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• #### Appendix F: Text of a Letter Endorsed by the Governing Boards of the Mathematical Association of America and the National Council of Teachers of Mathematics Concerning Calculus in the Secondary Schools 69-72

* Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form.

### II.Derivatives

Concept of the derivative.

• Derivative presented geometrically, numerically, and analytically.

• Derivative interpreted as an instantaneous rate of change.

• Derivative defined as the limit of the difference quotient.

• Relationship between differentiability and continuity.

Derivative at a point.

• Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.

• Tangent line to a curve at a point and local linear approximation.

• Instantaneous rate of change as the limit of average rate of change.

• Approximate rate of change from graphs and tables of values.

Derivative as a function.

• Corresponding characteristics of graphs of ƒ and ƒ’.

• Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’.

• The Mean Value Theorem and its geometric consequences.

• Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

Second derivatives.

• Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ’’.

• Relationship between the concavity of ƒ and the sign of ƒ’’.

• Points of inflection as places where concavity changes.

Applications of derivatives.

• Analysis of curves, including the notions of monotonicity and concavity.

• Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors.

• Optimization, both absolute (global) and relative (local) extrema.

• Modeling rates of change, including related rates problems.

• Use of implicit differentiation to find the derivative of an inverse function.

• Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

• Geometric interpretation of differential equations via slope fields and the relationship between slope fields and derivatives of implicitly defined functions.

• Numerical solution of differential equations using Euler’s method.

• L’Hô pital’s Rule and its use in determining convergence of improper integrals and series.

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