Math Before Calculus

 
(From: The Teacher's Guide AP Calculus 1997, The College Board pp34 36)  

The following is a list of topics students should have exposure to before taking calculus.

 

Algebra and Trigonometry:

Elementary set concepts and set notation

Real number properties; the real number line

Integers; rational and irrational numbers

Absolute value: equations, inequalities, and interpretation as distance

Linear and quadratic equations; the quadratic formula

Factoring techniques

Completion of the square

Direct and indirect variation

Polynomial equations and inequalities

Division of polynomials; rational expressions

Remainder, factor, and rational root theorems

Relationship between polynomial degree and the number of zeros

Rational equations and inequalities

Exponents and radicals; laws of exponents

Properties of logarithms

General functions: domain, range, zeros, inverse, graphs of y=f(x)

Function properties (increasing, decreasing, periodic, even, odd, one-to-one)

Domain, range, and graphical analysis of:

polynomial functions

rational functions

exponential functions ( base a>0, a 1); e x

logarithmic functions (base a>0, a 1); natural logarithms

trigonometric (circular) functions; radian measure

simple algebraic functions

functions involving absolute value

step functions (e.g., the greatest integer function)

functions define on split domain (piecewise functions)

Trigonometric identities

Trigonometric equations

The Binomial Theorem

Sequences and series; summation notation ( for BC course)

Forming functions from verbal descriptions ( in all contexts)

Problem solving (in all contexts)

 

Geometry

The Pythagorean Theorem

Congruent and similar triangles

Parallel and perpendicular lines

Polygons

Circles

Deductive proof

Indirect proof

Areas (rectangles, triangles, trapezoids, circles, parallelograms, regular polygons)

Volumes (regular prisms, cylinders, cones, spheres)

Total and lateral surface areas (regular prisms, cones, spheres)

 

Coordinate Geometry  

Lines (slopes, parallel and perpendicular)

Distance and midpoint formulas

Graphs of functions (see also Algebra and Trigonometry)

Symmetry and periodicity (function and graph behavior)

Translating and reflecting graphs; graphing inverse relations

Conic sections

Polar coordinate system; polar graphs (for BC course)

Vectors (for BC course)

 

Graphing Calculators

Basic calculation procedures and computations

Graphing functions; viewing windows

Finding roots of equations

Recognizing function behavior from graphical observations

Hidden behavior and limitations of calculator graphs

Error accumulation and roundoff procedures