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