University Calculus I

By Dr. M Categories: University Preparation
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About Course

University Calculus I is a complete first-year, single-variable calculus course for students who have finished MCV4U (or AP/IB calculus) and want to be ready for university on day one. It covers the Calculus I syllabus used across Ontario – UofT (MAT135/MAT137), Waterloo (MATH 137), McMaster (MATH 1A03), Western (CALC 1000), York (MATH 1300) – at the depth and speed universities actually use.

This is not a repeat of Grade 12 calculus. MCV4U introduces derivatives; this course finishes the job: limits done properly, the full differentiation toolkit, optimization and related rates at university level, integration from Riemann sums through the Fundamental Theorem, techniques of integration, and an introduction to differential equations and series.

What you will learn

  • Limits and continuity, including the precise (ε–δ) idea, one-sided limits, limits at infinity and the Squeeze Theorem
  • Derivatives of every standard function: polynomial, trigonometric, exponential, logarithmic, inverse trigonometric and implicit functions
  • Applications: related rates, linear approximation, curve sketching, optimization, L’Hôpital’s rule, Mean Value Theorem
  • Integration: Riemann sums, definite and indefinite integrals, the Fundamental Theorem of Calculus, substitution, integration by parts, partial fractions, improper integrals
  • Applications of integration: area between curves, volumes of revolution, arc length, work
  • Introduction to differential equations and infinite series

How the course works

  • Every lesson follows the same structure: a “Before You Start” check, key definitions with original figures, six fully worked examples, “Your Turn” problems, a Common Errors list, and four levels of practice with complete solutions.
  • Solutions are worked in symbols first, one step per line, with the reason beside each step – the way university graders expect to see it.
  • Each lesson has an interactive graph or simulation so you can see what the mathematics is doing.
  • A Mastery Check ends each lesson. Score 80% or more to move on; score below that and you can book a tutoring session with Dr. M directly from the page.

Who it is for

  • Grade 12 students heading into engineering, science, mathematics, computer science, economics or business
  • First-year students who want a structured second explanation alongside their university course
  • Students re-taking Calculus I or preparing for a placement test

Prerequisite: MCV4U, MHF4U, or equivalent (AP Calculus AB, IB Math AA SL/HL).

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What Will You Learn?

  • Evaluate limits rigorously, including the precise definition, one-sided limits, limits at infinity and the Squeeze Theorem
  • Differentiate every standard function: polynomial, trigonometric, exponential, logarithmic, inverse trigonometric and implicit
  • Solve related-rates, optimization and curve-sketching problems at the level of a university midterm
  • Apply the Mean Value Theorem, linear approximation and L'Hôpital's rule correctly
  • Build the definite integral from Riemann sums and use both parts of the Fundamental Theorem of Calculus
  • Integrate by substitution, by parts, by partial fractions and by trigonometric substitution; handle improper integrals
  • Find areas between curves, volumes of revolution, arc length and work
  • Solve first-order separable differential equations and test the convergence of basic series
  • Present every solution in symbols first, one step per line, with the reason beside each step

Course Content

Chapter 1 — Limits and Continuity
The foundation of calculus. Limits done properly: intuitive and precise definitions, limit laws, one-sided and infinite limits, the Squeeze Theorem, continuity and the Intermediate Value Theorem.

  • 1.1 The Idea of a Limit
  • 1.2 Limit Laws and Direct Substitution
  • 1.3 One-Sided Limits and Limits That Do Not Exist
  • 1.4 Infinite Limits and Limits at Infinity
  • 1.5 The Squeeze Theorem and Trigonometric Limits
  • 1.6 Continuity and the Intermediate Value Theorem
  • 1.7 The Precise Definition of a Limit (ε–δ)

Chapter 2 — Derivatives
What a derivative is and how to compute one for every standard function: definition as a limit, power, product, quotient and chain rules, trigonometric, exponential and logarithmic functions, implicit differentiation and higher derivatives.

Chapter 3 — Applications of Derivatives
Using the derivative to solve real problems: related rates, linear approximation, the Mean Value Theorem, curve sketching, optimization and L’Hôpital’s rule.

Chapter 4 — Integration
From area under a curve to the Fundamental Theorem of Calculus: Riemann sums, the definite integral, antiderivatives, the Fundamental Theorem and the substitution rule.

Chapter 5 — Techniques and Applications of Integration
The integration toolkit universities expect: integration by parts, trigonometric integrals and substitution, partial fractions, improper integrals, and applications to area, volume, arc length and work.

Chapter 6 — Differential Equations and Series
A first look at the two topics that open Calculus II: separable differential equations with growth and decay models, sequences, series and the basic convergence tests, and Taylor polynomials.

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