Calculus Curriculum

Below are skills needed, with links to resources to help with that skill. We also encourage plenty of exercises and book work. Curriculum Home

Important: this is a guide only.
Check with your local education authority to find out their requirements.

Calculus | Functions

☐ Introduction to continuity

☐ Intermediate Value Theorem

☐ How Polynomials Behave

☐ Continuous Functions

☐ Piecewise Functions

☐ Intermediate Value Theorem and Extreme Value Theorem

☐ Intermediate Value Theorem

☐ Understand how the behavior of the graphs of polynomials can be predicted from the equation, including: continuity, whether the leading term has an even or odd exponent, the size of the factor of the leading term, the number of turning points, and end behavior.

☐ Polynomials: Bounds on Zeros

☐ How Polynomials Behave

☐ Polynomials: The Rule of Signs

☐ Even and Odd Functions

☐ Understand what is meant by saying that a function is increasing, strictly increasing, decreasing or strictly decreasing.

☐ Increasing and Decreasing Functions

☐ Understand what is meant by the following terms for a function, and how to find them from the graph of the function: Local Maximum, Local Minimum, Global Maximum and Global Minimum.

☐ Maxima and Minima of Functions

☐ Graph of an Equation

☐ Activity: Soup Can

☐ Understand what is meant by a continuous function and how continuity can depend upon the domain.

☐ Continuous Functions

☐ Piecewise Functions

☐ Asymptote

Calculus | Infinite Series

☐ Express a function as a Taylor series.

☐ Taylor Series

Calculus | Derivatives

☐ Introduction to derivatives

☐ Derivatives as dy/dx

☐ Introduction to Derivatives

☐ Introduction to Calculus

☐ From average rate of change to instantaneous rate of change, derivatives as dy/dx

☐ Derivatives as dy/dx

☐ Derivative Rules

☐ Derivatives and continuity

☐ Increasing and Decreasing Functions

☐ Maxima and Minima of Functions

☐ Intermediate Value Theorem

☐ How Polynomials Behave

☐ Continuous Functions

☐ Differentiable

☐ Slope of a curve at a point: where there is a vertical tangent, or no tangents

☐ Slope (Gradient) of a Straight Line

☐ Differentiable

☐ Approximating rate of change (graphs and tables)

☐ Function Grapher and Calculator

☐ Introduction to Derivatives

☐ Differentiate functions using the Derivative Rules

☐ Derivative Rules

☐ Introduction to Derivatives

☐ Derivatives as dy/dx

☐ Find the second derivative of a function using the rules of differentiation

☐ Second Derivative

☐ Introduction to Derivatives

☐ Derivative Rules

☐ Find a maximum or minimum using derivatives and applying the second derivative test.

☐ Finding Maxima and Minima using Derivatives

☐ Second Derivative

☐ Introduction to Derivatives

☐ Understand what it means to say a function is differentiable, and how to choose an appropriate domain. Know that a differentiable function is continuous, but a continuous function is not necessarily differentiable.

☐ Differentiable

☐ Know how to use the Derivative Rules to differentiate a function implicitly.

☐ Implicit Differentiation

☐ Derivative Rules

☐ Find the first and second partial derivative of a function in two variables.

☐ Partial Derivatives

Calculus | Differential Equations

☐ Introduction to differential equations: 1. Order and (if appropriate) degree. 2. What is meant by a linear differential equation.

☐ Differential Equations

☐ Solve first order differential equations by the method of Separation of Variables.

☐ Separation of Variables

☐ Solve first order differential equations by the homogeneous method.

☐ Homogeneous Differential Equations

☐ Solve first order Linear differential equations.

☐ Solution of First Order Linear Differential Equations

☐ Solve first order Bernoulli differential equations.

☐ The Bernoulli Differential Equation

☐ Solve second order linear differential equations of the type y" + py' +qy = 0 where the characteristic equation has two distinct real roots.

☐ Second Order Differential Equations

☐ Solve second order linear differential equations of the type y" + py' +qy = 0 where the characteristic equation has one real root.

☐ Second Order Differential Equations

☐ Solve second order linear differential equations of the type y" + py' +qy = 0 where the characteristic equation has two complex roots.

☐ Second Order Differential Equations

☐ Solve second order linear differential equations of the type y" + py' +qy = f(x) using the method of undetermined coefficients.

☐ Method of Undetermined Coefficients

☐ Solve second order homogeneous differential equations using the method of Variation of Parameters.

☐ Integration by Parts

☐ The Method of Variation of Parameters

☐ Solve first order differential equations by the method of exact equations and integrating factors.

☐ Exact Equations and Integrating Factors

Calculus | Integrals

☐ Introduction to Integration. Understand that integration is the inverse of differentiation, and recognize the importance of the constant of integration.

☐ Introduction to Integration

☐ Integrate functions using the Integration rules.

☐ Introduction to Integration

☐ Integration Rules

☐ Integration by Parts

☐ Integration by Substitution

☐ Integrate products of functions using Integration by Parts, and know how this method can sometimes be applied to integrating single functions.

☐ Integration by Parts

☐ Integration by Substitution

☐ Calculate definite integrals and know how they relate to areas.

☐ Definite Integrals

☐ Use the arc length formula to find the length of an arc of a curve.