High School Block:

Calculus 120

Calculus 120 applies and extends knowledge developed through Pre-Calculus. The course includes limits, related rates and rates of change, derivatives, antiderivatives, integrals, optimization, and linearization. Learners will apply mathematical notation and mathematical rules, approximation, summation, and determine differentiability of functions.

Topics include graphs of functions and related equations; polynomial and rational functions; inverse, radical, exponential and logarithmic functions; limits, derivatives, applications of derivatives including related rates, linearization, optimization, and Newton’s method; antiderivatives, Riemann sums, the Fundamental Theorem of Calculus, the substitution method of integration, and applications of integration.

Pre-Calculus 110, Pre-Calculus 120A, and Pre-Calculus 120B are recommended prior to enrolling in Calculus 120.

For Educators:

➔ PL Hub: Mathematics

CONTEXTS AND CONCEPTS

Strategies

  • Using algorithms, thinking strategies, and reason to explore mathematical ideas
  • Using technology to pair numerical and visual exploration of calculus with symbolic work
  • Exploring model representations of dynamic phenomena
  • Planning, constructing, and verifying organized mathematical arguments
Processes

  • Deriving mathematical rules and algorithms
  • Using calculators and simulations
  • Squeeze Theorem
  • Constant, sum, difference, product, quotient rules
  • Chain rule and power chain rule
  • Outside-inside function rule
  • L’Hôpital’s rule
  • Optimization
  • Linearization
  • Rectangular approximation method
  • Antidifferentiation
Fluency

  • Conversion between formats, representations, and equivalents
  • Average rate of change
  • Instantaneous rate of change
  • Tangent and normal lines
  • Extreme Value Theorem
  • Mean Value Theorem
  • First, Second Derivative Test
  • Differentiability of functions
  • Velocity, speed, displacement, acceleration
  • Inverse functions
  • Left-hand and right-hand derivatives
  • First, second, third, higher order derivatives
  • Definite and indefinite integrals
  • Local and absolute values
Communication

  • Expressing mathematical arguments clearly, both written and orally
  • Using calculus to describe processes of change in the real world
  • Using a variety of notations with appropriateness and accuracy
  • Representing mathematical ideas using symbolic, graphical, and numerical forms.
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GRADE 12

Strand: Limits and Approximation

Big Idea: Approximation

Skill Descriptor: Apply approximation to solve contextual problems.

Global Competencies: CTPS

Achievement Indicators:

  • Interpret average rates of change from contextual situations.
  • Approximate area using different geometric shapes.
  • Approximate areas using finite sums.
  • Explore left, right, and midpoint rectangular approximations.

Big Idea: Limits

Skill Descriptor: Determine limits and derivatives of functions.

Global Competencies: CM, CTPS

Achievement Indicators:

  • Explain that the instantaneous rate of change of a function at a point is the limiting value of a sequence of average rates of change.
  • Use the Squeeze Theorem to indirectly find limits.
  • Determine the derivatives of polynomial, sine, cosine, rational, radical, and exponential functions 𝑦 = 𝑎x and 𝑦 = 𝑒x using the limit definition.

Strand: Relations and Functions

Big Idea: Derivatives as Relationships

Skill Descriptor: Differentiate functions.

Global Competencies: CTPS

Achievement Indicators:

  • Determine the derivative of polynomial, rational, radical, trigonometric, composite, exponential, and logarithmic functions.
  • Apply a combination of rules to differentiate functions.
  • Determine second and higher-order derivatives of functions.

Skill Descriptor: Interpret derivatives as relationships between changes in variables.

Global Competencies: CTPS

Achievement Indicators:

  • Sketch a graph of the derivative given graphs of functions.
  • Determine the first and second derivatives of relations using implicit differentiation.

Skill Descriptor: Apply differentiation to solve contextual problems.

Global Competencies: CTPS

Achievement Indicators:

  • Develop mathematical models for related rates problems.
  • Solve contextual problems involving related rates.
  • Interpret the solution of related rates problems.
  • Apply derivative rules to solve contextual problems involving derivatives.
  • Solve problems involving the derivative of composite functions.

Big Idea: Behaviour of Functions

Skill Descriptor: Determine local linearity of functions.

Global Competencies: CM, CTPS

Achievement Indicators:

  • Determine the slope of curves at a given point.
  • Determine whether curves have a tangent line at given points.
  • Sketch the graph of the function from the given graph of the derivative of functions.
  • Sketch the graph of the derivative of a function given a graph of the function.
  • Determine the equation of a tangent lines on curves at given points.
  • Explain Newton’s Method numerical technique.
  • Explain how the Mean Value Theorem connects slopes of secant and tangent lines.
  • Approximate numerical expressions using linearization.
  • Solve contextual linearization problems.

Skill Descriptor: Determine rate of growth and long-term behaviour of functions.

Global Competencies: CTPS

Achievement Indicators:

  • Evaluate limits involving functions using L’Hôpital’s Rule.
  • Compare asymptotic behaviour of functions using limits.

Skill Descriptor: Determine local behavior of functions.

Global Competencies: CTPS

Achievement Indicators:

  • Determine local extrema of functions.
  • Determine the critical points of functions.
  • Determine the intervals on which functions are increasing and decreasing.
  • Apply the First Derivative Test to locate the local extrema of functions.
  • Classify the local extrema of functions using the First Derivative Test.
  • Apply the Concavity Test to determine the intervals of concavity of functions.
  • Determine the points of inflection of functions.
  • Sketch the graph of functions using information about its derivative.

Skill Descriptor: Determine global behavior of functions.

Global Competencies: CTPS

Achievement Indicators:

  • Apply the Extreme Value Theorem to determine if absolute extrema exist on given functions.
  • Determine the absolute extrema on given functions

Skill Descriptor: Apply information about behaviour of functions to solve contextual problems.

Global Competencies: CTPS

Achievement Indicators:

  • Interpret the solution(s) of optimization problems.
  • Determine the equation of objective functions to be optimized.
  • Determine the equations of necessary parameters in optimization problems.
  • Solve contextual optimization problems requiring calculus techniques.

Skill Descriptor: Determine the differentiability of functions.

Global Competencies: CM, CTPS

Achievement Indicators:

  • Define and evaluate the derivative at 𝑥 = 𝑎.
  • Determine whether functions are differentiable at given points.
  • Explain why functions are not differentiable at given points including distinguishing among end points (restricted domain), corners, cusps, discontinuities, and vertical tangents.
  • Determine all values for which functions are differentiable.

Big Idea: Algebraic and Graphical Reasoning

Skill Descriptor: Solve problems involving inverse trigonometric functions.

Global Competencies: CM, CTPS

Achievement Indicators:

  • Explain the relationship between the primary trigonometric functions and the corresponding inverse trigonometric functions.
  • Determine the exact value of expressions involving inverse trigonometric functions.
  • Simplify expressions involving inverse trigonometric functions.
  • Determine the domain of inverse trigonometric functions.
  • Sketch the graph of inverse trigonometric functions.
  • Determine the derivative of inverse trigonometric functions including composites.
  • Solve problems involving the derivative of inverse trigonometric functions.

Strand: Connecting Concepts

Big Idea: Fundamental Theorem of Calculus Part 1

Skill Descriptor: Apply the Fundamental Theorem of Calculus as a differentiation formula.

Global Competencies: CTPS

Achievement Indicators:

  • Demonstrate an understanding of the relationship between the derivative and definite integral as expressed in the First Fundamental Theorem of Calculus (Part 1).
  • Apply both the Fundamental Theorem of Calculus and Chain Rule (and other derivative rules) to find derivatives.

Big Idea: Anti-differentiation

Skill Descriptor: Determine the antiderivative of functions.

Global Competencies: CM, CTPS

Achievement Indicators:

  • Define anti-differentiation in terms of differentiation.
  • Determine the general antiderivative of functions.
  • Apply antiderivative notation.
  • Determine the antiderivative of functions given initial conditions.
  • Apply substitution to determine the indefinite integral of functions.

Big Idea: Fundamental Theorem of Calculus Part 2

Skill Descriptor: Apply the Fundamental Theorem of Calculus as an integration formula.

Global Competencies: CTPS

Achievement Indicators:

  • Evaluate definite integrals from antiderivatives using the Second Fundamental Theorem of Calculus (Part 2).
  • Determine the average rate of change of functions over intervals.
  • Connect areas and average rates of change.

Strand: Summation

Big Idea: Area

Skill Descriptor: Apply rectangles for determining area.

Global Competencies: CTPS

Achievement Indicators:

  • Evaluate definite integrals geometrically using area formulae.
  • Relate Riemann sums to definite integrals.
  • Evaluate definite integrals using limits of Riemann sums.
  • Determine the area under functions and above the x-axis from 𝑥 = 𝑎 to 𝑥 = 𝑏 using definite integrals.
  • Determine the area between two functions.

Big Idea: Continuous Summation

Skill Descriptor: Apply continuous summation to solve contextual problems.

Global Competencies: CTPS

Achievement Indicators:

  • Model contextual problems involving continuous summation as integral functions.
  • Use integration to solve problems involving motion of particles along a line including displacement given the initial position and velocity as a function of time; velocity and/or displacement given suitable initial conditions and acceleration as a function of time.
  • Apply integration to solve contextual problems.
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