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The official Nova Scotia Calculus 12 curriculum
Nova Scotia defines Calculus 12 by strands and outcomes. MapleMind teaches the same curriculum reorganized for one-skill-at-a-time tutoring — the table shows exactly where every official strand lands, and the ministry's own wording is quoted under each unit below.
Official source Nova Scotia's official curriculumRead it on the government site — curriculum.novascotia.ca ↗| Official strand | Outcomes | Where MapleMind teaches it |
|---|---|---|
| Strand A | 4 | Calculus 12 — Rates of Change and the Derivative |
| Strand B | 19 | Calculus 12 — Differentiation Rules and Applications |
| Strand C | 10 | Calculus 12 — Curve Analysis, Tangents and Antiderivatives |
| Strand D | 6 | Calculus 12 — Integration and Area |
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How MapleMind teaches Calculus 12 — every unit, lesson, and skill
Every skill below runs as a short session: a plain-words lesson, a worked example, solving it together, then a five-question skill check that earns up to three stars. Guided Mode keeps it teaching instead of answer-handing — turning it off needs a parent's password.
Unit 1Calculus 12 — Rates of Change and the DerivativeOfficial strand · Strand A
Average and instantaneous rates of change, the secant-to-tangent limit, the formal definition of the derivative, and implicit differentiation.
Rates of change: secant and tangent slopes
- Average vs. instantaneous rate of changeA1 — Compute an average rate of change as the change in a quantity divided by the change in the input over an interval, and describe instantaneous rate of change as the value that average rate approaches as the interval shrinks.
- From secant slopes to the tangent slopeA1 — Interpret the slope of a secant line as an average rate of change and the slope of the tangent line as the limit of secant slopes as the two points merge, connecting the geometry to instantaneous rate.
The definition of the derivative
- The derivative as a limitA2 — State the definition of the derivative f'(x) = lim as h→0 of (f(x+h) − f(x)) / h, and use it to find the derivative of simple polynomial functions from first principles.
- Where a function is not differentiableB5 — Identify points where a function has no derivative — corners, cusps, discontinuities, and vertical tangents — and explain why the limit defining the derivative fails at each.
Implicit differentiation
- Implicit differentiationA3 — Differentiate both sides of an equation with respect to x, treating y as a function of x and applying the chain rule to y-terms, then solve for dy/dx; identify relations (like circles or implicitly defined curves) that require this method.
The official wording — 4 outcomes in this unit
- A1
Apply, understand, and explain average and instantaneous rates of change
- A2
Demonstrate an understanding of the definition of the derivative
- B5
Find where a function is not differentiable and distinguish between corners, cusps, discontinuities, and vertical tangents
- A3
Demonstrate an understanding of implicit differentiation and identify situations that require implicit differentiation
Unit 2Calculus 12 — Differentiation Rules and ApplicationsOfficial strand · Strand B
Limits and continuity of combined functions, the differentiation rules, derivatives of trigonometric, exponential and logarithmic functions, and applications: rates of change, related rates, differentials, optimization, and definite integrals.
Limits and continuity of combined functions
- Calculating limits and their propertiesB2 — Evaluate limits of function values using the limit laws (sum, difference, product, quotient, and power), including limits found by algebraic simplification, with and without technology.
- Removing removable discontinuitiesB3 — Identify a removable discontinuity (a hole) in a rational function and remove it by simplifying the function or by redefining its value at that point so the function becomes continuous.
- Continuity of combinations and compositesB4 — Apply the fact that sums, differences, products, quotients (where defined), and composites of continuous functions are continuous, to determine where a combined function is continuous.
The differentiation rules
- Power, sum, and difference rulesB6 — Derive and apply the power rule d/dx[xⁿ] = n·x^(n−1) together with the sum and difference rules to differentiate polynomial functions.
- Product and quotient rulesB6 — Derive and apply the product rule d/dx[uv] = u'v + uv' and the quotient rule d/dx[u/v] = (u'v − uv')/v² to differentiate products and quotients of functions.
- The chain ruleB7 — Apply the chain rule d/dx[f(g(x))] = f'(g(x))·g'(x) to differentiate composite functions, identifying the outer and inner functions.
- Derivatives of trigonometric functionsB9 — Apply the differentiation rules for the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent), combining them with the chain, product, and quotient rules.
- Derivatives of exponential and logarithmic functionsB11 — Calculate and apply derivatives of exponential functions (eˣ and aˣ) and logarithmic functions (ln x and log_a x), combined with the chain rule.
Applications of the derivative
- Calculating and interpreting rate of changeB1 — Calculate average and instantaneous rate of change from a function and interpret each in context, using the derivative for the instantaneous rate.
- Using derivatives to solve rate-of-change problemsB8 — Use derivatives to analyze and solve applied problems involving rates of change, such as velocity and acceleration from a position function.
- Estimating change with differentialsB13 — Estimate the change in a function using the differential dy = f'(x)·dx, and apply this linear approximation to real-world estimation problems.
- Critical points and absolute extremaB15 — Find critical points where the derivative is zero or undefined and determine the absolute maximum and minimum values of a function on a closed interval.
- Increasing and decreasing intervalsB16 — Find the intervals on which a function is increasing or decreasing by analyzing the sign of the first derivative.
- Optimization: maximum and minimum valuesB17 — Solve applied problems that require finding a maximum or minimum value of a quantity by modelling it with a function and using derivatives.
Definite integrals and net change
- Rules for definite integralsB18 — Apply the rules for definite integrals, including linearity, the reversal of limits, and splitting an interval, to evaluate and manipulate definite integrals.
- The Fundamental Theorem of CalculusB19 — Apply the Fundamental Theorem of Calculus to evaluate a definite integral by finding an antiderivative and computing F(b) − F(a).
- Integration by substitutionB20 — Compute indefinite and definite integrals by the method of substitution, choosing a substitution that simplifies the integrand and adjusting the limits for definite integrals.
- Integration by partsB21 — Apply integration by parts, ∫u dv = uv − ∫v du, to evaluate indefinite and definite integrals of products such as x·eˣ (this outcome is flagged optional in the ministry text).
- Net change from an integrated rateB22 — Solve problems in which a rate is integrated over time to find the net change of a quantity, such as distance from a velocity or accumulated water from a flow rate.
The official wording — 19 outcomes in this unit
- B2
Calculate limits for function values and apply the properties with and without technology
- B3
Remove removable discontinuities by extending or modifying a function
- B4
Apply the properties of algebraic combinations and composites of continuous functions
- B6
Derive, apply, and explain power, sum, difference, product and quotient rules
- B7
Apply the chain rule to composite functions
- B9
Apply the rules for differentiating the six trigonometric functions
- B11
Calculate and apply derivatives of exponential and logarithmic functions
- B1
Calculate and interpret average and instantaneous rate of change
- B8
Use derivatives to analyze and solve problems involving rates of change
- B13
Estimate the change in a function using differentials and apply them to real world situations
- B14
Solve and interpret related rate problems
- B15
Demonstrate an understanding of critical points and absolute extreme values of a function
- B16
Find the intervals on which a function is increasing or decreasing
- B17
Solve application problems involving maximum or minimum values of a function
- B18
Apply rules for definite integrals
- B19
Apply the Fundamental Theorem of Calculus
- B20
Compute indefinite and definite integrals by the method of substitution
- B21
Apply integration by parts to evaluate indefinite and definite integrals
- B22
Solve problems in which a rate is integrated to find the net change over time
Unit 3Calculus 12 — Curve Analysis, Tangents and AntiderivativesOfficial strand · Strand C
Continuity of functions, the tangent-from-secant development, tangent and normal lines, the graphical connection between f and f', the derivative tests, concavity and inflection, initial value problems, and antiderivatives via the Fundamental Theorem.
Continuity of a function
- Intervals of continuityC1 — Identify the intervals on which a given function is continuous and explain the meaning of a continuous function in terms of limits.
Tangent and normal lines
- Developing the tangent from the secantC2 — Understand how the slope of a tangent line arises as the limit of the slope of a secant line, connecting the geometric picture to the derivative.
- Equations of tangent and normal linesC3 — Find the equations of the tangent line (slope f'(a)) and the normal line (slope −1/f'(a)) to a curve at a given point.
Graphical analysis and the derivative tests
- The connection between the graphs of f and f'C4 — Demonstrate an understanding of the connection between the graph of a function f and the graph of its derivative f', reading increasing/decreasing behaviour and slopes from one to the other.
- First and Second Derivative Tests for local extremaC5 — Apply the First Derivative Test (sign change of f') and the Second Derivative Test (sign of f'' at a critical point) to determine the local extreme values of a function.
- Concavity and points of inflectionC6 — Determine the concavity of a function and locate its points of inflection by analyzing the sign of the second derivative.
Antiderivatives and initial value problems
- Initial value problemsC7 — Solve initial value problems of the form dy/dx = f(x) with y(x₀) = y₀ by finding the antiderivative and using the initial condition to determine the constant.
- The derivative–definite integral relationshipC8 — Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
- Constructing antiderivatives with the FTCC9 — Construct antiderivatives of a function using the Fundamental Theorem of Calculus, expressing an antiderivative as an accumulation integral.
- Antiderivatives of polynomialsC10 — Find antiderivatives of polynomial functions using the reverse power rule, adding a constant of integration.
- Antiderivatives of exponential and trigonometric functionsC10 — Find antiderivatives of exponential functions of the form e^(kx) and of selected trigonometric functions of kx, accounting for the 1/k factor.
The official wording — 10 outcomes in this unit
- C1
Identify the intervals upon which a given function is continuous and understand the meaning of a continuous function
- C2
Understand the development of the slope of a tangent line from the slope of a secant line
- C3
Find the equations of the tangent and normal lines at a given point
- C4
Demonstrate an understanding of the connection between the graphs of f and f’
- C5
Apply the First and Second Derivative Tests to determine the local extreme values of a function
- C6
Determine the concavity of a function and locate the points of inflection by analyzing the second derivative
- C7
Solve initial value problems of the form dy/dx = f(x) , y0 = f(x0), where f(x) is a function that students recognize as a derivative
- C8
Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus
- C9
Construct antiderivatives using the Fundamental Theorem of Calculus
- C10
Find antiderivatives of polynomials
Unit 4Calculus 12 — Integration and AreaOfficial strand · Strand D
Riemann sums, the meaning of area under a curve, the definite integral as area, numerical integration, areas between curves, and volumes of revolution.
Riemann sums and the area under a curve
- Riemann sums for area under a curveD1 — Apply and understand how a Riemann sum of rectangle areas approximates the area under a polynomial curve, improving as the rectangles narrow.
- The meaning of area under the curveD2 — Demonstrate an understanding of the meaning of the area under a curve as the accumulation of a quantity and as the limit of Riemann sums.
The definite integral as area
- Expressing area as a definite integralD3 — Express the area under a curve as a definite integral, translating the limit of Riemann sums into integral notation with limits of integration.
- Computing area by numerical integrationD4 — Compute the area under a curve using numerical integration procedures such as the trapezoidal rule when an exact antiderivative is unavailable.
Areas of regions and volumes
- Areas of regions in a planeD5 — Apply integration to calculate the area of a region in a plane, including the area between two curves, by integrating the difference of the top and bottom functions.
- Volumes by slices or shellsD6 — Apply integration by slices (disks/washers) or shells to calculate the volume of a solid of revolution (this outcome is flagged optional in the ministry text).
The official wording — 6 outcomes in this unit
- D1
Apply and understand how Riemann’s sum can be used to determine the area under a polynomial curve
- D2
Demonstrate an understanding of the meaning of area under the curve
- D3
Express the area under the curve as a definite integral
- D4
Compute the area under the curve using numerical integration procedures
- D5
Apply integration to calculate areas of regions in a plane
- D6
Apply integration (by slices or shells) to calculate volumes
Printable workbook · A keepsake of the year
A Calculus 12 workbook worth keeping
Built from the same official curriculum as this page. Before and after each skill above, your student colours in how sure they feel — so the two of you can see, on one page, what's clicking and what needs another look. It's a quiet way to follow how the year is really going.
By June it's full of their own handwriting: units worked through, confidence grown, a mid-year check-in, notes from parent-teacher night, and a certificate at the end. Less a worksheet, more a record of the year worth keeping on the shelf.
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Can MapleMind help me with Calculus 12?
Yes. MapleMind's AI tutor covers all 42 skills in Nova Scotia's Calculus 12 — you pick the exact skill, and the tutor teaches it step by step: a short lesson, a worked example, solving together, then a skill check to show it stuck.
Is MapleMind aligned to Nova Scotia's official curriculum?
Yes. Every skill in this course maps to an official outcome code from Nova Scotia's Grade 12 Mathematics curriculum, and the ministry's own wording is quoted under each unit on this page — with the official government source linked so you can check it yourself.
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That's the point of skill-level tutoring: open Calculus 12 in the app, tap the exact skill from the list on this page, and the tutor teaches just that — no wading through lessons you don't need.
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Yes — 14 languages, including French. Both the app and the tutor's explanations switch to the language you choose.
Where can I see the official Nova Scotia curriculum for Calculus 12?
The official source is linked on this page — Nova Scotia's official curriculum. The outline here follows it: every MapleMind skill carries its official outcome code, and the ministry's own wording is quoted under each unit.
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