Calculus Cheat Sheet - Lamar University

1 Calculus Cheat Sheet Visit for a complete set of Calculus notes. 2005 Paul Dawkins Limits Definitions Precise Definition : We say ()limxafxL = if for every 0e> there is a 0d>such that whenever 0xad<-< then ()fxLe-<. Working Definition : We say ()limxafxL = if we can make ()fx as close to L as we want by taking x sufficiently close to a (on either side of a) without letting xa=. Right hand limit : ()limxafxL+ =. This has the same definition as the limit except it requires xa>. Left hand limit : ()limxafxL- =. This has the same definition as the limit except it requires xa<. Limit at Infinity : We say()limxfxL = if we can make ()fx as close to L as we want by taking x large enough and positive.

2 There is a similar definition for ()limxfxL - = except we require x large and negative. Infinite Limit : We say ()limxafx = if we can make ()fx arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a) without letting xa=. There is a similar definition for ()limxafx =- except we make ()fx arbitrarily large and between the limit and one-sided limits ()limxafxL = fi ()()limlimxaxafxfxL+- == ()()limlimxaxafxfxL+- == fi ()limxafxL = ()()limlimxaxafxfx+- fi ()limxafx Does Not Exist Properties Assume ()limxafx and ()limxagx both exist and c is any number then, 1. ()()limlimxaxacfxcfx = 2. ()()()()limlimlimxaxaxafxgxfxgx = 3.

3 ()()()()limlimlimxaxaxafxgxfxgx = 4. ()()()()limlimlimxaxaxafxfxgxgx = provided ()lim0xagx 5. ()()limlimnnxaxafxfx = 6. ()()limlimnnxaxafxfx = Basic Limit Evaluations at Note : ()sgn1a= if 0a> and ()sgn1a=- if 0a<.1. limxx = e & lim0xx - =e 2. ()limlnxx = & ()0limlnxx+ =- 3. If 0r>thenlim0rxbx = 4. If 0r> and rxis real for negative x thenlim0rxbx - = 5. n even : limnxx = 6. n odd : limnxx = & limnxx - =- 7. n even : ()limsgnnxaxbxca +++= L 8. n odd : ()limsgnnxaxbxca +++= L 9. n odd : ()limsgnnxaxcxda - +++=- L Calculus Cheat Sheet Visit for a complete set of Calculus notes. 2005 Paul Dawkins Evaluation Techniques Continuous Functions If()fxis continuous at a then()()limxafxfa = Continuous Functions and Composition ()fx is continuous at b and ()limxagxb = then ()()()()()limlimxaxafgxfgxfb == Factor and Cancel ()()()2222226412limlim2268lim42xxxxxxxxx xxxx -++-=--+=== Rationalize Numerator/Denominator ()()()()()()2299299333limlim8181391limli m8139311186108xxxxxxxxxxxxxxx --+=--+--==-+++-==- Combine Rational Expressions ()()()()002001111limlim111limlimhhhhxxhh xhxhxxhhhxxhxxhx -+ -= ++ --===- ++ L Hospital s Rule If ()()0lim0xafxgx = or ()()limxafxgx = then, ()()()()

4 Limlimxaxafxfxgxgx = a is a number, or - Polynomials at Infinity ()px and ()qx are polynomials. To compute ()()limxpxqx factor largest power of x in ()qxout of both ()px and ()qx then compute limit. ()()222222445533343limlimlim52222xxxxxxx xxxxx - - - ---===----Piecewise Function ()2limxgx - where ()25if 213if 2xxgxxx +<-= - - Compute two one sided limits, ()222limlim59xxgxx-- - -=+= ()22limlim137xxgxx++ - -=-= One sided limits are different so ()2limxgx - doesn t exist. If the two one sided limits had been equal then ()2limxgx - would have existed and had the same value. Some Continuous Functions Partial list of continuous functions and the values of x for which they are Polynomials for all x.

5 2. Rational function, except for x s that give division by zero. 3. nx(n odd) for all x. 4. nx(n even) for all 0x . 5. xe for all x. 6. lnx for 0x>. 7. ()cosx and ()sinx for all x. 8. ()tanx and ()secx provided 33,,,,,2222xpppp --LL 9. ()cotx and ()cscx provided ,2,,0,,2,xpppp --LL Intermediate Value Theorem Suppose that ()fx is continuous on [a, b] and let M be any number between ()fa and ()fb. Then there exists a number c such that acb<< and ()fcM=. Calculus Cheat Sheet Visit for a complete set of Calculus notes. 2005 Paul Dawkins Derivatives Definition and Notation If ()yfx= then the derivative is defined to be()()()0limhfxhfxfxh +- =. If ()yfx= then all of the following are equivalent notations for the derivative.

6 ()()()()dfdydfxyfxDfxdxdxdx ===== If()yfx=all of the following are equivalent notations for derivative evaluated at xa=. ()()xaxaxadfdyfayDfadxdx=== ==== Interpretation of the Derivative If ()yfx= then, 1. ()mfa = is the slope of the tangent line to ()yfx= at xa=and the equation of the tangent line at xa= is given by ()()()yfafaxa =+-. 2. ()fa is the instantaneous rate of change of ()fx at xa=. 3. If ()fx is the position of an object at time x then ()fa is the velocity of the object at xa=. Basic Properties and Formulas If ()fx and ()gx are differentiable functions (the derivative exists), c and n are any real numbers, 1. ()()cfcfx = 2. ()()()fgfxgx = 3. ()fgfgfg =+ Product Rule 4.

7 2ffgfggg -= Quotient Rule 5. ()0dcdx= 6. ()1nndxnxdx-= Power Rule 7. ()()()()()()dfgxfgxgxdx = This is the Chain Rule Common Derivatives ()1dxdx= ()sincosdxxdx= ()cossindxxdx=- ()2tansecdxxdx= ()secsectandxxxdx= ()csccsccotdxxxdx=- ()2cotcscdxxdx=- ()121sin1dxdxx-=- ()121cos1dxdxx-=-- ()121tan1dxdxx-=+ ()()lnxxdaaadx=()xxddx=ee ()()1ln,0dxxdxx=>()1ln,0dxxdxx= ()()1log,0lnadxxdxxa=> Calculus Cheat Sheet Visit for a complete set of Calculus notes. 2005 Paul Dawkins Chain Rule Variants The chain rule applied to some specific functions. 1. ()()()()1nndfxnfxfxdx- = 2. ()()()()fxfxdfxdx =ee 3. ()()()()lnfxdfxdxfx = 4. ()()()()sincosdfxfxfxdx = 5.

8 ()()()()cossindfxfxfxdx =- 6. ()()()()2tansecdfxfxfxdx = 7. []()[][]()()()()secsectanfxfxfxfxddx = 8. ()()()()12tan1fxdfxdxfx- = + Higher Order Derivatives The Second Derivative is denoted as ()()()222dffxfxdx == and is defined as ()()()fxfx =, the derivative of the first derivative,()fx . The nth Derivative is denoted as ()()nnndffxdx= and is defined as ()()()()()1nnfxfx- =, the derivative of the (n-1)st derivative,()()1nfx-. Implicit Differentiation Find y if ()2932sin11xyxyyx-+=+e. Remember()yyx= here, so products/quotients of x and y will use the product/quotient rule and derivatives of y will use the chain rule. The trick is to differentiate as normal and every time you differentiate a y you tack on a y (from the chain rule).

9 After differentiating solve for y . ()()()()()()2922329222929223329329292229 32cos1111232932cos1129cos29cos1123xyxyxy xyxyxyxyyxyxyyyyxyyxyxyyyyyxyyxyyyxy---- --- -++=+-- -++=+fi=-- --=--eeeeeee Increasing/Decreasing Concave Up/Concave Down Critical Points xc= is a critical point of ()fx provided either 1. ()0fc = or 2. ()fc doesn t exist. Increasing/Decreasing 1. If ()0fx > for all x in an interval I then ()fx is increasing on the interval I. 2. If ()0fx < for all x in an interval I then ()fx is decreasing on the interval I. 3. If ()0fx = for all x in an interval I then ()fx is constant on the interval I. Concave Up/Concave Down 1. If ()0fx > for all x in an interval I then ()fx is concave up on the interval I.

10 2. If ()0fx < for all x in an interval I then ()fx is concave down on the interval I. Inflection Points xc= is a inflection point of ()fx if the concavity changes at xc=. Calculus Cheat Sheet Visit for a complete set of Calculus notes. 2005 Paul Dawkins Extrema Absolute Extrema 1. xc=is an absolute maximum of()fx if()()fcfx for all x in the domain. 2. xc= is an absolute minimum of()fx if()()fcfx for all x in the domain. Fermat s Theorem If ()fx has a relative (or local) extrema at xc=, then xc= is a critical point of ()fx. Extreme Value Theorem If ()fx is continuous on the closed interval [],ab then there exist numbers c and d so that, 1. ,acdb , 2. ()fc is the abs. max. in [],ab, 3. ()fd is the abs.