Calculus Cheat Sheet - Multivariate Calculus | MA 26100, Study notes of Calculus

Download Calculus Cheat Sheet - Multivariate Calculus | MA 26100 and more Study notes Calculus in PDF only on Docsity! Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Limits Definitions Precise Definition : We say ( )lim x a f x L → = if for every 0ε > there is a 0δ > such that whenever 0 x a δ< − < then ( )f x L ε− < . “Working” Definition : We say ( )lim x a f x L → = if we can make ( )f x as close to L as we want by taking x sufficiently close to a (on either side of a) without letting x a= . Right hand limit : ( )lim x a f x L +→ = . This has the same definition as the limit except it requires x a> . Left hand limit : ( )lim x a f x L −→ = . This has the same definition as the limit except it requires x a< . Limit at Infinity : We say ( )lim x f x L →∞ = if we can make ( )f x as close to L as we want by taking x large enough and positive. There is a similar definition for ( )lim x f x L →−∞ = except we require x large and negative. Infinite Limit : We say ( )lim x a f x → = ∞ if we can make ( )f x arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a) without letting x a= . There is a similar definition for ( )lim x a f x → = −∞ except we make ( )f x arbitrarily large and negative. Relationship between the limit and one-sided limits ( )lim x a f x L → = ⇒ ( ) ( )lim lim x a x a f x f x L + −→ → = = ( ) ( )lim lim x a x a f x f x L + −→ → = = ⇒ ( )lim x a f x L → = ( ) ( )lim lim x a x a f x f x + −→ → ≠ ⇒ ( )lim x a f x → Does Not Exist Properties Assume ( )lim x a f x → and ( )lim x a g x → both exist and c is any number then, 1. ( ) ( )lim lim x a x a cf x c f x → → =   2. ( ) ( ) ( ) ( )lim lim lim x a x a x a f x g x f x g x → → → ± = ±   3. ( ) ( ) ( ) ( )lim lim lim x a x a x a f x g x f x g x → → → =   4. ( ) ( ) ( ) ( ) lim lim lim x a x a x a f xf x g x g x → → →   =    provided ( )lim 0 x a g x → ≠ 5. ( ) ( )lim lim nn x a x a f x f x → →  =     6. ( ) ( )lim limn n x a x a f x f x → →   =  Basic Limit Evaluations at ± ∞ Note : ( )sgn 1a = if 0a > and ( )sgn 1a = − if 0a < . 1. lim x x→∞ = ∞e & lim 0x x→− ∞ =e 2. ( )lim ln x x →∞ = ∞ & ( ) 0 lim ln x x −→ = − ∞ 3. If 0r > then lim 0rx b x→∞ = 4. If 0r > and rx is real for negative x then lim 0rx b x→−∞ = 5. n even : lim n x x →±∞ = ∞ 6. n odd : lim n x x →∞ = ∞ & lim n x x →− ∞ = −∞ 7. n even : ( )lim sgnn x a x b x c a →±∞ + + + = ∞L 8. n odd : ( )lim sgnn x a x b x c a →∞ + + + = ∞L 9. n odd : ( )lim sgnn x a x c x d a →−∞ + + + = − ∞L Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Evaluation Techniques Continuous Functions If ( )f x is continuous at a then ( ) ( )lim x a f x f a → = Continuous Functions and Composition ( )f x is continuous at b and ( )lim x a g x b → = then ( )( ) ( )( ) ( )lim limx a x af g x f g x f b→ →= = Factor and Cancel ( ) ( ) ( ) 2 22 2 2 2 64 12lim lim 2 2 6 8lim 4 2 x x x x xx x x x x x x x → → → − ++ − = − − + = = = Rationalize Numerator/Denominator ( )( ) ( ) ( ) ( ) ( ) 2 29 9 29 9 3 3 3lim lim 81 81 3 9 1lim lim 81 3 9 3 1 1 18 6 108 x x x x x x x x x x x x x x x → → → → − − + = − − + − − = = − + + + − = = − Combine Rational Expressions ( ) ( ) ( ) ( ) 0 0 20 0 1 1 1 1lim lim 1 1 1lim lim h h h h x x h h x h x h x x h h h x x h x x h x → → → →  − + − =     + +     − − = = = −  + +  L’Hospital’s Rule If ( ) ( ) 0lim 0x a f x g x→ = or ( ) ( ) lim x a f x g x→ ± ∞ = ± ∞ then, ( ) ( ) ( ) ( ) lim lim x a x a f x f x g x g x→ → ′ = ′ a is a number, ∞ or −∞ Polynomials at Infinity ( )p x and ( )q x are polynomials. To compute ( ) ( ) lim x p x q x→±∞ factor largest power of x out of both ( )p x and ( )q x and then compute limit. ( ) ( ) 2 2 2 2 2 2 4 4 55 3 33 4 3lim lim lim 5 2 2 22x x x xx x x xx x x x→−∞ →− ∞ →−∞ − −− = = = − − −− Piecewise Function ( ) 2 lim x g x →− where ( ) 2 5 if 2 1 3 if 2 x x g x x x  + < − =  − ≥ − Compute two one sided limits, ( ) 2 2 2 lim lim 5 9 x x g x x − −→− →− = + = ( ) 2 2 lim lim 1 3 7 x x g x x + +→− →− = − = One sided limits are different so ( ) 2 lim x g x →− doesn’t exist. If the two one sided limits had been equal then ( ) 2 lim x g x →− 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 continuous. 1. Polynomials for all x. 2. Rational function, except for x’s that give division by zero. 3. n x (n odd) for all x. 4. n x (n even) for all 0x ≥ . 5. xe for all x. 6. ln x for 0x > . 7. ( )cos x and ( )sin x for all x. 8. ( )tan x and ( )sec x provided 3 3, , , , , 2 2 2 2 x π π π π≠ − −L L 9. ( )cot x and ( )csc x provided , 2 , ,0, ,2 ,x π π π π≠ − −L L Intermediate Value Theorem Suppose that ( )f x is continuous on [a, b] and let M be any number between ( )f a and ( )f b . Then there exists a number c such that a c b< < and ( )f c M= . Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Derivatives Definition and Notation If ( )y f x= then the derivative is defined to be ( ) ( ) ( ) 0 lim h f x h f x f x h→ + − ′ = . If ( )y f x= then all of the following are equivalent notations for the derivative. ( ) ( )( ) ( )df dy df x y f x Df x dx dx dx ′ ′= = = = = If ( )y f x= all of the following are equivalent notations for derivative evaluated at x a= . ( ) ( )x a x a x a df dyf a y Df a dx dx= = = ′ ′= = = = Interpretation of the Derivative If ( )y f x= then, 1. ( )m f a′= is the slope of the tangent line to ( )y f x= at x a= and the equation of the tangent line at x a= is given by ( ) ( )( )y f a f a x a′= + − . 2. ( )f a′ is the instantaneous rate of change of ( )f x at x a= . 3. If ( )f x is the position of an object at time x then ( )f a′ is the velocity of the object at x a= . Basic Properties and Formulas If ( )f x and ( )g x are differentiable functions (the derivative exists), c and n are any real numbers, 1. ( ) ( )c f c f x′ ′= 2. ( ) ( ) ( )f g f x g x′ ′ ′± = ± 3. ( )f g f g f g′ ′ ′= + – Product Rule 4. 2 f f g f g g g ′ ′ ′  − =    – Quotient Rule 5. ( ) 0d c dx = 6. ( ) 1n nd x n xdx −= – Power Rule 7. ( )( )( ) ( )( ) ( )d f g x f g x g xdx ′ ′= This is the Chain Rule Common Derivatives ( ) 1d x dx = ( )sin cosd x x dx = ( )cos sind x x dx = − ( ) 2tan secd x x dx = ( )sec sec tand x x x dx = ( )csc csc cotd x x x dx = − ( ) 2cot cscd x x dx = − ( )1 2 1sin 1 d x dx x − = − ( )1 2 1cos 1 d x dx x − = − − ( )1 21tan 1 d x dx x − = + ( ) ( )lnx xd a a adx = ( )x xddx =e e ( )( ) 1ln , 0d x xdx x= > ( ) 1ln , 0d x x dx x = ≠ ( )( ) 1log , 0 lna d x x dx x a = > Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Chain Rule Variants The chain rule applied to some specific functions. 1. ( )( ) ( ) ( )1n nd f x n f x f xdx − ′=       2. ( )( ) ( ) ( )f x f xd f xdx ′=e e 3. ( )( ) ( )( )ln f xd f x dx f x ′ =   4. ( )( ) ( ) ( )sin cosd f x f x f xdx ′=       5. ( )( ) ( ) ( )cos sind f x f x f xdx ′= −       6. ( )( ) ( ) ( )2tan secd f x f x f xdx ′=       7. [ ]( ) [ ] [ ]( ) ( ) ( ) ( )sec sec tanf x f x f x f xd dx ′= 8. ( )( ) ( ) ( ) 1 2tan 1 f xd f x dx f x − ′=   +    Higher Order Derivatives The Second Derivative is denoted as ( ) ( ) ( ) 2 2 2 d ff x f x dx ′′ = = and is defined as ( ) ( )( )f x f x ′′′ ′= , i.e. the derivative of the first derivative, ( )f x′ . The nth Derivative is denoted as ( ) ( ) n n n d ff x dx = and is defined as ( ) ( ) ( ) ( )( )1n nf x f x− ′= , i.e. the derivative of the (n-1)st derivative, ( ) ( )1nf x− . Implicit Differentiation Find y′ if ( )2 9 3 2 sin 11x y x y y x− + = +e . Remember ( )y y x= 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). After differentiating solve for y′ . ( ) ( ) ( ) ( )( ) ( ) 2 9 2 2 3 2 9 2 2 2 9 2 9 2 2 3 3 2 9 3 2 9 2 9 2 2 2 9 3 2 cos 11 11 2 32 9 3 2 cos 11 2 9 cos 2 9 cos 11 2 3 x y x y x y x y x y x y x y y x y x y y y y x yy x y x y y y y y x y y x y y y x y − − − − − − − ′ ′ ′− + + = + − −′ ′ ′ ′− + + = + ⇒ = − − ′− − = − − e ee e e e e Increasing/Decreasing – Concave Up/Concave Down Critical Points x c= is a critical point of ( )f x provided either 1. ( ) 0f c′ = or 2. ( )f c′ doesn’t exist. Increasing/Decreasing 1. If ( ) 0f x′ > for all x in an interval I then ( )f x is increasing on the interval I. 2. If ( ) 0f x′ < for all x in an interval I then ( )f x is decreasing on the interval I. 3. If ( ) 0f x′ = for all x in an interval I then ( )f x is constant on the interval I. Concave Up/Concave Down 1. If ( ) 0f x′′ > for all x in an interval I then ( )f x is concave up on the interval I. 2. If ( ) 0f x′′ < for all x in an interval I then ( )f x is concave down on the interval I. Inflection Points x c= is a inflection point of ( )f x if the concavity changes at x c= . Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Trig Substitutions : If the integral contains the following root use the given substitution and formula to convert into an integral involving trig functions. 2 2 2 sinaba b x x θ− ⇒ = 2 2cos 1 sinθ θ= − 2 2 2 secabb x a x θ− ⇒ = 2 2tan sec 1θ θ= − 2 2 2 tanaba b x x θ+ ⇒ = 2 2sec 1 tanθ θ= + Ex. 2 2 16 4 9x x dx −∫ 2 2 3 3sin cosx dx dθ θ θ= ⇒ = 2 22 4 4 sin 4cos 2 cos4 9x θ θ θ= − = =− Recall 2x x= . Because we have an indefinite integral we’ll assume positive and drop absolute value bars. If we had a definite integral we’d need to compute θ ’s and remove absolute value bars based on that and, if 0 if 0 x x x x x ≥ = − < In this case we have 2 2cos4 9x θ=− . ( ) ( )23sin 2cos 2 224 9 16 12 sin cos 12csc 12cot d d d c θ θ θ θ θ θ θ θ = = = − + ⌠ ⌡ ∫ ∫ Use Right Triangle Trig to go back to x’s. From substitution we have 32sin xθ = so, From this we see that 24 93cot x xθ −= . So, 2 2 2 16 4 4 9 4 9 x xx x dx c− − = − +∫ Partial Fractions : If integrating ( ) ( ) P x Q x dx∫ where the degree of ( )P x is smaller than the degree of ( )Q x . Factor denominator as completely as possible and find the partial fraction decomposition of the rational expression. Integrate the partial fraction decomposition (P.F.D.). For each factor in the denominator we get term(s) in the decomposition according to the following table. Factor in ( )Q x Term in P.F.D Factor in ( )Q x Term in P.F.D ax b+ A ax b+ ( )kax b+ ( ) ( ) 1 2 2 k k AA A ax b ax b ax b + + + + + + L 2ax bx c+ + 2 Ax B ax bx c + + + ( )2 kax bx c+ + ( ) 1 1 2 2 k k k A x BA x B ax bx c ax bx c ++ + + + + + + L Ex. 2( )( ) 2 1 4 7 13 x x x x dx − + +∫ ( ) ( ) 2 2 2 2 ( )( ) 2 13 2 2 2 3 164 11 4 4 3 164 1 4 4 7 13 4 ln 1 ln 4 8 tan x xx x x x x x x x x x dx dx dx x x − + −− + + − + + + = + = + + = − + + + ∫ ∫ ∫ Here is partial fraction form and recombined. 2 2 2 2 4) ( ) ( ) ( )( ) ( )( ) 2 1 11 4 4 1 4 (7 13 Bx C x xx x x x x A xBx CAx x + + + − −− + + − + ++ = + = Set numerators equal and collect like terms. ( ) ( )2 27 13 4x x A B x C B x A C+ = + + − + − Set coefficients equal to get a system and solve to get constants. 7 13 4 0 4 3 16 A B C B A C A B C + = − = − = = = = An alternate method that sometimes works to find constants. Start with setting numerators equal in previous example : ( ) ( ) ( )2 27 13 4 1x x A x Bx C x+ = + + + − . Chose nice values of x and plug in. For example if 1x = we get 20 5A= which gives 4A = . This won’t always work easily. Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Applications of Integrals Net Area : ( ) b a f x dx∫ represents the net area between ( )f x and the x-axis with area above x-axis positive and area below x-axis negative. Area Between Curves : The general formulas for the two main cases for each are, ( ) upper function lower function b a y f x A dx     = ⇒ = −∫ & ( ) right function left function d c x f y A dy     = ⇒ = −∫ If the curves intersect then the area of each portion must be found individually. Here are some sketches of a couple possible situations and formulas for a couple of possible cases. ( ) ( ) b a A f x g x dx= −∫ ( ) ( ) d c A f y g y dy= −∫ ( ) ( ) ( ) ( ) c b a c A f x g x dx g x f x dx= − + −∫ ∫ Volumes of Revolution : The two main formulas are ( )V A x dx= ∫ and ( )V A y dy= ∫ . Here is some general information about each method of computing and some examples. Rings Cylinders ( ) ( )( )2 2outer radius inner radiusA π= − ( ) ( )radius width / height2A π= Limits: x/y of right/bot ring to x/y of left/top ring Limits : x/y of inner cyl. to x/y of outer cyl. Horz. Axis use ( )f x , ( )g x , ( )A x and dx. Vert. Axis use ( )f y , ( )g y , ( )A y and dy. Horz. Axis use ( )f y , ( )g y , ( )A y and dy. Vert. Axis use ( )f x , ( )g x , ( )A x and dx. Ex. Axis : 0y a= > Ex. Axis : 0y a= ≤ Ex. Axis : 0y a= > Ex. Axis : 0y a= ≤ outer radius : ( )a f x− inner radius : ( )a g x− outer radius: ( )a g x+ inner radius: ( )a f x+ radius : a y− width : ( ) ( )f y g y− radius : a y+ width : ( ) ( )f y g y− These are only a few cases for horizontal axis of rotation. If axis of rotation is the x-axis use the 0y a= ≤ case with 0a = . For vertical axis of rotation ( 0x a= > and 0x a= ≤ ) interchange x and y to get appropriate formulas. Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Work : If a force of ( )F x moves an object in a x b≤ ≤ , the work done is ( ) b a W F x dx= ∫ Average Function Value : The average value of ( )f x on a x b≤ ≤ is ( )1 b avg ab a f f x dx − = ∫ Arc Length Surface Area : Note that this is often a Calc II topic. The three basic formulas are, b a L ds= ∫ 2 b a SA y dsπ= ∫ (rotate about x-axis) 2 b a SA x dsπ= ∫ (rotate about y-axis) where ds is dependent upon the form of the function being worked with as follows. ( ) ( )21 if ,dydxds dx y f x a x b= + = ≤ ≤ ( ) ( )21 if ,dxdyds dy x f y a y b= + = ≤ ≤ ( ) ( ) ( ) ( )22 if , ,dydx dtdtds dt x f t y g t a t b= + = = ≤ ≤ ( ) ( )22 if ,drdds r d r f a bθ θ θ θ= + = ≤ ≤ With surface area you may have to substitute in for the x or y depending on your choice of ds to match the differential in the ds. With parametric and polar you will always need to substitute. Improper Integral An improper integral is an integral with one or more infinite limits and/or discontinuous integrands. Integral is called convergent if the limit exists and has a finite value and divergent if the limit doesn’t exist or has infinite value. This is typically a Calc II topic. Infinite Limit 1. ( ) ( )lim t a at f x dx f x dx →∞ ∞ =∫ ∫ 2. ( ) ( )lim b b tt f x dx f x dx − →−∞∞ =∫ ∫ 3. ( ) ( ) ( ) c c f x dx f x dx f x dx − − ∞ ∞ ∞ ∞ = +∫ ∫ ∫ provided BOTH integrals are convergent. Discontinuous Integrand 1. Discont. at a: ( ) ( )lim b b a tt a f x dx f x dx +→ =∫ ∫ 2. Discont. at b : ( ) ( )lim b t a at b f x dx f x dx −→ =∫ ∫ 3. Discontinuity at a c b< < : ( ) ( ) ( ) b c b a a c f x dx f x dx f x dx= +∫ ∫ ∫ provided both are convergent. Comparison Test for Improper Integrals : If ( ) ( ) 0f x g x≥ ≥ on [ ),a ∞ then, 1. If ( ) a f x dx ∞ ∫ conv. then ( )a g x dx ∞ ∫ conv. 2. If ( )a g x dx ∞ ∫ divg. then ( )a f x dx ∞ ∫ divg. Useful fact : If 0a > then 1 a px dx ∞ ∫ converges if 1p > and diverges for 1p ≤ . Approximating Definite Integrals For given integral ( ) b a f x dx∫ and a n (must be even for Simpson’s Rule) define b anx −∆ = and divide [ ],a b into n subintervals [ ]0 1,x x , [ ]1 2,x x , … , [ ]1, nnx x− with 0x a= and nx b= then, Midpoint Rule : ( ) ( ) ( ) ( )* * *1 2 b na f x dx x f x f x f x ≈ ∆ + + + ∫ L , *ix is midpoint [ ]1, iix x− Trapezoid Rule : ( ) ( ) ( ) ( ) ( ) ( )0 1 2 12 2 22 b n na xf x dx f x f x f x f x f x− ∆ ≈ + + + + + +  ∫ L Simpson’s Rule : ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 2 2 14 2 2 43 b n n na xf x dx f x f x f x f x f x f x− − ∆ ≈ + + + + + +  ∫ L