Math Calculus Review CHSN Review Project Contents Limits 3 Derivatives 14 Integrals 33 Appendix 42 This review guide was written by Dara Adib Portions of the Limits and Derivatives chapters are based off the Calculus Wikibook available on the Internet at http en wikibooks org wiki Calculus CHSN Review Project contributors Dara Adib and Paul Sieradzki contributed to the Limits section of the Calculus Wikibook This is a development version of the text that should be considered a work in progress This review guide is developed by the CHSN Review Project To download this review guide and other review guides visit chsntech org Copyright 2008 2009 Dara Adib and other contributors to the Calculus Wikibook This is a freely licensed work as explained in the Definition of Free Cultural Works freedomdefined org Except as noted under Graphic Credits on this page it is licensed under the Creative Commons Attribution Share Alike 3 0 Unported License To view a copy of this license visit http creativecommons org licenses by sa 3 0 or send a letter to Creative Commons 171 Second Street Suite 300 San Francisco California 94105 USA This review guide is provided as is without warranty of any kind either expressed or implied You should not assume that this review guide is error free or that it will be suitable for the particular purpose which you have in mind when using it In no event shall the CHSN Review Project be liable for any special incidental indirect or consequential damages of any kind or any damages whatsoever including without limitation those resulting from loss of use data or profits whether or not advised of the possibility of damage and on any theory of liability arising out of or in connection with the use or performance of this review guide or other documents which are referenced by or linked to in this review guide Graphic Credits Figure 0 1 on page 24 is a public domain graphic by Inductiveload http commons wikimedia org wiki File Maxima and Minima svg Figure 0 2 on page 26 is a public domain graphic by Inductiveload http commons wikimedia org wiki File X cubed narrow svg 2 Limits This chapter was originally designed for a test on limits administered by Jeanine Lennon to her Math 12H 4H Precalculus class on April 2 2008 It was later updated with an Addendum section page 12 for a test on limits administered by Jonathan Chernick to his AP1 Calculus BC class on September 18 2008 Introduction A limit looks at what happens to a function when the input approaches but does not necessarily reach a certain value The general notation for a limit is below lim f x L x c This is read as the limit of f x as x approaches c is L Informal Definition of a Limit L is the limit of f x as x approaches c The value of f x comes close to L when x is close but not necessarily equal to c It can be represented by either of the following forms with the former being far more common lim f x L x c f x L as x c Rules Now that a limit has been informally defined some rules that are useful for manipulating a limit are listed Identities The following identities assume lim f x L and lim g x M Using these identities other rules can be deduced x c x c 1 AP is a registered trademark of the College Board which was not involved in the production of and does not endorse this product 3 Scalar Multiplication A scalar is a constant When a function is multiplied by a constant scalar multiplication is performed lim kf x k lim f x kL x c x c Addition lim f x g x lim f x lim g x L M x c x c x c Subtraction lim f x g x lim f x lim g x L M x c x c x c Multiplication lim f x g x lim f x lim g x L M x c x c x c Division lim x c limx c f x L f x where M 6 0 g x limx c g x M Constant Rule The constant rule states that if f x k is constant for all x then the limit as x approaches c must be equal to k lim k k x c Identity Rule The identity rule states that if f x x then the limit as x approaches c is equal to c lim x c x c 4 Power Rule The rule for products many times results in determining the power rule n lim f x n lim f x x c x c Finding Limits If c is in the domain of the function and the function can be built out of rational trigonometric logarithmic and exponential functions then the limit is simply the value of the function at c If c is not in the domain of the function then in many cases as with rational functions the domain of the function includes all of the points near c but not c An example would be if one wanted to x find lim where the domain includes all real numbers except 0 In that case one would want to x 0 x find a similar function with the hole filled in The limit of this function at c will be the same while the function is the same at all points not equal to c The limit definition depends on f x only at the points where x is close to c but not equal to it And since the domain of the new function includes c one can now assuming it s still built out of rational trigonometric logarithmic and exponential functions just evaluate the function at c as before In the above example this is easy canceling the x s gives 1 which equals x x at all points except 0 x Thus lim lim 1 1 In general when computing limits of rational functions it s a good idea to x 0 x x 0 look for common factors in the numerator and denominator Does Not Exist Note that the limit might not exist at all There are a number of ways in which this can occur Not Same from Both Sides A left handed limit is different from the right handed limit of the same variable value and function Since the left handed limit 6 right handed limit the limit does not exist This includes cases in which the limit of a certain side does not exist e g lim x 2 which has no left handed limit x 2 Gap There is a gap more than a point wide in the function where the function is not defined As an example in f x x2 16 f x does not have any limit when 4 x 4 There is no way to approach the middle of the graph Note also that the function also has no limit at the endpoints of the two …