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University of Toronto · St. George campus

MAT135H1 Applied calculus: differential foundations

Practise limits, derivatives and applications from the St. George applied-calculus route. MAT135H5, MAT135H3 and proof-based courses are different courses.

40 original question families · 120 parameter instances

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Course alignment and coverage

Aligned to the current official course-calendar description, checked 25 October 2026. A current section syllabus was not publicly verified. No weekly order or assessment weighting is claimed.

The four pathways share topic families. Across the whole pilot there are 80 distinct families and 240 parameter instances, presented in English and French; translations and shared course mappings are not extra questions.

Supported learning objectives

  • Limits

    Evaluate a limit using substitution, factoring or rationalization.

  • Continuity

    Find missing values and compare one-sided limits.

  • Polynomial derivatives

    Apply the power rule and interpret a derivative.

  • Product rule

    Differentiate polynomial products and identify missing terms.

  • Chain rule

    Differentiate composite polynomial functions.

  • Tangent lines

    Connect slope, point and tangent-line equation.

  • Implicit differentiation

    Differentiate a curve and evaluate a local slope.

  • Related rates

    Relate changing lengths, areas, volumes and distances.

  • Linear approximation

    Use a tangent model and estimate a differential.

  • Optimization

    Locate and check a maximum or minimum in a stated domain.

What this pilot does not cover

This pilot does not cover the full course: logarithmic and trigonometric derivatives, slope fields, Euler’s method, differential equations and longer modelling arguments remain outside this bank.

An original worked example

Find lim as x→2 of (x² + 3x - 1).

Worked solution

Answer: 9

  1. Polynomials are continuous everywhere.
  2. 2² + 3(2) - 1 = 9.
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Official course sources

Original independent self-study only. This is not a university service, an official exam, an assignment-solving service or a prediction of your course grade. Follow your instructor’s rules. No live language model is used.