Overview: This guide is intended as a supplement to the course. The assumption is made that the student has already seen much of the material presented here, and has a solid grasp of the basics of limits, integrals and derivatives. The goal of this guide is to organize the major course concepts in an easy-to-understand format, and to provide the student with problem solving techniques and an understanding of how to applywhat they have learned in class to the actual problems they will see. Contents: Pg 1: Overview and Table of ContentsPg 2: Formula sheet, Numerical Integration:Essential identities, formulae, and basic integrals and derivatives Pg 3: Integration:How to recognize which method is appropriate for a given integral Pg 4: Differential Equations: A step by step problem solving guide Pg 5: Sequences: Proving monotone convergence and finding limits Pg 6: Series: Showing convergence. How to recognize which test is appropriate for a given series Pg 7: Power Series and Taylor Series: Intervals of convergence, converting between series and functions, error approximation. Pg 10:Polar and Parametric Equations: Curve sketching, tangents, converting from x-y and back.
Essential Identities, Formulae, Integrals, and Derivatives Trigonometric Identities ( ) ( ) ( ) ( ) ( ) ( ) These can all be simply derived from each other by multiplying or dividing by cos2(x) or sin2(x), so it is only necessary to remember one. ( ) ( ) ( ) ( ) ( )( ) ( )Derivatives ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) √ ( ) √ ( ) Above are the derivatives of the sine, tangent, and secant functions. These functions have counterparts; cosine, cotangent, and cosecant. The derivatives of these functions are below. There is a simple pattern determining the derivatives below. Knowing this pattern halves the amount of memorization required. The mnemonic “sex tanks!” may be used to remember the derivative of secant.( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) √ ( ) √ ( ) ( ( ) ) ( )( )( ) ( )( ) ( )Integrals If you have memorized the derivatives above, then you also have memorized most of the important integrals. ∫ ( ) ( ) ( ) ∫ ( ) ( ) ( ) Numerical Integration For all methods. For trapezoid and midpoint approximations, K is determined by ( ) For Simpson’s rule, ( )( ) is used. The value used for these derivatives is the max value they achieve over the interval [b-a]. Trapezoid Rule: ( ( ) ( ) ( ) ( ) ( )) ( )Midpoint Rule: ∑() ( )Simpson’s Rule:( ( ) ( ) ( ) ( ) ( ) ) ( )Power Series: Multiplying Series: Taylor’s InequalityCommon Series:Taylor’s Theorem∑ ∑ ∑ ∑ ( )( ) ( ) ∑ ∑( ) ∑( )( ) ( ) ∑( )( ) ( ) ∑( )( )∑ () () ( )( ) ( )( ) ∑( )( )( )
Integration Integration by Parts Integration by Substitution Integration by Trigonometric Substitution ∫ ∫ ( ) ( )( ) ( )Partial Fractions Decomposition This is a method for splitting up fractions. Usually, the degree of the bottom should be 1 greater than the degree of the top. E.g. ( )( )or ( )( )( ). Terms in the denominator which are to a power are handled like so: ( )( )( )( )( )( )Long Division or Synthetic Division If we have a fraction of polynomials, where the degree of the top is greater than the degree of the bottom, we use long division.