Mathematics: Provincial Level—Calculus

Goal Statement

ABE Provincial Level Calculus is designed to (1) provide students with the mathematical knowledge and skills needed for post-secondary academic and career programs and (2) ease the transition from Provincial level Mathematics to first year calculus at college/ university.

1. Prelude to Calculus

It is expected that learners will be able to:

• demonstrate an understanding of the concept of the limit and notation used in expressing the limit of a function
• evaluate the limit of a function analytically, graphically and 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 and two-sided limits
• evaluate limits at infinity
• determine vertical and horizontal asymptotes using limits
• determine continuity of functions at a point x = a
• determine discontinuities and removable discontinuities
• determine continuity of polynomial, rational, and composite functions

Optional Outcomes:

• determine continuity of trigonometric functions
• determine limits of trigonometric functions

2. The Derivative

It is expected that learners will be able to:

• define and evaluate the derivative at x = a as: [latex]\displaystyle f'(x) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}[/latex]
• distinguish between continuity and differentiability of a function
• determine the slope of a tangent line to a curve at a given point
• calculate derivatives of elementary, rational and algebraic functions
• distinguish between rate of change and instantaneous rate of change
• apply differentiation rules to applied problems
• use Chain Rule to compute derivatives of composite functions
• solve rate of change application problems
• determine local and global extreme values of a function
• solve applied optimization (max/min) problems

Optional Outcomes:

• calculate derivatives of trigonometric functions and their inverses
• calculate derivatives of exponential and logarithmic functions
• use logarithmic differentiation
• calculate derivatives of functions defined implicitly
• solve related rates problems
• use Newton’s Method

3. Applications of the Derivative

It is expected that learners will be able to:

• determine critical numbers and inflection points of a function
• compute differentials
• use the First and Second Derivative Tests to sketch graphs of functions
• use concavity and asymptotes to sketch graphs of functions

Optional Outcomes:

• differentiate implicitly
• understand and use the Mean Value Theorem
• apply L’Hopital’s Rule to study the behaviour of functions

4. Antiderivatives

It is expected that learners will be able to:

• compute antiderivatives of linear combinations of functions
• use antidifferentiation to solve rectilinear motion problems
• use antidifferentiation to find the area under a curve
• evaluate integrals using integral tables and substitutions

Optional Outcomes:

• use antidifferentiation to find the area between two curves
• compute Riemann sums
• apply the Trapezoidal Rule
• solve initial value problems

5. Differential Equations

It is expected that learners will be able to:

• derive a general solution of differential equations and find a particular solution satisfying initial conditions
• derive differential equations that explain mathematical models in the applied sciences