Appendix
A: Calculus 12 Prescribed Learning Outcomes
The organizers for Calculus
12 are as follows:
Problem
Solving
Overview and History of Calculus (Overview of
Calculus)
Overview and History of Calculus (Historical
Development of Calculus)
Functions, Graphs, and Limits (Functions and
their Graphs)
Functions, Graphs, and Limits (Limits)
The Derivative (Concept and Interpretations)
The Derivative (Computing Derivatives)
Applications of Derivatives (Applied Problems)
Applications of Derivatives (Derivatives and the
Graph of the Function)
Antidifferentiation (Recovering Functions from
their Derivatives)
Antidifferentiation (Applications of Antidifferentiation)
Problem
Solving
It is expected that students
will use a variety of methods to solve real-life, practical, technical, and
theoretical problems.
It is expected that students
will:
- solve problems that
involve a specific content area (e.g., geometry, algebra, trigonometry statistics,
probability)
- solve problems that
involve more than one content area
- solve problems that
involve mathematics within other disciplines
- analyse problems and
identify the significant elements
- develop specific skills
in selecting and using an appropriate problem-solving strategy or combination
of strategies chosen from, but not restricted to, the following:
- guess and check
- look for a pattern
- make a systematic
list
- make and use a drawing
or model
- eliminate possibilities
- work backward
- simplify the original
problem
- develop alternative
original approaches
- analyse keywords
- demonstrate the ability
to work individually and co-operatively to solve problems
- determine that their
solutions are correct and reasonable
- clearly communicate
a solution to a problem and justify the process used to solve it
- use appropriate technology
to assist in problem solving
Overview
and History of Calculus (Overview of Calculus)
It is expected that students
will understand that calculus was developed to help model dynamic situations.
It is expected that students
will:
- distinguish between
static situations and dynamic situations.
- identify the two classical
problems that were solved by the discovery of calculus:
- the tangent problem
- the area problem
- describe the two main
branches of calculus:
- differential calculus
- integral calculus
- understand the limit
process and that calculus centers around this concept
It is expected that students
will understand the historical background and problems that led to the development
of calculus.
It is expected that students
will:
- describe the contributions made by various mathematicians and philosophers
to the development of calculus, including:
- Archimedes
- Fermat
- Descartes
- Barrow
- Newton
- Leibniz
- Jakob and Johann Bernoulli
- Euler
- L’Hopital
Functions,
Graphs, and Limits (Functions and their Graphs)
It is expected that students
will represent and analyze rational, inverse trigonometric, base e exponential,
natural logarithmic, elementary implicit, and composite functions, using technology
as appropriate.
It is expected that students
will:
- model and apply inverse
trigonometric, base e exponential, natural logarithmic, elementary
implicit and composite functions to solve problems
- draw (using technology),
sketch and analyze the graphs of rational, inverse trigonometric, base e
exponential, natural logarithmic, elementary implicit, and composite functions
for:
- domain and range
- intercepts
- recognize the relationship
between a base a exponential function (a > 0) and the equivalent base e
exponential function
- determine, using the
appropriate method (analytic or graphing utility) the points where
Functions,
Graphs, and Limits (Limits)
It is expected that students
will understand the concept of a limit of a function, notation used, and be
able to evaluate the limit of a function.
It is expected that students
will:
- demonstrate an understanding
of the concept of limit and notation used in expressing the limit of a function
as x approaches
- evaluate the limit of
a function
- analytically
- graphically
- numerically
- distinguish between
the limit of a function as x approaches a and the value of the function
at x = a
- demonstrate an understanding
of the concept of one-sided limits and evaluate one-sided limits
- determine limits that
result in infinity (infinite limits)
- evaluate limits of functions
as x approaches infinity (limits at infinity)
- determine vertical and
horizontal asymptotes of a function, using limits
- determine whether a
function is continuous at x = a
The
Derivative (Concept and Interpretations)
It is expected that students
will understand the concept of a derivative and evaluate derivatives of a function
using the definition of derivative.
It is expected that students
will:
- describe geometrically
a secant line and a tangent line for the graph of a function at x = a.
- define and evaluate the
derivative at x = a as:
and 
- define and calculate
the derivative of a function using:
- use alternate notation
interchangeably to express derivatives
- compute derivatives
using the definition of derivative
- distinguish between
continuity and differentiability of a function at a point
- determine when a function
is non-differentiable, and explain why
- determine the slope
of a tangent line to a curve at a given point
- determine the equation
of the tangent line to a curve at a given point
- for a displacement function
,
calculate the average velocity over a given time interval and the instantaneous
velocity at a given time
- distinguish between
average and instantaneous rate of change
The
Derivative (Computing Derivatives)
It is expected that students
will determine derivatives of functions using a variety of techniques.
It is expected that students
will:
- compute and recall the
derivatives of elementary functions including:
- use the following derivative
formulas to compute derivatives for the corresponding types of functions:constant
times a function:
- constant times a
function:
- use the Chain Rule to
compute the derifative of a composite function:
- compute the derivative
of an implicit function.
- use the technique of
logarithmic differentiation.
- compute higher order
derivatives
Applications
of Derivatives (Applied Problems)
It is expected that students
will solve applied problems from a variety of fields including the Physical
and Biological Sciences, Economics, and Business.
It is expected that students
will:
- solve problems involving
displacement, velocity, acceleration
- solve related rates
problems
- solve optimization problems
(applied maximum/minimum problems)
Applications
of Derivatives (Derivatives and the Graph of the Function)
It is expected that students
will use the first and second derivatives to describe the characteristic of
the graph of a function.
It is exptected that
students will:
- given the graph of
:
-
- relate the sign
of the derivative on an interval to whether the function is increasing
or decreasing over that interval.
- relate the sign
of the second derivative to the concavity of a function.
- determine the critical
numbers and inflection points of a function.
- determine the maximum
and minimum values of a function and use the first and/or second derivative
test(s) to justify their solutions
- use Newton’s iterative
formula (with technology) to find the solution of given equations,
- use the tangent line
approximation to estimate values of a function near a point and analyze the
approximation using the second derivative.
Antidfferentiation
(Recovering Functions from their Derivatives)
It is expected that students
will recognize antidfferentiaion (indefinite integral) as the reverse of the
differentiation process.
It is expected that students
will:
-
- compute the antiderivatives
of linear combinations of functions whose individual antiderivatives are known
including:
-
- create integration formulas
from the known differentiation formulas
- solve initial value problems
using the concept that if
on an interval, then 
differ by a constant on that interval.
Antidifferentiation
(Applications of Antidifferentiation)
It is expected that students
will use antidifferentiation to solve a variety of problems.
It is expected that students
will:
- use antidifferentiation
to solve problems about motion of a particle along a line that involve:
- computing the displacement
given initial position and velocity as a function of time
- computing velocity
and/or displacement given suitable initial conditions and acceleration
as a function of time
- use antidifferentiation
to find the area under the curve
,
above the x-axis, from x = a to x = b.
- use differentiation
to determine whether a given function or family of functions is a solution
of a given differential equation
- use correct notation
and form when writing the general and particular solution for differential
equations.
- model and solve exponential
growth and decay problems using a differential equation of the form:
- model and solve problems
involving Newton’s Law of Cooling using a differential equation of the form:
©
2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: November 28, 2000
Ministry of Education Home Page
Next