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