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### 52 Cards in this Set

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 The Taylor Polynomial of order n based at a real number a for functions Rn (look up) When does a function have an inverse? When it's 1-1 » if x=x2 then F(x) = f(x2) How do you find the inverse? Switch x and y » solve for y How do you prove a function is one to one? f(x) is strictly monotomic if it is increasing or decreasing on an interval I. If f(x) is monotomic it is 1-1. What is the relationship between a function and its inverse? the inverse is the unique function with domain equal to the range of f that satisfies the equation: f(f¯¹(x))=x inverse(b)= 1/f(a) Domain and Range of ln(x) Domain:(0, ) Range: all Real numbers Know graph of ln(x) ¡ Derivative of ln(x) 1/x The definition of ln(x) or L(x) L(x)= çdt/t (integral from 1 to x) x>0 Logrithmic Differentiation g(x)=g(x)[g1(x)/g1(x) + g2(x)/g2(x)+. . .g'n(x)/gn(x)] relationship between e^x and ln(x) ln(e^x)=x defininition of the number e e is where L(x)=1 Domain and Range of e^x Domain: all Real numbers Range:(0, ) know graph of e^x __/ derivate/integral of e^x dy/dx e^x = e^x ç e^x = e^x dy/dx Definition of exponential function to the base "a" a function of the form F(x)=p^x Domain and Range of a^x Domain:all real numbers Range:(0,) dy/dx of a^x a^x (ln x) du/dx integral of a^x 1/ln a (a^x) + C dy/dx of x^n where n is a variable set f(x)=x^n use log diff the logarithm of x to the base p log x = ln x/ln p p log base p of p raised to the t = t dy/dx log x a 1/xlnp logarithm to the base e ln=log e restrictions of domain for sine tan sec sin [¨ö¬á, -¨ö¬á] tan (¨ö¬á, -¨ö¬á) sec [0,¨ö¬á)U(-¨ö¬á,¬á] domain and range of inverse sin of x D: [-1,1] R: [¬á¨ö, -¬á¨ö] domain and range of inverse tangent of x D: (-¡Ä, ¡Ä) R: (-¨ö¬á, ¨ö¬á) domain and range of inverse secant of x D:[0, ¨ö¬á) U (¨ö¬á, ¬á] R:(-¡Ä, -1] U [1, ¡Ä) Relation between [r, ө] and (x,y) x=rcosө y=rsinө polar coordinates for a circle with a radius a centered at the origin r=a area of 2 parametric equations A= ¡ò¨ö([p2ө]©÷-[p1ө]©÷)dө Length of a curve (parametric) L=¡ò¡î([x¡¯(t)]©÷+[y¡¯(t)]©÷) Length of a curve (cartesian) L= ¡ò¡î(1+[f¡¯(x)]©÷)dx Length of a curve (polar coordinates) L= ¡ò¡î([p(Ө)]©÷+[p¡¯(Ө)©÷]dӨ Surface Area (parametric) SA=¡ò2¬áy(t)¡î([x¡¯(t)]©÷+[y¡¯(t)]©÷)dt Surface Area (cartesean) SA=¡ò2¬áf(x)¡î(1+[f'(x)]©÷)dx Greatest upper bound highest number a set approaches GUB of (-4,-1]=-1 The limit as n goes to infinity of x^n is ¡Þ x>1 1 x=1 0 -1 or = 1, then diverges Formula for the Partial Sum of a Geometric Series 1/1-x p-series 1/k^p p-series converge? diverge? converge if p>1 Basic Divergence Test If Ak does not got to O, then the sum/series of parital sums diverges harmonic series 1/k Integral Test the sum of f(k) converges if the integral from 1 to infinity of f(x) converges Basic Comparison Test non-negative terms Ak