# 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 and Extreme 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.

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

☐ 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.

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

Calculus | Infinite Series

☐ Express a function as a Taylor series.

Calculus | Derivatives

☐ Introduction to derivatives

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

☐ Derivatives and continuity

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

☐ Approximating rate of change (graphs and tables)

☐ Differentiate functions using the Derivative Rules

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

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

☐ 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.

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

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

Calculus | Differential Equations

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

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

☐ Solve first order differential equations by the homogeneous method.

☐ Solve first order Linear differential equations.

☐ Solve first order Bernoulli differential equations.

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

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

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

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

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

☐ Solve first order differential equations by the method of 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.

☐ Integrate functions using the Integration rules.

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

☐ Integration by Substitution

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

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

☐ Use approximate methods - LRAM, RRAM, MRAM, Trapezoidal and Simpson's Rule - to find the values of integrals.

☐ Calculate the volumes of solids of revolution using disks, washers or shells.

Calculus | Limits

☐ Introduction to limits

☐ Evaluating limits

☐ Formal definition of limits

☐ Estimating limits (graphs and tables)

☐ Continuity and Limits - how to interpret the limit of a function at a discontinuity ("hole", "pointy change" or "jump").

☐ Use Cavalieri's Principle and informal limit arguments to find areas and volumes.

☐ Use L'Hopital's Rule to evaluate limits.