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Calculus Cheat Sheet All Reduced
Differential & Integral Calculus I (MATH 203)
Concordia University
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Calculus Cheat Sheet Calculus Cheat Sheet Limits Definitions Limit at Infinity : We say lim f x L if we Precise Definition : We say lim f x L if x for every 0 there is a 0 such that whenever 0 x a then f x L . can make f x as close to L as we want Definition : We say lim f x L There is a similar definition for lim f x L if we can make f x as close to L as we want except we require x large and negative. taking x sufficiently close to a (on either side of a) without letting x a . Infinite Limit : We say lim f x if we a Right hand limit : f x L . This has a the same definition as the limit except it requires x a . Left hand limit : f x L . This has the taking x large enough and positive. can make f x arbitrarily large (and positive) taking x sufficiently close to a (on either side of a) without letting x a . There is a similar definition for lim f x a except we make f x arbitrarily large and same definition as the limit except it requires negative. Relationship between the limit and limits lim f x L f x f x L f x f x L lim f x L a a a lim f x f x lim f x Does Not Exist Properties Assume lim f x and lim g x both exist and c is any number then, a a lim f x 1. lim x c lim f x 2. lim f x g x lim f x lim g x a 3. lim f x g x lim f x lim g x x f x a 4. lim provided lim g x 0 a a g g x lim a n n 5. lim f x lim f x a 6. lim n f x n lim f x a Basic Limit Evaluations at Note : sgn a 1 if a 0 and sgn a if a 0 . 1. lim e x 2. lim ln x x lim e x 0 lim ln x b 3. If r 0 then lim r 0 x x 4. If r 0 and x r is real for negative x b then lim r 0 x 5. n even : lim x n 6. n odd : lim x n lim x n 7. n even : lim a x n b x c sgn a 8. n odd : lim a x n b x c sgn a x Evaluation Techniques Rule Continuous Functions f 0 f If f x is continuous at a then lim f x f a If lim or lim then, a g x a g x 0 Continuous Functions and Composition f f lim lim a is a number, or f x is continuous at b and lim g x b then g x x x lim f g x f lim g x f b Polynomials at Infinity p x and q x are polynomials. To compute Factor and Cancel x 2 x 6 x 2 4 x 12 lim lim 2 x2 2 x x x lim 8 2 2 x Rationalize x x x lim 2 lim 2 x 9 x 81 x 9 x 81 x x lim lim 2 x x 3 x x 3 x 6 x 2 3 42 3 42 3x2 4 3 x x lim lim 5 5x 2 x2 2 x 2 5x 2 x x 2 lim Piecewise Function 1 108 Combine Rational Expressions 1 1 x x lim lim h h 0 h x x h factor largest power of x in q x out of both p x and q x then compute limit. lim p q 1 1 lim 2 lim h x x h x x h x x 2 5 if x lim g x where g x x 3 x if x Compute two one sided limits, g x x 2 5 9 x x x x g x 1 3x 7 One sided limits are different so lim g x x exist. If the two one sided limits had been equal then lim g x would have existed x 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. 7. cos x and sin x for all x. 2. Rational function, except for that give 8. tan x and sec x provided division zero. n 3. x (n odd) for all x. x , , , , 2 2 2 2 n x (n even) for all x 0 . 4. 9. cot x and csc x provided x 5. e for all x. x , , , 0, , 6. ln x for x 0 . Intermediate Value Theorem Suppose that f x is continuous on 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 . 9. n odd : lim a x c x d sgn a Visit for a complete set of Calculus notes. n x 2005 Paul Dawkins Visit for a complete set of Calculus notes. 2005 Paul Dawkins Calculus Cheat Sheet Calculus Cheat Sheet Derivatives Definition and Notation f x f . If y f x then the derivative is defined to be f x lim h If y f x all of the following are equivalent equivalent notations for the derivative. df dy d f x f x Df x dx dx dx notations for derivative evaluated at x a . df dy Df a f a x a dx x a dx x a Interpretation of the Derivative 2. f a is the instantaneous rate of 1. m f a is the slope of the tangent change of f x at x a . line to y f x at x a and the 3. If f x is the position of an object at time x then f a is the velocity of equation of the tangent line at x a is given y f a f a x a . 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 3. f g f 4. d 0 dx d n 6. x n x n Power Rule dx d 7. f g x f g x g x dx This is the Chain Rule 5. 2. f g f x g x f g f g Product Rule f g f g Quotient Rule g2 d x a a x ln a dx d x ex dx d 1 ln x x , x 0 dx d 1 ln x x , x 0 dx d 1 log a x x ln a , x 0 dx Higher Order Derivatives The nth Derivative is denoted as The Second Derivative is denoted as 2 d f dn f 2 n f x f x 2 and is defined as f x n and is defined as dx dx f x f x , i. the derivative of the f n x f n x , i. the derivative of n first derivative, f x . the derivative, f x . Implicit Differentiation Find if e 2 x 9 y x3 y 2 sin y 11x . Remember y y x here, so of x and y will use the rule and derivatives of y will use the chain rule. The is to differentiate as normal and every time you differentiate a y you tack on a (from the chain rule). After differentiating solve for . e 2 x 9 y 2 9 3 x 2 y 2 2 x 3 y y cos y 11 2e2 x 9 y 9 2 x y 3x 2 y 2 2 x 3 y cos y 11 3 d csc x csc x cot x dx d cot x csc2 x dx d sin x 1 2 dx x d 1 cos x dx 1 x2 d tan x 1 2 dx 2 x y 9e x Common Derivatives d 1 dx d sin x cos x dx d cos x sin x dx d tan x sec 2 x dx d sec x sec x tan x dx If y f x then all of the following are If y f x then, Chain Rule Variants The chain rule applied to some specific functions. n n d d 5. f x n f x f x cos f x f x sin f x 1. dx dx d f d 2. 6. e f x e f tan f x f x sec 2 f x dx dx d f d 7. sec f ( x ) f x ) sec f ( x ) tan f ( x 3. ln f x dx dx f f d d tan f x 8. 2 4. sin f x f x cos f x dx 1 f x dx 2 y cos y 11 2e2 x 9 y 3x 2 y 2 11 2e 2 x 9 y 3 x 2 y 2 2 x3 y 9e 2 x 9 y cos y Concave Down Critical Points Concave Down x c is a critical point of f x provided either 1. If f x 0 for all x in an interval I then 1. f c 0 or 2. f c exist. f x is concave up on the interval I. 1. If f x 0 for all x in an interval I then f x is increasing on the interval I. 2. If f x 0 for all x in an interval I then f x is decreasing on the interval I. 3. If f x 0 for all x in an interval I then 2. If f x 0 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 . f x is constant on the interval I. Visit for a complete set of Calculus notes. 2005 Paul Dawkins Visit for a complete set of Calculus notes. 2005 Paul Dawkins Calculus Cheat Sheet Calculus Cheat Sheet Integrals Definitions : An of f x Definite Integral: Suppose f x is continuous Standard Integration Techniques Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class. g a f g x g x dx g a f u du on a, . Divide a, into n subintervals of is a function, F x , such that F x f x . u Substitution : The substitution u g x will convert width x and choose from each interval. Indefinite Integral : f x dx F x c du g x dx . For indefinite integrals drop the limits of integration. Then n where F x is an of f x . f x . a f x dx nlim i b i Fundamental Theorem of Calculus Variants of Part I : Part I : If f x is continuous on a, then d x f t dt u x f x g x f t dt is also continuous on a, dx a a d b d x f t dt x f x and g x f t dt f x . dx v x dx a d Part II : f x is continuous on a, , F x is f t dt u x f ( x ) x f v ( x ) dx v x an of f x (i. F x f x dx ) then f x dx F b F a . b Ex. 2 5x 2 cos x3 dx 2 5x b b b a f x g x dx a f x dx a g x dx cf x dx c f x dx , c is a constant a f x dx 0 a cf x dx c a f x dx , c is a constant b a b a b c dx c b a f x dx a f x dx f x dx a b b a a b f x dx a a c If f x 0 on a x b then b b a a f x dx 0 b a If m f x M on a x b then m b a f x dx M b a b a k dx k x c Common Integrals cos u du sin u c tan u du ln sec u c n x dx sin u du cos u c sec u du ln sec u tan u c x n c, n n 1 x dx x dx ln x c a x b dx a ln ax b c ln u du u ln u u c e du e c 1 1 u 1 u sec u du tan u c sec u tan u du sec u c csc u cot udu csc u c csc u du cot u c 2 u a u du a tan a c u 1 a u du sin a c 1 2 1 2 2 2 8 5 1 3 8 3 Integration Parts : u dv uv v du and b a u dv uv b a using cos u du 53 sin u 1 x 1 u 1 1 :: x 2 u 2 8 3 5 3 sin b v du . Choose u and dv from a integral and compute du differentiating u and compute v using v dv . Ex. xe xe dx Ex. dv x dx xe du dx v x e dx xe 5 ln x dx u ln x dv dx du 1x dx v x 5 5 5 ln x dx x ln x 3 dx x ln x x 5 3 5ln 3ln 2 Products and (some) Quotients of Trig Functions For sin n x cos m x dx we have the following : For tan n x sec m x dx we have the following : 1. n odd. Strip 1 sine out and convert rest to 1. cosines using sin 2 x 1 cos 2 x , then use the substitution u cos x . 2. m odd. Strip 1 cosine out and convert rest 2. to sines using cos 2 x 1 sin 2 x , then use the substitution u sin x . 3. n and m both odd. Use either 1. or 2. 3. 4. n and m both even. Use double angle 4. half angle formulas to reduce the integral into a form that can be integrated. Trig Formulas : sin 2 x 2sin x cos x , cos 2 x f x dx f x dx f x dx for any value of c. If f x g x on a x b then f x dx g x dx c cos x3 dx Properties f x g x dx f x dx g x dx b 2 u x 3 du 3x 2 dx x 2 dx 13 du a b b Ex. tan 3 x sec5 x dx sec2 x sec 4 x tan x sec xdx u 2 u 4 du u sec x sec x sec x c 7 sin 5 x cos x dx (sin x ) sin x sin x sin x sin x cos x dx cos x dx cos x dx cos x ) sin x dx u cos x cos x ) du 2u du u u Ex. 3 5 2 4 tan x sec xdx tan x sec x tan x sec xdx 1 7 n odd. Strip 1 tangent and 1 secant out and convert the rest to secants using tan 2 x sec 2 x 1 , then use the substitution u sec x . m even. Strip 2 secants out and convert rest to tangents using sec2 x 1 tan 2 x , then use the substitution u tan x . n odd and m even. Use either 1. or 2. n even and m odd. Each integral will be dealt with differently. 1 1 cos 2 x , sin 2 x 12 cos 2 x 1 5 5 3 5 3 2 4 3 2 3 2 2 3 2 2 2 3 3 4 12 sec2 x 2 ln cos x 12 cos 2 x c 2 Visit for a complete set of Calculus notes. 2005 Paul Dawkins Visit for a complete set of Calculus notes. 2005 Paul Dawkins Calculus Cheat Sheet Calculus Cheat Sheet Trig Substitutions : If the integral contains the following root use the given substitution and formula to convert into an integral involving trig functions. a 2 b 2 x 2 x ab sin b 2 x 2 a 2 x ba sec cos 2 1 sin 2 Ex. 16 2 4 x 2 tan 2 sec 2 1 dx x 23 sin dx 23 cos Recall sec2 1 tan 2 16 4 sin 2 2 cos 9 2 x x . Because we have an indefinite 2 integral assume positive and drop absolute value bars. If we had a definite integral need to compute and remove absolute value bars based on that and, x if x 0 x x if x 0 From this we see that cot 16 2 4 x 2 dx 4 4 x 2 3x 4 x 2 x Net Area : a f x dx represents the net area between f x and the with area above positive and area below negative. Area Between Curves : The general formulas for the two main cases for each are, Use Right Triangle Trig to go back to From substitution we have sin 32x so, 4 9x 2 2 cos . In this case we have 23 cos sin122 12 csc 2 cot c 4 9x 4 4sin 4 cos 2 cos 2 2 a 2 b 2 x 2 x ab tan Applications of Integrals b y f A b a upper function Q x dx where the degree of P x is smaller than the degree of the rational expression. Integrate the partial fraction decomposition (P.F.). For each factor in the denominator we get term(s) in the decomposition according to the following table. Ex. Term in P.F Factor in Q x ax b A ax b ax 2 bx c Ax B ax bx c 7 x 2 x ( x ( x 2 4 ) 7 x 2 x ( x ( x 2 4 ) 2 ax b ax 2 bx c k 7 x 2 x dx dx k Term in P.F Ak A1 A2 k ax b ax b ax b Ak x Bk A1 x B1 k ax 2 bx c ax 2 bx c ( x ( x 2 4 ) 4 x 3x x2 4 3 x x2 4 dx 4 ln x 1 ln x 4 8 tan 3 2 2 A x Bx x2 4 d c right function function dy A f x g x dx b A f y g y dy d A f x g x dx g x f x dx c c b a c 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 2 A outer radius inner radius A radius width height Limits: of ring to of ring Limits : of inner cyl. to of outer cyl. Vert. Axis use f y , Horz. Axis use f y , Vert. Axis use f x , Horz. Axis use f x , g x , A x and dx. g y , A y and dy. g y , A y and dy. g x , A x and dx. Ex. Axis : y a 0 Ex. Axis : y a 0 Ex. Axis : y a 0 Ex. Axis : y a 0 outer radius : a f x outer radius: a g x radius : a y radius : a y inner radius : a g x inner radius: a f x A ( x 2 4) ( Bx C ) ( x ( x ( x 2 4 ) Set numerators equal and collect like terms. 7 x 2 13 x A B x 2 C B x 4 A C dx 16 x2 4 dx x f y A . So, Q x . Factor denominator as completely as possible and find the partial fraction decomposition of Factor in Q x function 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. a Partial Fractions : If integrating x Here is partial fraction form and recombined. Set coefficients equal to get a system and solve to get constants. B 7 C B 13 4A 0 C 16 An alternate method that sometimes works to find constants. Start with setting numerators equal in previous example : 7 x 2 13 x A x 2 4 Bx C x . Chose nice values of x and plug in. For example if x 1 we get 20 5 A which gives A 4 . This always work easily. Visit for a complete set of Calculus notes. 2005 Paul Dawkins width : f y g y width : f y g y These are only a few cases for horizontal axis of rotation. If axis of rotation is the use the y a 0 case with a 0 . For vertical axis of rotation ( x a 0 and x a 0 ) interchange x and y to get appropriate formulas. Visit for a complete set of Calculus notes. 2005 Paul Dawkins
Calculus Cheat Sheet All Reduced
Course: Differential & Integral Calculus I (MATH 203)
University: Concordia University
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