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Precalculus Functions Anecdotes

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Precalculus Functions Anecdotes

1. Identity Function-This is the only function thats acts on every real number by leaving it alone 2. Square Root Function-Put any positive number into your calculator. Take the square root. Then take the square root again, and so on. Eventually you will always get 1. 3. Squaring Function-The graph of this function, called a parabola, had a reflection property that is useful in making flashlights and satellite dishes. 4. Cubing Function-The origin is called a ?point of inflection? for this curve because the graph changes the curvature at the point. 5. Reciprocal Function-This curve, called a hyperbola, also has a reflection property that is useful in satellite dishes. 6. Natural Log Function-This function increases very slowly. If the x-axis and y-axis

Pre-Calculus Functions Chart

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Pre-Calculus Functions Chart

Sheet1 Formula Name Domain Range Increasing Interval Decreasing Interval Maximum Minimum x-Intercept y-Intercept End-Behavior VA HA Symmetry Continuity 1a(Parent) f(x)=x Identity Function (-?,?) (-?,?) (-?,?) None None None (0,0) (0,0) y-->? None None Odd Continuous 1b f(x)=x-6 None (-?,?) (-?,?) (-?,?) None None None (6,0) (0,-6) y-->? None None Neither Continuous 1c f(x)=x+3 None (-?,?) (-?,?) (-?,?) None None None (-3,0) (0,3) y-->? None None Neither Continuous 1d f(x)=3x None (-?,?) (-?,?) (-?,?) None None None (0,0) (0,0) y-->? None None Neither Continuous 1e f(x)=-x None (-?,?) (-?,?) None (-?,?) None None (0,0) (0,0) y-->-? None None Neither Continuous 1f f(x)=-3x+6 None (-?,?) (-?,?) None (-?,?) None None (2,0) (0,6) y-->-? None None Neither Continuous 1g y=a(x-h) +k

Handbook

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Algebra
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Analytic functions
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Hyperbolic function
Bessel function
Euler's formula
Integration by parts
Differential equation

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Bibliography; Physical Constants 1. Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Arithmetic and Geometric progressions; Convergence of series: the ratio test; Convergence of series: the comparison test; Binomial expansion; Taylor and Maclaurin Series; Power series with real variables; Integer series; Plane wave expansion

AP Calculus Cheat Sheet

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Differentiation rules
Integration by parts
Differential equation
derivative
Quotient Rule
Integration by substitution
Antiderivative
Product rule

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 ( )limx a f x L? = if for every 0e > there is a 0d > such that whenever 0 x a d< - < then ( )f x L e- < . ?Working? Definition : We say ( )limx 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 : ( )limx a f x L+? = . This has the same definition as the limit except it requires x a> . Left hand limit : ( )limx a f x L-? = . This has the same definition as the limit except it requires x a< . Limit at Infinity : We say ( )limx f x L?? = if we

New Work

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MasterMathMentor.com!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!Q#!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Stu Schwartz Function Analysis - Classwork =*!+70!,1'+!,7!&+&(.S)+:!61+5,)7+2!A)&!5&(51(12;!=*!3)3!27!)+!D'*5&(51(12!4.!3*,*'9)+)+:!,-*!S*'72!76!,-*!61+5,)7+ X0-*'*!),!5'722*2!,-*!x"&E)2G!&+3!,-*!2):+!76!,-*!61+5,)7+!4*,0**+!S*'72;!L-&,!:&A*!12!279*!)+67'9&,)7+!&471,!,-* 61+5,)7+!41,!(),,(*!5(1*!&2!,7!),2!&5,1&(!2-&D*;!@&(51(12!0)((!D'7A)3*!,-*!9)22)+:!D)*5*2!,7!,-*!D1SS(*; =*!0)((!3*6)+*!&!61+5,)7+!&2!)+5'*&2)+:!)6/!&2!x!97A*2!,7!,-*!'):-,/!,-*!y"A&(1*!:7*2!1D;!N!61+5,)7+!)2!3*5'*&2)+:!)6/ &2!x!97A*2!,7!,-*!'):-,/!,-*!y"A&(1*!:7*2!370+;!N!61+5,)7+!)2!57+2,&+,!)6/!&2!x!97A*2!,7!,-*!'):-,/!,-*!y"A&(1* 37*2+`,!5-&+:*;

Summary of Trig Inverse Function

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Trigonometry
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Inverse function

Trig Inverse Function Summary Function Meaning Domain Range Quadrants of Found on the (i.e., possible (i.e., possible the Unit Circle calculator by values for x) values for y) from Which Range Values Come j=sin!x y is the angle in the first or [-1, 1] [-nI2, n12] I and IV sin" (X) j=arcsin x fourth quadrant whose sine value is x j=cosir Y is the angle in the first or [-1, 1] [0, n] I and II cos-I(X) y=arccos x second quadrant whose cosine value is x j=tanix y is the angle in the first or (-00, (0) (-nl2, n12) I and IV tan" (X) y=arctanx fourth quadrant whose , tangent value is x

Trig formula sheet

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Inverse function
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Inverse trigonometric functions
education

All Rights Reserved: http://regentsprep.org Algebra 2 ? Things to Remember! Exponents: 0 1x ? 1m m x x ? ? ?m n m nx x x ?? ?( )n m n mx x? m m n n x x x ?? n n n x x y y ? ? ?? ?? ? ( ) ?n n nxy x y? Complex Numbers: 1 i? ? ; 0a i a a? ? ? 2 1i ? ? 14 2 1i i? ? ? divide exponent by 4, use remainder, solve. ( ) conjugate ( )a bi a bi? ? 2 2( )( )a bi a bi a b? ? ? ? 2 2a bi a b? ? ? absolute value=magnitude Logarithms log yby x x b? ? ? ln logex x? natural log e = 2.71828? 10log logx x? common log Change of base formula: log log logb a a b

Ch9 SG

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Calculus
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Law
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Calculus
Mathematical analysis
Mathematics
Ordinary differential equations
Differential calculus
Differentiation rules
Differential equations
Separation of variables
Natural logarithm
derivative
Differential of a function
Chain rule
law

372 CHAPTER 9 Mathematical Modeling with Differential Equations EXERCISE SET 9.1 1. y? = 2x2ex 3/3 = x2y and y(0) = 2 by inspection. 2. y? = x3 ? 2 sinx, y(0) = 3 by inspection. 3. (a) ?rst order; dy dx = c; (1 + x) dy dx = (1 + x)c = y (b) second order; y? = c1 cos t? c2 sin t, y?? + y = ?c1 sin t? c2 cos t+ (c1 sin t+ c2 cos t) = 0 4. (a) ?rst order; 2 dy dx + y = 2 ( ? c 2 e?x/2 + 1 ) + ce?x/2 + x? 3 = x? 1 (b) second order; y? = c1et ? c2e?t, y?? ? y = c1et + c2e?t ? ( c1et + c2e?t ) = 0 5. 1 y dy dx = x dy dx + y, dy dx (1? xy) = y2, dy dx = y2 1? xy 6. 2x+ y2 + 2xy dy dx = 0, by inspection. 7. (a) IF: ? = e3 ? dx = e3x, d dx [ ye3x ] = 0, ye3x = C, y = Ce?3x separation of variables: dy y = ?3dx, ln |y| = ?3x+ C1, y = ?e?3xeC1 = Ce?3x

Ch8 SG

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U2
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Calculus
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Integral calculus
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Natural logarithm
Hyperbolic function
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Integration by substitution
derivative
Integral of secant cubed
Sum rule in integration

317 CHAPTER 8 Principles of Integral Evaluation EXERCISE SET 8.1 1. u = 3? 2x, du = ?2dx, ? 1 2 ? u3 du = ?1 8 u4 + C = ?1 8 (3? 2x)4 + C 2. u = 4 + 9x, du = 9dx, 1 9 ? u1/2 du = 2 3 ? 9u 3/2 + C = 2 27 (4 + 9x)3/2 + C 3. u = x2, du = 2xdx, 1 2 ? sec2 u du = 1 2 tanu+ C = 1 2 tan(x2) + C 4. u = x2, du = 2xdx, 2 ? tanu du = ?2 ln | cosu |+ C = ?2 ln | cos(x2)|+ C 5. u = 2 + cos 3x, du = ?3 sin 3xdx, ? 1 3 ? du u = ?1 3 ln |u|+ C = ?1 3 ln(2 + cos 3x) + C 6. u = 3x 2 , du = 3 2 dx, 2 3 ? du 4 + 4u2 = 1 6 ? du 1 + u2 = 1 6 tan?1 u+ C = 1 6 tan?1(3x/2) + C 7. u = ex, du = exdx, ? sinhu du = coshu+ C = cosh ex + C 8. u = lnx, du = 1 x dx, ? secu tanu du = secu+ C = sec(lnx) + C 9. u = cotx, du = ? csc2 xdx, ? ? eu du = ?eu + C = ?ecot x + C

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