Calculus Curriculum

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.

Express a function as a Taylor series.

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.

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.

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.

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.