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Grade 12 · Mathematics · Advanced

Calculus 12

Limits, differentiation, applications and an introduction to integration.

Contents

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1

Limits & Continuity

5 of 5 published

  1. 1.1
    Limits

    Describe what a function heads toward — the idea every derivative and integral rests on.

    Published

  2. 1.2
    One-sided limits

    Compare the left and right approach to decide whether a limit exists.

    Published

  3. 1.3
    Continuity

    Pin down when a graph has no breaks, so the big theorems apply.

    Published

  4. 1.4
    Limit laws & indeterminate forms

    Evaluating limits with the limit laws and resolving the 0/0 form by factoring, rationalising or combining fractions.

    Published

  5. 1.5
    Limits at infinity & horizontal asymptotes

    End behaviour of functions: evaluating limits as x → ±∞ and finding horizontal asymptotes of rational and root functions.

    Published

2

Functions and Graphs

0 of 2 published

  1. 2.1
    Piecewise functions

    Defining, evaluating and graphing a function given by different rules on different parts of its domain.

    In review

  2. 2.2
    Inverse trigonometric functions

    Defining arcsin, arccos and arctan, their restricted domains and ranges, and evaluating them.

    In review

3

Derivatives

3 of 5 published

  1. 3.1
    Definition

    Instantaneous rate of change.

    Published

  2. 3.2
    Power & product rules

    Differentiation rules.

    Published

  3. 3.3
    Chain rule

    Composite functions.

    Published

  4. 3.4
    Derivatives of transcendental functions

    Differentiating logarithmic, exponential and trigonometric functions.

    In review

  5. 3.5
    Implicit differentiation

    Differentiating a relation that is not solved for y.

    In review

4

Applications of Derivatives

3 of 5 published

  1. 4.1
    Related rates

    Connected rates of change.

    Published

  2. 4.2
    Optimization

    Max and min problems.

    Published

  3. 4.3
    Curve sketching

    Using derivatives.

    Published

  4. 4.4
    Differentiability & the Mean Value Theorem

    When a derivative exists, and what the Mean Value Theorem guarantees about average and instantaneous rates.

    In preparation

  5. 4.5
    Newton's method

    Approximating a root of a function using tangent lines.

    In preparation

5

Integration

3 of 4 published

  1. 5.1
    Antiderivatives

    Reversing differentiation.

    Published

  2. 5.2
    Definite integrals

    Area under a curve.

    Published

  3. 5.3
    Fundamental theorem

    Linking derivatives and integrals.

    Published

  4. 5.4
    The trapezoidal rule

    Estimating a definite integral by averaging left and right Riemann sums.

    In preparation

6

Methods of Integration

0 of 2 published

  1. 6.1
    Integration by substitution

    Reversing the chain rule to integrate a composite function.

    In review

  2. 6.2
    Integration by parts

    Reversing the product rule to integrate a product of functions.

    In review

7

Applications of Integration

0 of 4 published

  1. 7.1
    Area between curves

    Finding the area of a region bounded by two curves.

    In review

  2. 7.2
    Volumes of solids

    Finding the volume of a solid of revolution by slicing.

    In preparation

  3. 7.3
    Average value of a function

    Finding the mean height of a function over an interval using its integral.

    In preparation

  4. 7.4
    Differential equations & slope fields

    Solving a simple initial-value differential equation and visualising its solutions with a slope field.

    In preparation

What is Calculus 12 in British Columbia?

Short answer

Calculus 12 is the BC senior-math course that introduces single-variable calculus: limits and continuity, derivatives and the rules for computing them, applications of derivatives (rates, optimization, curve analysis), and an introduction to integration. It is excellent preparation for AP Calculus and first-year university calculus.

Course overview

Calculus 12 develops two big ideas — the derivative as an instantaneous rate of change, and the integral as accumulation. You build limits and continuity first, learn to differentiate efficiently, apply derivatives to real problems like optimization and motion, and finish with the basics of integration and the connection between the two.

What you will learn

  • Evaluate limits and reason about continuity
  • Compute derivatives using the standard rules
  • Apply derivatives to rates of change, optimization, and curve sketching
  • Understand integration as accumulation and its link to the derivative

Who this course is for

  • • BC students who want a strong first exposure to calculus
  • • Students preparing for AP Calculus or first-year university calculus
  • • Anyone who wants to understand derivatives conceptually, not just mechanically

Commonly difficult topics

  • • Limits — understanding what a limit describes rather than just plugging in
  • • The chain rule — identifying the inner and outer functions
  • • Optimization — translating a word problem into a function to maximize or minimize
  • • Connecting derivatives and integrals conceptually

How Dr. Shreyank Educare teaches this course

  1. 01 · Diagnose

    Find the exact skill causing difficulty — not the chapter, the skill.

  2. 02 · Understand

    Learn it through explanations, worked examples and visual reasoning.

  3. 03 · Practice

    Work problems that adapt to how you are actually doing.

  4. 04 · Fix

    A wrong answer becomes targeted remediation for the misconception behind it.

  5. 05 · Prove

    Solve new, unseen problems independently before mastery is awarded.

Course curriculum

1Limits & Continuity5 topics
  • — Limits
  • — One-sided limits
  • — Continuity
  • — Limit laws & indeterminate forms
  • — Limits at infinity & horizontal asymptotes
2Functions and Graphs2 topics
  • — Piecewise functions
  • — Inverse trigonometric functions
3Derivatives5 topics
  • — Definition
  • — Power & product rules
  • — Chain rule
  • — Derivatives of transcendental functions
  • — Implicit differentiation
4Applications of Derivatives5 topics
  • — Related rates
  • — Optimization
  • — Curve sketching
  • — Differentiability & the Mean Value Theorem
  • — Newton's method
5Integration4 topics
  • — Antiderivatives
  • — Definite integrals
  • — Fundamental theorem
  • — The trapezoidal rule
6Methods of Integration2 topics
  • — Integration by substitution
  • — Integration by parts
7Applications of Integration4 topics
  • — Area between curves
  • — Volumes of solids
  • — Average value of a function
  • — Differential equations & slope fields

A sample learning experience

Find the derivative of f(x) = 3x² and use it to give the instantaneous rate of change at x = 2.

  1. 1. Apply the power rule: the derivative of xⁿ is n·xⁿ⁻¹.
  2. 2. So f′(x) = 3 · 2x = 6x.
  3. 3. Evaluate at x = 2: f′(2) = 6(2) = 12.
  4. 4. Interpretation: at x = 2 the function is increasing at a rate of 12 units of output per unit of input.

Frequently asked questions

Is Calculus 12 the same as AP Calculus?
No. Calculus 12 is a strong BC introduction to calculus. AP Calculus (AB/BC) covers more, at greater depth and difficulty, and prepares specifically for the College Board AP exam.
Do I need Pre-Calculus 12 before Calculus 12?
Yes — Calculus 12 assumes fluency with functions, logarithms, and trigonometry from Pre-Calculus 12.
Is Calculus 12 good preparation for AP Calculus?
Very good. The limits, derivatives, and applications overlap substantially with AP Calculus AB, so it lightens the AP workload considerably.
How much integration is in Calculus 12?
Currently, the published lessons cover definite and indefinite integrals, Riemann sums, and the Fundamental Theorem of Calculus. The course is being expanded to add integration methods (substitution, integration by parts) and applications (area between curves, volumes of solids, average value, differential equations) once those lessons are authored. AP Calculus goes further still.
What is the best way to study Calculus 12?
Make sure each new rule is tied to its meaning (a rate, a slope, an area). Practise setting up problems — especially optimization — before rushing to differentiate.

Related courses

Pre-Calculus 12 →AP Calculus AB/BC →

Explore all Mathematics courses →

Curriculum & sources

Content source: original lessons, worked examples and diagrams created for Dr. Shreyank Educare.

Curriculum alignment: Grade 12 Mathematics (BC curriculum).

Curriculum-aligned learning support — not an official school, ministry, or College Board course, and not a replacement for official credit.

Last updated . Authored and maintained in-house by the Dr. Shreyank Educare teaching team — see our editorial standards, methodology and source policy.