Nova Scotia · Grade 12
Calculus 12
Nova Scotia’s Grade 12 math course, from Calculus 12 — Rates of Change and the Derivative to Calculus 12 — Integration and Area.
- Units
- 4
- Lessons
- 14
- Skills
- 42
- Provincial exam
- None
- Official curriculum
- Nova Scotia Department of Education and Early Childhood Development
Unit by unit
Every skill in the course, with its official code.
Unit 1 · 5 skills
Calculus 12 — Rates of Change and the Derivative
Rates of change: secant and tangent slopes
Average vs. instantaneous rate of change
I can compute an average rate of change and explain what instantaneous rate of change means.
Official code: A1
From secant slopes to the tangent slope
I can explain the tangent slope as the limit of secant slopes at a point.
Official code: A1
The definition of the derivative
The derivative as a limit
I can find a derivative from the limit definition (first principles).
Official code: A2
Where a function is not differentiable
I can find points of non-differentiability and name the reason (corner, cusp, discontinuity, vertical tangent).
Official code: B5
Implicit differentiation
Implicit differentiation
I can find dy/dx by implicit differentiation and recognize when it is needed.
Official code: A3
The official wording · 5 outcomes
- A1
- Apply, understand, and explain average and instantaneous rates of change
- A1
- extend these concepts to secant line and tangent line slopes
- 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
Key ideas
Average rate of change of f over [a, b] is (f(b) − f(a)) / (b − a).
The derivative is f'(x) = lim (h→0) [f(x+h) − f(x)] / h — the limiting secant slope.
Differentiate each side with respect to x, treating y as a function of x.
Instantaneous rate of change is the value the average rate approaches as the interval shrinks toward a single point.
f'(x) is itself a function giving the tangent slope at every x.
Watch out for
- Thinking instantaneous rate is just the function value f(a) rather than how fast f is changing at a.
- Letting h = 0 before simplifying the difference quotient, producing 0/0.
- Forgetting the dy/dx factor when differentiating a term containing y.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 2 · 20 skills
Calculus 12 — Differentiation Rules and Applications
Limits and continuity of combined functions
Calculating limits and their properties
I can calculate a limit using the limit properties, with or without technology.
Official code: B2
Removing removable discontinuities
I can remove a removable discontinuity by extending or redefining a function.
Official code: B3
Continuity of combinations and composites
I can determine continuity of a function built from continuous pieces.
Official code: B4
The differentiation rules
Power, sum, and difference rules
I can differentiate polynomials using the power, sum, and difference rules.
Official code: B6
Product and quotient rules
I can differentiate products and quotients using the product and quotient rules.
Official code: B6
The chain rule
I can differentiate a composite function using the chain rule.
Official code: B7
Derivatives of trigonometric functions
I can differentiate the six trigonometric functions, including with the chain rule.
Official code: B9
Derivatives of exponential and logarithmic functions
I can differentiate exponential and logarithmic functions.
Official code: B11
Applications of the derivative
Calculating and interpreting rate of change
I can calculate and interpret average and instantaneous rates of change in context.
Official code: B1
Using derivatives to solve rate-of-change problems
I can use derivatives to solve applied rate-of-change problems.
Official code: B8
Estimating change with differentials
I can estimate a small change in a function using differentials.
Official code: B13
Related rates
I can solve a related-rates problem by differentiating with respect to time.
Official code: B14
Critical points and absolute extrema
I can find critical points and the absolute extreme values of a function.
Official code: B15
Increasing and decreasing intervals
I can find where a function is increasing or decreasing from the sign of f'(x).
Official code: B16
Optimization: maximum and minimum values
I can solve an optimization problem by maximizing or minimizing a modelled function.
Official code: B17
Definite integrals and net change
Rules for definite integrals
I can apply the rules for definite integrals.
Official code: B18
The Fundamental Theorem of Calculus
I can evaluate a definite integral using the Fundamental Theorem of Calculus.
Official code: B19
Integration by substitution
I can integrate by substitution for indefinite and definite integrals.
Official code: B20
Integration by parts
I can evaluate an integral of a product using integration by parts.
Official code: B21
Net change from an integrated rate
I can integrate a rate over time to find a net change.
Official code: B22
The official wording · 20 outcomes
- 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
- 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
Key ideas
lim notation: lim (x→a) f(x) is the value f(x) approaches as x approaches a.
Power rule: d/dx[xⁿ] = n·x^(n−1).
Average rate of change over an interval is a difference quotient; instantaneous rate is the derivative.
A definite integral is written with an integral sign and lower/upper limits.
The limit of a sum/product/quotient is the sum/product/quotient of the limits (when defined).
Watch out for
- Substituting to get 0/0 and concluding the limit does not exist without simplifying first.
- Forgetting to reduce the exponent by one after multiplying by it.
- Reporting the function value as the rate of change.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 3 · 11 skills
Calculus 12 — Curve Analysis, Tangents and Antiderivatives
Continuity of a function
Intervals of continuity
I can identify where a function is continuous and explain what continuity means.
Official code: C1
Tangent and normal lines
Developing the tangent from the secant
I can explain how the tangent slope develops from secant slopes as a limit.
Official code: C2
Equations of tangent and normal lines
I can write the equations of the tangent and normal lines at a point on a curve.
Official code: C3
Graphical analysis and the derivative tests
The connection between the graphs of f and f'
I can relate the graph of f to the graph of its derivative f'.
Official code: C4
First and Second Derivative Tests for local extrema
I can classify local extrema with the First and Second Derivative Tests.
Official code: C5
Concavity and points of inflection
I can find concavity and points of inflection from the second derivative.
Official code: C6
Antiderivatives and initial value problems
Initial value problems
I can solve an initial value problem by antidifferentiating and fixing the constant.
Official code: C7
The derivative–definite integral relationship
I can explain how differentiation and definite integration are inverse processes.
Official code: C8
Constructing antiderivatives with the FTC
I can construct an antiderivative using the Fundamental Theorem of Calculus.
Official code: C9
Antiderivatives of polynomials
I can find antiderivatives of polynomials with the reverse power rule.
Official code: C10
Antiderivatives of exponential and trigonometric functions
I can find antiderivatives of e^(kx) and trigonometric functions of kx.
Official code: C10
The official wording · 11 outcomes
- 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
- C10
- Find antiderivatives of polynomials, 𝑒𝑒𝑘𝑘𝑘𝑘, and selected trigonometric functions of kx
Key ideas
A function is continuous at a point if the limit there exists and equals the function value.
The secant slope over [x, x+h] approaches the tangent slope as h→0.
Where f is increasing, f' is positive; where f is decreasing, f' is negative.
Antidifferentiate dy/dx = f(x) to get y = F(x) + C.
Continuity on an interval means continuity at every point of it.
Watch out for
- Calling a function continuous at a hole where the value is undefined.
- Treating the tangent slope as an average slope rather than a limiting slope.
- Reading a maximum of f as a maximum of f' (it is a zero of f').
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 4 · 6 skills
Calculus 12 — Integration and Area
Riemann sums and the area under a curve
Riemann sums for area under a curve
I can approximate area under a curve with a Riemann sum.
Official code: D1
The meaning of area under the curve
I can explain what the area under a curve represents.
Official code: D2
The definite integral as area
Expressing area as a definite integral
I can write the area under a curve as a definite integral.
Official code: D3
Computing area by numerical integration
I can estimate a definite integral with a numerical integration procedure.
Official code: D4
Areas of regions and volumes
Areas of regions in a plane
I can find the area of a plane region, including between two curves, using integration.
Official code: D5
Volumes by slices or shells
I can find the volume of a solid of revolution using integration by slices or shells.
Official code: D6
The official wording · 6 outcomes
- 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
Key ideas
A Riemann sum adds the areas of rectangles under a curve on an interval.
The definite integral ∫ from a to b of f(x) dx is the limit of Riemann sums.
The area between two curves is ∫ (top function − bottom function) dx over the interval.
Narrower rectangles (more of them) give a better approximation.
The lower and upper limits a and b mark the interval's ends.
Watch out for
- Assuming a Riemann sum with a few rectangles gives the exact area.
- Swapping the limits of integration.
- Integrating the sum instead of the difference of the two functions.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Plan the term
Pick your term and the days you study. Each unit gets its share of the weeks and comes back a week later for a review. The last two weeks go over the whole course.
Which course?
Nova Scotia has 4 math courses for Grade 12. Here they are side by side.
- Calculus 12This course4 units · 42 skillsNo provincial examWhat it coversCalculus 12 — Rates of Change and the Derivative · Calculus 12 — Differentiation Rules and Applications · Calculus 12 — Curve Analysis, Tangents and Antiderivatives · Calculus 12 — Integration and Area
- Mathematics 124 units · 30 skillsNo provincial examWhat it coversFinancial Mathematics · Logical Reasoning · Probability · Relations and Functions
- Mathematics at Work 125 units · 22 skillsNo provincial examWhat it coversGeometry · Number · Algebra · Statistics · Probability
- Pre-Calculus 122 units · 26 skillsNo provincial examWhat it coversTrigonometry · Relations and Functions
What a university, college or program asks for is theirs to say: check its admission page, and your school’s course guide.
The official curriculum
This page follows Nova Scotia’s own curriculum for Calculus 12.
Study it in MapleMind
MapleMind teaches Calculus 12 one skill at a time. Each skill gets a short lesson, practice as you go and a check worth up to three stars. It remembers where you stopped.
Course Companion
Your year in one workbook
A fun way to keep Calculus 12 organized, get to know your child better and see early when they need a hand. Fill it in on a screen, or print it.
- Pages
- 32
- Fillable fields
- 1,601
- Price
- $4.99
Questions about Calculus 12
What is Calculus 12?
Nova Scotia’s Grade 12 math course: 4 units and 42 skills, from Calculus 12 — Rates of Change and the Derivative to Calculus 12 — Integration and Area.
What units are in Calculus 12?
1. Calculus 12 — Rates of Change and the Derivative, 2. Calculus 12 — Differentiation Rules and Applications, 3. Calculus 12 — Curve Analysis, Tangents and Antiderivatives and 4. Calculus 12 — Integration and Area.
Is there a provincial exam for Calculus 12?
No. Nova Scotia gives no provincial exam for Calculus 12.
Can MapleMind help with Calculus 12?
Yes. MapleMind teaches every skill of Calculus 12, step by step, with practice and checks, and its tutor answers questions about any of them. It’s free with daily limits, on iPhone, Android and the web.
Get ready for Calculus 12
Practise every skill of Calculus 12 in MapleMind, free with daily limits.
