Techniques and Formulas From Precalculus and Calculus

That Students Should Have Memorized

Mathematics Department - September 3, 1999

Mathematics students should memorize the following and be able to use them in all future courses with only pencil and paper - no books, notes, calculators, etc. In addition, professors can require additional facts to be memorized during their specific courses.

Precalculus

- Algebra

Add, subtract, multiply, and divide polynomials and fractions

Factor algebraic expressions

Exponents and radicals

Solving linear and quadratic equations in one variable

Quadratic formula

Complete the square

Solving linear systems of two equations and two unknowns

Solving easy linear systems that are larger than 2 by 2

- Inequalities

Solving linear and quadratic inequalities in one variable

Graphing linear inequalities in one and two variables

- Geometry

Area and perimeter of triangles, quadrilaterals, circles

Volume and surface area of spheres, cylinders, boxes

- Analytic Geometry

Distance formula, midpoint formula, and slope formula

Equations and graphs of straight lines

Equations of circles, ellipses, parabolas, hyperbolas

- Trigonometry

Conversion between degrees and radians

Definition of trig functions (both right angle and unit circle)

Values of trig functions at special angles; reference angles

Graphs of trig functions

Basic trig identities:

- Properties and graphs of logarithmic and exponential functions

Calculus I

- Limits

Intuitive definition of limits

Estimating limits from a table and a graph

Limits that fail to exist

Limits of polynomial and rational functions

Limits of functions involving radicals

One-sided limits

Definition of continuous functions

and

Limits at infinity; infinite limits

- Derivatives

Definition of derivative

Geometric interpretation of derivative

Derivatives that fail to exist

Basic differentiation rules and techniques:

Power rule

Constant Multiple Rule

Sum and Difference Rules

Product rule

Quotient rule

Chain rule

Implicit differentiation

Derivatives of the six trigonometric functions

- Applications of derivatives

Position, velocity, and acceleration

Definition of relative and absolute extrema of a function

Extreme Value Theorem

Definition of critical number

Definition of increasing and decreasing functions

Test for increasing and decreasing functions

First derivative test

Definition of concavity

Test for concavity

Definition of inflection points

Second derivative test

- Integration

Definition of antiderivative (indefinite integral)

Integrals of sums, differences, and constant multiples

Integrals yielding the six trigonometric functions

Sigma notation

Definition of definite integral

Relationship between integrals and areas of regions

Fundamental Theorem of Calculus

Second Fundamental Theorem of Calculus

Integration by u-substitution

Calculus II

- Other Transcendental Functions

Definition of inverse function

Existence of an inverse function

Definition of logarithmic and exponential functions

Conversion between logarithmic and exponential forms

Properties of logarithmic and exponential functions:

and

Graphs of logarithmic and exponential functions

Derivatives of logarithmic and exponential functions

Integrals yielding logarithmic and exponential functions

Definition of the six inverse trigonometric functions

Derivatives of the six inverse trigonometric functions

Integrals yielding inverse trigonometric functions

- Applications of integration

Areas of regions between curves

Volumes by disc, washer, and shell methods

- Integration techniques

Integration by parts

Integration using trigonometric substitution

Integration using partial fraction decomposition

- LÕH™pitalÕs Rule and indeterminate forms for limits

- Improper integrals

- Sequences and series

Meaning of convergence of sequences and series

Sum of a geometric series

Convergence or divergence of p-series

Power series and Taylor series

- Parametric equations and polar coordinates

Graphs of parametric equations and polar equations

Conversion of parametric and polar equations to and from rectangular coordinate equations

Calculus III

- Vectors

Component form of a vector

Sum, scalar multiple, length of vectors

Definition of unit vectors

Parallel vectors

Dot product and cross product of vectors

Angle between vectors

Orthogonal vectors

Tangent vectors and normal vectors

Differentiation and integration of vector-valued functions

- Plotting in cylindrical and spherical coordinates

- Functions of Several Variables

Meaning and techniques of partial differentiation

Chain Rule

Definition of directional derivatives and gradients

- Multiple integration

Evaluate iterated integrals

Evaluate multiple integrals

Double integrals and volume

Changing the order of integration