Manitoba · Grade 12
Introduction to Calculus, Grade 12 (40S)
- 45S
Manitoba’s Grade 12 math course, from Limits to Integration (Integrals).
- Units
- 4
- Lessons
- 12
- Skills
- 19
- Provincial exam
- None
- Official curriculum
- Manitoba Education
Unit by unit
Every skill in the course, with its official code.
Unit 1 · 5 skills
Limits
The Concept of a Limit
Understanding limits graphically
I can estimate the limit a function approaches near a point.
Official code: 1C.1.1
One-sided limits and existence
I can find one-sided limits and decide whether a limit exists.
Official code: 1C.1.1
Evaluating Limits
Evaluating limits algebraically
I can evaluate a limit algebraically.
Official code: 1C.1.2
Limits at infinity
I can evaluate a limit at infinity and find a horizontal asymptote.
Official code: 1C.1.2
Limits and Continuity
Continuity of a function
I can use limits to decide whether a function is continuous.
Official code: 1C.1.3
The official wording · 5 outcomes
- 1C.1.1
- Demonstrate an understanding of the concept of the limit.
- 1C.1.1
- Demonstrate an understanding of the concept of the limit.
- 1C.1.2
- Evaluate limits to analyze functions.
- 1C.1.2
- Evaluate limits to analyze functions.
- 1C.1.3
- Apply the concept of limit to the continuity of a function.
Key ideas
A limit is the value a function approaches as x gets close to a point.
Direct substitution evaluates a limit when the function is defined there.
A function is continuous at a point when the limit exists and equals the function's value.
The limit can exist even where the function itself is undefined.
An indeterminate form like 0 over 0 is resolved by factoring or rationalizing.
Watch out for
- Confusing the limit with the function's value at the point.
- Stopping at 0 over 0 instead of simplifying.
- Checking only the function value and ignoring the limit.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 2 · 6 skills
Derivatives
The Derivative as a Slope
From secant slope to tangent slope
I can explain the tangent slope as the limit of secant slopes.
Official code: 1C.2.1
The limit definition of the derivative
I can find a derivative from the limit definition.
Official code: 1C.2.1
Differentiation Rules
Power, constant, and sum rules
I can differentiate polynomials using the basic rules.
Official code: 1C.2.2
Product and quotient rules
I can differentiate products and quotients of functions.
Official code: 1C.2.2
The chain rule
I can differentiate a composite function using the chain rule.
Official code: 1C.2.2
Implicit Differentiation
Implicit differentiation
I can find dy/dx by implicit differentiation.
Official code: 1C.2.3
The official wording · 6 outcomes
- 1C.2.1
- Develop the definition of the derivative as the slope of a curve at a point.
- 1C.2.1
- Develop the definition of the derivative as the slope of a curve at a point.
- 1C.2.2
- Develop and apply differentiation rules.
- 1C.2.2
- Develop and apply differentiation rules.
- 1C.2.2
- Develop and apply differentiation rules.
- 1C.2.3
- Demonstrate an understanding of implicit differentiation.
Key ideas
A secant line's slope is the average rate of change between two points.
The power rule: the derivative of x to the n is n times x to the (n - 1).
Implicit differentiation differentiates both sides of a relation with respect to x.
As the second point approaches the first, the secant slope approaches the tangent slope.
Derivatives distribute over sums and pull out constant factors.
Watch out for
- Treating the tangent slope as an average over an interval.
- Forgetting to reduce the exponent by one in the power rule.
- Forgetting the dy/dx factor when differentiating a y term.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 3 · 5 skills
Applications of Derivatives
Motion of Particles
Velocity and acceleration
I can find velocity and acceleration from a position function.
Official code: 1C.3.1
Curve Sketching
Increasing, decreasing, and extrema
I can use the first derivative to find where a function rises, falls, and turns.
Official code: 1C.3.2
Concavity and sketching the curve
I can use the second derivative to find concavity and sketch the curve.
Official code: 1C.3.2
Optimization and Related Rates
Optimization
I can solve an optimization problem using derivatives.
Official code: 1C.3.3
Related rates
I can solve a related-rates problem using derivatives.
Official code: 1C.3.3
The official wording · 5 outcomes
- 1C.3.1
- Apply derivatives to solve problems involving the motion of particles.
- 1C.3.2
- Determine features of a function using derivatives to sketch the function
- 1C.3.2
- Determine features of a function using derivatives to sketch the function
- 1C.3.3
- Apply derivatives to solve optimization and related rates problems.
- 1C.3.3
- Apply derivatives to solve optimization and related rates problems.
Key ideas
Velocity is the derivative of position with respect to time.
A function increases where f'(x) is positive and decreases where it is negative.
Optimization models a quantity, then finds its maximum or minimum with the derivative.
Acceleration is the derivative of velocity, the second derivative of position.
A local extremum occurs where the first derivative changes sign.
Watch out for
- Confusing velocity with speed's sign.
- Assuming every critical point is a maximum or minimum.
- Forgetting to check endpoints for the optimum.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 4 · 3 skills
Integration (Integrals)
Antidifferentiation
Antiderivatives and integration
I can find an antiderivative and write an indefinite integral.
Official code: 1C.4.1
The Definite Integral
Understanding the definite integral
I can evaluate a definite integral and interpret it as an area.
Official code: 1C.4.2
Applying Integration
Solving problems with integration
I can apply integration to solve a problem.
Official code: 1C.4.3
The official wording · 3 outcomes
- 1C.4.1
- Demonstrate an understanding of the relationship between anti- differentiation and integration of functions.
- 1C.4.2
- Demonstrate and apply an understanding of the definite integral.
- 1C.4.3
- Apply integration to solve problems.
Key ideas
Antidifferentiation reverses differentiation to recover a function.
A definite integral gives the accumulated area under a curve between two bounds.
Integration recovers a total from a rate of change.
An indefinite integral includes a constant of integration, C.
The Fundamental Theorem evaluates it as the antiderivative at the bounds.
Watch out for
- Omitting the constant of integration.
- Forgetting to subtract the antiderivative at the lower bound.
- Integrating the wrong difference for the area between curves.
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?
Manitoba has 4 math courses for Grade 12. Here they are side by side.
- Applied Mathematics, Grade 12 (40S)6 units · 27 skillsGrade 12 Applied Mathematics Achievement TestWhat it coversFinancial Mathematics · Logical Reasoning · Probability · Relations and Functions · Mathematics Research Project · Design and Measurement
- Essential Mathematics, Grade 12 (40S)9 units · 24 skillsGrade 12 Essential Mathematics Achievement TestWhat it coversAnalysis of Games and Numbers · Vehicle Finance · Statistics · Precision Measurement · Career Life · Home Finance · Geometry and Trigonometry · Business Finance · Probability
- Introduction to Calculus, Grade 12 (40S)This course45S4 units · 19 skillsNo provincial examWhat it coversLimits · Derivatives · Applications of Derivatives · Integration (Integrals)
- Pre-Calculus Mathematics, Grade 12 (40S)3 units · 34 skillsGrade 12 Pre-Calculus Mathematics Achievement TestWhat it coversTrigonometry · Relations and Functions · Permutations, Combinations and Binomial Theorem
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 Manitoba’s own curriculum for Introduction to Calculus, Grade 12 (40S).
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Questions about Introduction to Calculus, Grade 12 (40S)
What is Introduction to Calculus, Grade 12 (40S)?
Manitoba’s Grade 12 math course: 4 units and 19 skills, from Limits to Integration (Integrals).
What units are in Introduction to Calculus, Grade 12 (40S)?
1. Limits, 2. Derivatives, 3. Applications of Derivatives and 4. Integration (Integrals).
Is there a provincial exam for Introduction to Calculus, Grade 12 (40S)?
No. Manitoba gives no provincial exam for Introduction to Calculus, Grade 12 (40S).
Can MapleMind help with Introduction to Calculus, Grade 12 (40S)?
Yes. MapleMind teaches every skill of Introduction to Calculus, Grade 12 (40S), 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.
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