STANDARDS
OBJECTIVES
Specific Performance Criteria
Intermediate Learning Objectives
1. Functions
1.1 Recognition of functions
1.1.1 Decide whether a given relation is a function from its graphical representation.
1.1.2 Recognize and name the following functions from their symbolic representations:
constant function
linear function
quadratic function
absolute value function
exponential function
logarithmic function
sine function
cosine function
tangent function
cosecant function
secant function
cotangent function
square root function
1.1.3 Recognize and name the following function from its symbolic representation:
nth root function
1.1.4 Recognize and name the functions listed in 1.1.2, from their graphical representations.
1.2 Finding domain, range and intercepts
1.2.1 Find and state the domain of functions listed in 1.1.2 from both their graphical and their symbolic representations
1.2.2 Find and state the range of functions listed in 1.1.2 and 1.1.3 from both their graphical and their symbolic representations.
1.2.3 Find and state the x and y intercepts, if they exist, of functions listed in 1.1.2 from both their graphical and their symbolic representations.
1.3 Graphing of functions
1.3.1 Graph the functions listed in 1.1.2.
1.3.2 Graph piecewise defined functions whose pieces are made up of the functions listed in 1.1.2.
1.3.3 Apply vertical and horizontal shifts and reflections about the horizontal and vertical axes, and any combination of these to the functions listed in 1.1.2.
1.4 Operations on functions
1.4.1 Perform addition, subtraction, multiplication, division and composition of functions.
1.4.2 Divide two polynomial functions and express the answer in the form 
.
1.4.3 Find the value of a function at a point in its domain.
1.4.4 Evaluate 
(the difference quotient) for linear, quadratic and simple rational functions.
1.5 Appropriate use of functions to represent given situations
1.5.1 Given an applied problem, decide which function best represents the situation and express the relationship using appropriate notation.
2. Limits, Continuity and Derivatives
2.1 Determination of Limits
2.1.1 Give an intuitive description of the limit of a function at a point.
2.1.2 Evaluate a limit of a function by viewing the graph of the function.
2.1.3 Estimate a limit numerically by using successive approximations (using a table of values).
2.1.4 Evaluate a limit analytically by direct substitution, factoring, rationalizing or simplifying rational expressions.
2.1.5 Evaluate analytically limits at infinity.
Evaluate one-sided limits.
2.1.7 Recognize and evaluate infinite limits.
2.2 Determination of whether a function is continuous at a point or on an interval
2.2.1 Define continuity of a function at a point; that is, state the three conditions which must be satisfied in order that a function be continuous at a point.
2.2.2 Use the definition of continuity to determine if a function is continuous at a specific point.
2.2.3 Determine on which interval(s) a function is continuous.
2.3 Use of the limit definition of the derivative
2.3.1 Define the derivative of a function as
i) the limit of a difference quotient,
ii) the slope of the tangent line, and
iii) the rate of change (in particular the velocity function associated with a position function).
2.3.2 Use the limit definition of the derivative to determine the derivative of polynomials of degree 1,2 or 3, square root and simple rational functions. 2.3.3 Use the limit definition of the derivative to determine the numerical value of the derivative at a given point.
2.3.4 Use the limit definition of the derivative to determine the slope of the tangent line to a curve at a specific point.
2.3.5 Use the limit definition of the derivative to determine the equation of the tangent line to a curve at a specific point.
2.4 Use of the graph of a function to determine whether a function is differentiable at a point or on an interval
2.4.1 Determine if the derivative of a function exists at a point or on an interval by examining the graph of the function.
3. Rules and techniques of differentiation
3.1 Recognition of the equivalence of various derivative notations
3.1.1 Recognize different notations for the derivative of y with respect to x.
3.2 Use of basic differentiation formulas and rules and proof of simple propositions
3.2.1 Recognize when and how to use the basic differentiation formulas:
3.2.2 Recognize when and how to use the following differentiation formulas derived from the chain rule:
3.2.3 Recognize when and how to use the following rules: power rule, constant multiple rule, sum and difference rule.
3.2.4 Recognize when and how to use the product and quotient rules.
3.2.5 Prove a selection of the rules in 3.2.1 using the limit definition of the derivative.
3.3 Determination of whether a function is differentiable at a point or on an interval
3.3.1 Determine whether a function is differentiable at a specified point or on an interval using graphical, numerical, or analytical methods.
3.4 Use of differentiation rules to perform implicit and logarithmic differentiation
3.4.1 Recognize when and how to use implicit differentiation.
3.4.2 Recognize when and how to use logarithmic differentiation.
3.5 Evaluation and application of higher order derivatives
3.5.1 Find higher order derivatives.
3.5.2 Use higher-order derivatives to solve position, velocity and acceleration problems.
3.6 Use of derivatives to find the slope of a tangent (normal) line to a curve at a point
3.6.1 Use the differentiation rules listed in 3.2.1 and 3.2.2 to find the slope of the tangent line to a curve at a point.
3.6.2 Use the differentiation rules listed in 3.2.1 and 3.2.2 to find the equation of the tangent line to a curve at a point.
3.6.3 Use the differentiation rules listed in 3.2.1 and 3.2.2 to find the equation of the normal line to a curve at a point.
4. Graphing of functions
4.1 Use of the derivative and related concepts to analyse the variations of a function and to sketch a graph of the function.
4.1.1 Find critical numbers.
4.1.2 Find intervals on which a function is increasing and decreasing using the sign of the first derivative.
4.1.3 Find relative and absolute extrema.
4.1.4 Use the first or second derivative test to decide whether the critical points represent relative maxima or relative minima.
4.1.5 Find inflection points.
4.1.6 Find intervals on which a function is concave up or concave down using the sign of the second derivative.
4.1.7 Use limits to find all vertical and horizontal asymptotes.
4.1.8 Use 4.1.1 – 4.1.7 to graph polynomial, rational, trigonometric, logarithmic and exponential functions.
4.2 Demonstration of the ability to understand abstract properties of continuous and differentiable functions, as illustrated by two simple standard theorems.
4.2.1 State the conditions necessary for Rolle’s Theorem.
4.2.2 State the conclusion of Rolle’s Theorem.
4.2.3 State the conditions necessary for the Mean Value Theorem.
4.2.4 State the conclusion of the Mean Value Theorem
5. Optimization and rate-of-change problems
5.1 Solution of optimization problems
5.1.1 Represent an optimization word problem in functional form
5.1.2 Determine the quantity, P, to be maximized or minimized and identify the variables which are involved.
5.1.3 Draw a diagram, if possible, to illustrate the problem and list any other relationship(s) between the variables.
5.1.4 Express P as a function of one variable.
5.1.5 Take the derivative of the function, P, in 5.1.4.
5.1.6 Find all the possible critical values by solving the equation P’ = 0.
5.1.7 Test the critical value(s) and interval endpoints for absolute maximum or minimum.
5.1.8 Interpret (explain) the results found in the optimization problem.
5.2 Solution of problems involving related rates
5.2.1 Represent a word problem involving related rates in functional form.
5.2.2 Identify the variables and rates in the problem.
5.2.3 Draw a diagram, if possible, to illustrate the problem.
5.2.4 Determine the equation relating the variables.
5.2.5 Differentiate the equation in 5.2.4 with respect to time, t.
5.2.6 Solve the equation in 5.2.5 for the required rate.
5.2.7 Interpret (explain) the results found in the problem involving related rates.
6. Integration
6.1 Evaluation of the Indefinite Integral
6.1.1 Give the definition of the indefinite integral as an antiderivative.
6.1.2 Express the basic differentiation formulas listed in 3.2.1 as antidifferentiation formulas.
6.1.3 Recognize when and how to use the constant multiple rule and the sum and difference rule in the evaluation of integrals.
6.1.4 Use the antidifferentiation formulas from 6.1.2 and the rules in 6.1.3 to evaluate indefinite integrals.
6.2 Evaluation of the Definite Integral
6.2.1 State the definition of the definite integral.
6.2.2 State the Fundamental Theorem of Calculus.
6.2.3 Find the definite integral of functions described in 6.1.
6.2.4 Use the Fundamental Theorem of Calculus to find the area of a region under a curve on a closed interval.
6.3 Solution of simple Differential Equations
6.3.1 Find a general solution to a differential equation of the form 
.
6.3.2 Find a particular solution to a differential equation of the form 
given an initial condition 
.
7. Integration, Comprehensive Assessment, and Exit Profile Goals
7.1 Demonstration of the ability to solve problems systematically
7.1.1 Solve both drill-type and word problems.
7.1.2 Analyze a problem by:
i) selecting principles that apply to the problem,
ii) selecting relevant formulae and mathematical tools that apply to the problem,
iii) translating the written description of a problem into the appropriate mathematical, graphical, vectorial and diagrammatic representation,
iv) solving mathematical equations that arise in the problem, and
v) presenting the solution logically with a clear listing of the various steps.
7.2 Demonstration of the ability to learn in an autonomous manner
7.2.1 Organize course materials such as notes, solved examples, assigned problems, worksheets, quizzes and class tests for a binder inspection.
7.2.2 Develop learning strategies in the areas of understanding the concepts taught, reviewing for tests, seeking help outside of class and accessing pertinent information from the Mathematics Lab.
7.2.3 Solve problems by referring to 7.2.1.
7.2.4 Give a summary of the topic that was presented in the previous class.
7.2.5 Undertake to learn a certain topic without the help of the teacher by:
i) reading about the topic in the textbook,
ii) writing a summary of the topic,
iii) giving solved examples with a clear method of solution,
iv) using these examples to solve problems from the exercises, and
v) finding an application of the topic in another course in the Science Program.
7.3 Placement in context of the emergence and development of scientific concepts
7.3.1 Write one or more historically informed essays on topic(s) relevant to the project material, such as a biographical, descriptive, or analytical account of events related to the project.
7.4 Clear demonstration of links between mathematics and at least one other science discipline
7.4.1 Apply knowledge or skills that have been acquired in Calculus I to topic(s) in Physics, Chemistry or Biology.