By Eli Maor

He proof On dossier Calculus instruction manual brings the basics of calculus to scholars in a simple and obtainable layout. an excellent primer for the topic, this e-book takes all of the fundamentals of calculus and is helping scholars comprehend and practice them of their paintings.

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Thus the set of real numbers (and also the set of imaginary numbers) is a subset of the set of complex numbers. See also POLAR FORM OF A COMPLEX NUMBER. composite function A combination of two or more functions so that the output of one function is the input to the other. Symbolically, if y = f(u) and u = g(x), then y = f(g(x)) is the composition of g and ƒ (in that order). — For example, the function y = √1 + x can be regarded as a composition of the functions u = g(x) = 1 + x and y = f(u) = √– u.

For example, the expression y = xx, defined for x > 0, takes the form 00 when x = 0, so we take the natural logarithm of both sides and get ln y = x ln x = (ln x)/(1/x), which takes the form (–∞)/∞ at x = 0 (it can be shown that lim xx = 1 x→0 as x → 0 through positive values). See also L’HOSPITAL’S RULE. n Σ ai. The choice of n n letter for the summation index is immaterial; that is, Σ ai= Σ aj; for i=1 j=1 index of summation The subscript i in the expression GLOSSARY 50 improper integral – index of summation i=1 inequality, triangle – inflection point GLOSSARY this reason, the summation index is sometimes called a dummy index.

See also LOGARITHMIC FUNCTION. exponential function GLOSSARY 35 GLOSSARY exponential growth – family of curves Exponential function: (a) growing; (b) decaying y y y = cekx c > 0, k >0 y = ce kx c > 0, k< 0 c 0 c x (a) y Exponential growth (b) exponential growth A term applied to a variable y that increases (often with time) according to the formula y = y0 ekt, where y0 is the initial value of y (the value when t = 0), and k is a positive constant. For example, the growth of a population with time is approximately exponential, from which the popular phrase “exponential growth” comes.