By Eli Maor
One of the issues lined are: Absolute worth, Binomial Theorem, spinoff, Exponential progress, maximum integer functionality, Implicit differentiation, Integration, legislations of Cosines, Linear functionality, Parabola, Polynomial capabilities, moment spinoff, speed. Biographies comprise: Jakob Bernoulli (1654-1705) Rene Descartes (1596-1650) Eudoxus of Cnidus (ca. 408-355 BCE) Jean Baptiste Joseph Fourier (1768-1830) Carl Friedrich Gauss (1777-1855) Sir Isaac Newton (1642-1727) Blaise Pascal (1623-1662) Francois Viete (1540-1603) John Wallis (1616-1703).
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Extra info for The Facts on File Calculus Handbook
Example text
If the characteristic equation has repeated roots, for example a double solution r, then the solution of the differential equation is a linear combination of the functions erx and xerx. For example, the differential equation y″ – 4y + 4 = 0 has the characteristic equation r2 – 4r + 4 = 0, which has the double root r = 2. The general solution of the differential equation is y = Ae2x + Bxe2x = (A + Bx)e2x. See also DIFFERENTIAL EQUATION; LINEAR COMBINATION. chaos – characteristic equation GLOSSARY 17 GLOSSARY circle, general equation of – comparison tests for proper integrals circle, general equation of The equation Ax2 + Ay2 + Bx + Cy + D = 0 represents a circle; depending on the values of the coefficients, this circle can be real, imaginary, or degenerate (a single point).
See also IMPLICIT DIFFERENTIATION. identities, trigonometric – implicit function GLOSSARY 49 GLOSSARY improper integral – index of summation b improper integral A definite integral a ∫ f(x) dx whose interval of integration (a, b) is unbounded (that is, either a or b or both are at infinity), or whose integrand f(x) has a finite number of infinite discontinuities on (a, b). Such an integral can only be evaluated as a limit. For example, ∞ –x b e dx = lim [0 ∫ e–x dx] = lim [1 – e–b] = 1, because lim e–b = 0 for 0∫ b→∞ b→∞ 1 b→∞ b > 0.
As an example, consider the function y = f(x) = x2. Its derivative is lim [(x + h)2 – x2]/h. Of course, we cannot simply substitute h = 0 h→0 in this limit, because this will give us the indeterminate expression 0/0. We go around this by first simplifying the expression inside the limit: [(x + h)2 – x2]/h = [x2 + 2xh + h2 – x2]/h = (2xh + h2)/h = h(2x + h)/h = 2x + h. Now we let h → 0, resulting in the expression 2x. Thus f′(x) = 2x. The derivative can be interpreted in two ways: as the rate of change of the independent variable y with respect to the dependent variable x, or as the slope of the tangent line to the graph of y = f(x) at the point x.