The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how the position of the object changes when time advances; the derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is the best linear approximation to the graph of the original function.
The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector; the process of finding a derivative is called differentiation. The reverse process is called antidifferentiation; the fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative; the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x. If x and y are real numbers, if the graph of f is plotted against x, the derivative is the slope of this graph at each point.
The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y is a line. In this case, y = f = mx + b, for real numbers m and b, the slope m is given by m = change in y change in x = Δ y Δ x, where the symbol Δ is an abbreviation for "change in", the combinations Δ x and Δ y refer to corresponding changes, i.e.: Δ y = f − f. The above formula holds because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x, thus Δ y = m Δ x. This gives the value for the slope of a line. If the function f is not linear the change in y divided by the change in x varies over the considered range: differentiation is a method to find a unique value for this rate of change, not across a certain range, but at any given value of x; the idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences Δy / Δx as Δx becomes infinitely small. Two distinct notations are used for the derivative, one deriving from Leibniz and the other from Joseph Louis Lagrange.
In Leibniz's notation, an infinitesimal change in x is denoted by dx, the derivative of y with respect to x is written d y d x suggesting the ratio of two infinitesimal quantities. In Lagrange's notation, the derivative with respect to x of a function f is denoted f' or fx′, in case of ambiguity of the variable implied by the differentiation. Lagrange's notation is sometimes incorrectly attributed to Newton; the most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit of difference quotients of real numbers. This is the approach described below. Let f be a real valued function defined in an open neighborhood of a real number a. In classical geometry, the tangent
Wolf Hall is a historical novel by English author Hilary Mantel, published by Fourth Estate, named after the Seymour family’s seat of Wolfhall or Wulfhall in Wiltshire. Set in the period from 1500 to 1535, Wolf Hall is a sympathetic fictionalised biography documenting the rapid rise to power of Thomas Cromwell in the court of Henry VIII through to the death of Sir Thomas More; the novel won both the National Book Critics Circle Award. In 2012, The Observer named it as one of "The 10 best historical novels"; the book is the first in a trilogy. The last book in the trilogy will be called The Mirror and the Light and is expected to cover the last four years of Cromwell's life. Born to a working-class family of no position or name, Cromwell rose to become the right-hand man of Cardinal Thomas Wolsey, adviser to the King, he survived Wolsey's fall from grace to take his place as the most powerful of Henry's ministers. In that role, he observed turning points of English history, as Henry asserted his authority to declare his marriage annulled from Catherine of Aragon, married Anne Boleyn, broke from Rome, established the independence of the Church of England, called for the dissolution of the monasteries.
The novel is a reenvisioning of literary records. Mantel's novel offers an alternative to that characterization, a more intimate portrait of Cromwell as a tolerant and talented man attempting to serve king and family amid the political machinations of Henry's court and the religious upheavals of the Reformation, in contrast to More's viciously punitive adherence to the old Roman Catholic order that Henry is sweeping away. Mantel said she spent five years researching and writing the book, trying to match her fiction to the historical record. To avoid contradicting history, she created a card catalogue, organised alphabetically by character, with each card containing notes indicating where a particular historical figure was on relevant dates. "You need to know, where is the Duke of Suffolk at the moment? You can't have him in London if he's supposed to be somewhere else", she explained. In an interview with The Guardian, Mantel stated her aim to place the reader in "that time and that place, putting you into Henry's entourage.
The essence of the thing is not to judge with hindsight, not to pass judgment from the lofty perch of the 21st century when we know what happened. It's to be there with them in that hunting party at Wolf Hall, moving forward with imperfect information and wrong expectations, but in any case moving forward into a future, not predetermined, but where chance and hazard will play a terrific role." Wolf Hall includes a large cast of fictionalised historical persons. In addition to those mentioned, prominent characters include: Stephen Gardiner, Master Secretary to King Henry Princess Mary, the daughter and only surviving child of Henry and Catherine Queen Mary I of England. Mary Boleyn, sister of Anne Thomas Boleyn, father of Anne and Mary Thomas Howard, 3rd Duke of Norfolk, Anne's uncle Thomas Cranmer, Archbishop of Canterbury Jane Seymour, who became the third of Henry's six wives Rafe Sadler, Thomas Cromwell's ward The title comes from the name of the Seymour family seat at Wolfhall or Wulfhall in Wiltshire.
Wolf Hall succeeds on its own terms and some, both as a non-frothy historical novel and as a display of Mantel's extraordinary talent. Lyrically yet cleanly and written, solidly imagined yet filled with spooky resonances, funny at times, it's not like much else in contemporary British fiction. A sequel is in the works, it's not the least of Mantel's achievements that the reader finishes this 650-page book wanting more....dreadfully badly written... Mantel wrote. I have yet to meet anyone outside the Booker panel who managed to get to the end of this tedious tome. God forbid. Over two decades, she has gained a reputation as an elegant anatomiser of cruelty. From the French Revolution of A Place of Greater Safety to the Middle England of Beyond Black, hers are scrupulously moral – and scrupulously unmoralistic – books that refuse to shy away from the underside of life, finding in disaster a kind of bleak and unconsoling humour, it is that supple movement between laughter and horror that makes this rich pageant of Tudor life her most humane and bewitching novel.... as soon as I opened the book I was gripped.
I read it non-stop. When I did have to put it down, I was full of a regret I still feel; this is a wonderful and intelligently imagined retelling of a familiar tale from an unfamiliar angle – one that makes the drama unfolding nearly five centuries ago look new again, shocking again, too. In a poll of literary experts by the Independent Bath Literature Festival, Wolf Hall was voted the greatest novel from 1995-2015, it ranked third in a BBC Culture poll of the best novels since 2000. In 2019, Wolf Hall was ranked first in The Guardian's list of the 100 best books of the 21st century. Winner – 2009 Man Booker Prize. James Naughtie, the chairman of the Booker prize judges, said the decision to give Wolf Hall the award was "based on the sheer bigness of the book; the boldness of its narrative, its scene setting... The extraordinary way that Hilary Mantel h
Janis Carter was a film and television actress working in the 1940s and 1950s. Carter was born Janis Elinore Dremann in Ohio, she changed her last name because people had trouble pronouncing it and spelling it, choosing her grandmother's maiden name as her new last name. After initial training as a pianist, Carter changed to singing, her elementary and secondary education was provided by schools in Ohio. After that, she attended Western Reserve University, graduating with two degrees — bachelor of arts and bachelor of music, she participated in dramatics in college. After attending Mather College in Cleveland, Carter headed to New York in an attempt to start a career in opera. Although that goal was unsuccessful, when she was subsequently working on Broadway she was spotted on stage by Darryl F. Zanuck, who signed her to a movie deal, her Broadway credits included Du Barry Was Virginia. After moving to Hollywood, she appeared in over 30 films beginning in 1941 for 20th Century Fox, MGM, RKO, she appeared in the films Night Editor and Framed with Glenn Ford, Flying Leathernecks with John Wayne.
After leaving Los Angeles, Carter returned to New York and found work in television in comedies and dramas and as hostess for the quiz show Feather Your Nest opposite Bud Collyer. She was married to Carl Prager, a musician and composer, from 1942 to 1951. In 1956, she married Julius Stulman, a New York lumber and shipping tycoon, quit show business, they remained together until her death. She died on July 30, 1994, aged 80, in Durham, North Carolina. Janis Carter on IMDb Janis Carter at the Internet Broadway Database Janis Carter at Find a Grave Photos of Janis Carter in'A Thousand and One Nights' by Ned Scott