how to run it.

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Audience
University students taking any of Calculus I, II or III, and engineers or scientists who need a dependable working command of calculus.
Modules
6, in order
Level
University

Prerequisites

  • University Foundations or equivalent preparation
  • Confident use of functions, exponentials, logarithms and trigonometry
  • Comfort with algebraic manipulation of rational expressions

You will be able to

  • Differentiate using the product, quotient and chain rules and implicit differentiation, and use derivatives to find extrema and intervals of monotonicity and concavity
  • Evaluate integrals by substitution, integration by parts and partial fractions, and decide whether an improper integral converges
  • Compute Taylor polynomials and bound the remainder for a given accuracy
  • Compute partial derivatives, gradients and directional derivatives, and classify critical points of functions of two variables
  • Set up and evaluate double and triple integrals, including reversing the order of integration and using polar, cylindrical and spherical coordinates
  • State conditions for differentiating under the integral sign and apply it to a parameter-dependent integral

Modules

Work through these in order.

  1. 01

    Limits and continuity

    Make limits precise and use continuity to reason about functions on intervals.

    • ε–δ limits
    • Limit laws
    • Continuity and the intermediate value theorem
    • Asymptotic behavior
  2. 02

    Differentiation and its uses

    Build the rules of differentiation and use them to analyze and optimize functions.

    • Derivative rules
    • Implicit differentiation
    • Mean value theorem
    • Optimization and curve sketching
    • Higher-order derivatives
  3. 03

    Integration

    Connect integrals to antiderivatives and develop the main techniques, including improper integrals.

    • Fundamental theorem of calculus
    • Substitution and integration by parts
    • Partial fractions
    • Improper integrals
  4. 04

    Sequences, series and Taylor expansions

    Decide convergence and represent functions by power series.

    • Convergence tests
    • Power series and radius of convergence
    • Taylor polynomials and remainders
    • Series of standard functions
  5. 05

    Multivariable differential calculus

    Extend differentiation to functions of several variables.

    • Partial derivatives
    • Gradient and directional derivatives
    • Chain rule in several variables
    • Critical points and the Hessian test
  6. 06

    Multiple integrals and parameter integrals

    Integrate over regions in two and three dimensions and differentiate integrals with respect to a parameter.

    • Double and triple integrals
    • Change of variables and the Jacobian
    • Fubini's theorem and order of integration
    • Differentiation under the integral sign

Next step

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Rida can follow this pathway with you one-to-one or in a small group, with diagnostics and feedback.

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