Misplaced Pages

#769230

122-404: Meccano Magazine was an English monthly hobby magazine published by Meccano Ltd between 1916 and 1963, and by other publishers between 1963 and 1981. The magazine was initially created for Meccano builders, but it soon became a general hobby magazine aimed at "boys of all ages". The magazine was launched by Frank Hornby , the inventor of Meccano , as a bi-monthly publication in 1916 in

244-601: A {\displaystyle a} can be denoted ⁠ f ′ ( a ) {\displaystyle f'(a)} ⁠ , read as " ⁠ f {\displaystyle f} ⁠ prime of ⁠ a {\displaystyle a} ⁠ "; or it can be denoted ⁠ d f d x ( a ) {\displaystyle \textstyle {\frac {df}{dx}}(a)} ⁠ , read as "the derivative of f {\displaystyle f} with respect to x {\displaystyle x} at ⁠

366-400: A {\displaystyle a} ⁠ " or " ⁠ d f {\displaystyle df} ⁠ by (or over) d x {\displaystyle dx} at ⁠ a {\displaystyle a} ⁠ ". See § Notation below. If f {\displaystyle f} is a function that has a derivative at every point in its domain , then

488-404: A {\displaystyle a} ⁠ , and returns a different value 10 for all x {\displaystyle x} greater than or equal to a {\displaystyle a} . The function f {\displaystyle f} cannot have a derivative at a {\displaystyle a} . If h {\displaystyle h} is negative, then

610-540: A ) v {\displaystyle f'(\mathbf {a} )\mathbf {v} } is the directional derivative of f {\displaystyle f} in the direction ⁠ v {\displaystyle \mathbf {v} } ⁠ . If f {\displaystyle f} is written using coordinate functions, so that ⁠ f = ( f 1 , f 2 , … , f m ) {\displaystyle f=(f_{1},f_{2},\dots ,f_{m})} ⁠ , then

732-437: A + h ) − ( f ( a ) + f ′ ( a ) h ) ‖ ‖ h ‖ = 0. {\displaystyle \lim _{\mathbf {h} \to 0}{\frac {\lVert f(\mathbf {a} +\mathbf {h} )-(f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {h} )\rVert }{\lVert \mathbf {h} \rVert }}=0.} Here h {\displaystyle \mathbf {h} }

854-424: A + v ) ≈ f ( a ) + f ′ ( a ) v . {\displaystyle f(\mathbf {a} +\mathbf {v} )\approx f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {v} .} Similarly with the single-variable derivative, f ′ ( a ) {\displaystyle f'(\mathbf {a} )} is chosen so that the error in this approximation

976-576: A 1 , … , a n ) = lim h → 0 f ( a 1 , … , a i + h , … , a n ) − f ( a 1 , … , a i , … , a n ) h . {\displaystyle {\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i}+h,\ldots ,a_{n})-f(a_{1},\ldots ,a_{i},\ldots ,a_{n})}{h}}.} This

1098-476: A n ) {\displaystyle (a_{1},\dots ,a_{n})} to the vector ∇ f ( a 1 , … , a n ) {\displaystyle \nabla f(a_{1},\dots ,a_{n})} . Consequently, the gradient determines a vector field . If f {\displaystyle f} is a real-valued function on ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ , then

1220-403: A n ) , … , ∂ f ∂ x n ( a 1 , … , a n ) ) , {\displaystyle \nabla f(a_{1},\ldots ,a_{n})=\left({\frac {\partial f}{\partial x_{1}}}(a_{1},\ldots ,a_{n}),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a_{1},\ldots ,a_{n})\right),} which is called

1342-408: A ) h = ( a + h ) 2 − a 2 h = a 2 + 2 a h + h 2 − a 2 h = 2 a + h . {\displaystyle {\frac {f(a+h)-f(a)}{h}}={\frac {(a+h)^{2}-a^{2}}{h}}={\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}=2a+h.} The division in the last step

SECTION 10

#1732858812770

1464-434: A + h {\displaystyle a+h} is on the low part of the step, so the secant line from a {\displaystyle a} to a + h {\displaystyle a+h} is very steep; as h {\displaystyle h} tends to zero, the slope tends to infinity. If h {\displaystyle h} is positive, then a + h {\displaystyle a+h}

1586-423: A + h ) {\displaystyle f(a+h)} is defined, and | L − f ( a + h ) − f ( a ) h | < ε , {\displaystyle \left|L-{\frac {f(a+h)-f(a)}{h}}\right|<\varepsilon ,} where the vertical bars denote the absolute value . This is an example of the (ε, δ)-definition of limit . If

1708-642: A + h ) − f ( a ) h {\displaystyle L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}} exists. This means that, for every positive real number ⁠ ε {\displaystyle \varepsilon } ⁠ , there exists a positive real number δ {\displaystyle \delta } such that, for every h {\displaystyle h} such that | h | < δ {\displaystyle |h|<\delta } and h ≠ 0 {\displaystyle h\neq 0} then f (

1830-402: A complete picture of the behavior of f {\displaystyle f} . The total derivative gives a complete picture by considering all directions at once. That is, for any vector v {\displaystyle \mathbf {v} } starting at ⁠ a {\displaystyle \mathbf {a} } ⁠ , the linear approximation formula holds: f (

1952-536: A costume based on an already existing creative property), creating models out of card stock or paper – called papercraft . Many of these fall under the category visual arts . Writing is often taken up as a hobby by aspiring writers and usually appears in the form of personal blog , guest posting or fan fiction (literary art resulting in creation of written content based on already existing, licensed creative property under specified terms). Reading books, eBooks , magazines, comics, or newspapers, along with browsing

2074-445: A derivative at most, but not all, points of its domain. The function whose value at a {\displaystyle a} equals f ′ ( a ) {\displaystyle f'(a)} whenever f ′ ( a ) {\displaystyle f'(a)} is defined and elsewhere is undefined is also called the derivative of ⁠ f {\displaystyle f} ⁠ . It

2196-404: A derivative. Most functions that occur in practice have derivatives at all points or almost every point. Early in the history of calculus , many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions (for example, if the function is a monotone or a Lipschitz function ), this is true. However, in 1872, Weierstrass found the first example of

2318-723: A framework which distinguishes the terms in a useful categorization of leisure in which casual leisure is separated from serious Leisure . He describes serious leisure as undertaken by amateurs , hobbyists and volunteers . Amateurs engage in pursuits that have a professional counterpart, such as playing an instrument or astronomy . Hobbyists engage in five broad types of activity: collecting , making and tinkering (like embroidery and car restoration), activity participation (like fishing and singing), sports and games , and liberal-arts hobbies (like languages, cuisine, literature). Volunteers commit to organizations where they work as guides, counsellors, gardeners and so on. The separation of

2440-511: A function can be defined by mapping every point x {\displaystyle x} to the value of the derivative of f {\displaystyle f} at x {\displaystyle x} . This function is written f ′ {\displaystyle f'} and is called the derivative function or the derivative of ⁠ f {\displaystyle f} ⁠ . The function f {\displaystyle f} sometimes has

2562-419: A function of ⁠ t {\displaystyle t} ⁠ , then the first and second derivatives can be written as y ˙ {\displaystyle {\dot {y}}} and ⁠ y ¨ {\displaystyle {\ddot {y}}} ⁠ , respectively. This notation is used exclusively for derivatives with respect to time or arc length . It

SECTION 20

#1732858812770

2684-409: A function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in vector calculus and differential geometry . As with ordinary derivatives, multiple notations exist: the partial derivative of a function f ( x , y , … ) {\displaystyle f(x,y,\dots )} with respect to

2806-468: A function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function . In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point. One common way of writing

2928-437: A function with a smooth graph is not differentiable at a point where its tangent is vertical : For instance, the function given by f ( x ) = x 1 / 3 {\displaystyle f(x)=x^{1/3}} is not differentiable at x = 0 {\displaystyle x=0} . In summary, a function that has a derivative is continuous, but there are continuous functions that do not have

3050-404: A logical manner. The following categorization of hobbies was developed by Stebbins. Collecting includes seeking, locating, acquiring, organizing, cataloging, displaying and storing. Collecting is appealing to many people due to their interest in a particular subject and a desire to categorize and make order out of complexity. Some collectors are generalists, accumulating items from countries of

3172-413: A long period of time. Making and Tinkering hobbies include higher-end projects, such as building or restoring a car or building a computer from individual parts, like CPUs and SSDs. For computer savvy do-it-yourself hobbyists, CNC (Computer Numerical Control) machining may also be popular. A CNC machine can be assembled and programmed to make different parts from wood or metal. Tinkering is 'dabbling' with

3294-468: A real horse . By 1816 the derivative , "hobby", was introduced into the vocabulary of a number of English people. Over the course of subsequent centuries, the term came to be associated with recreation and leisure . In the 17th century, the term was used in a pejorative sense by suggesting that a hobby was a childish pursuit, however, in the 18th century with more industrial society and more leisure time, hobbies took on greater respectability. A hobby

3416-586: A real-life object in a smaller scale and dates back to prehistoric times with small clay "dolls" and other children's toys that have been found near known populated areas. Some of the earliest scale models of residences were found in Cucuteni–Trypillia culture in Eastern Europe. These artifacts were dated to be around 3000–6000 BC. Similar models dating back to the same period were found in ancient Egypt, India, China and Mesopotamia archaeological sites. At

3538-420: A small sample. derivative In mathematics , the derivative is a fundamental tool that quantifies the sensitivity of change of a function 's output with respect to its input. 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

3660-504: A space referred to as the garden. Although a garden typically is located on the land near a residence, it may also be located on a roof , in an atrium , on a balcony , in a windowbox , or on a patio or vivarium . Gardening also takes place in non-residential green areas, such as parks, public or semi-public gardens ( botanical gardens or zoological gardens ), amusement and theme parks , along transportation corridors, and around tourist attractions and hotels . In these situations,

3782-464: A staff of gardeners or groundskeepers maintains the gardens. Indoor gardening is concerned with growing houseplants within a residence or building, in a conservatory , or in a greenhouse . Indoor gardens are sometimes incorporated into air conditioning or heating systems. Water gardening is concerned with growing plants that have adapted to pools and ponds, along with aqua-scaping in planted aquariums . Bog gardens are also considered

Meccano Magazine - Misplaced Pages Continue

3904-520: A system of rules for manipulating infinitesimal quantities is required. The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals are an extension of the real numbers that contain numbers greater than anything of the form 1 + 1 + ⋯ + 1 {\displaystyle 1+1+\cdots +1} for any finite number of terms. Such numbers are infinite, and their reciprocals are infinitesimals. The application of hyperreal numbers to

4026-406: A traditional hobby, such as collecting , to flourish and support trading in a new environment. Hobbyists are a part of a wider group of people engaged in leisure pursuits where the boundaries of each group overlap to some extent. The Serious Leisure Perspective groups hobbyists with amateurs and volunteers and identifies three broad groups of leisure activity with hobbies being found mainly in

4148-722: A type of water garden. A simple water garden may consist solely of a tub containing the water and plants. Container gardening is concerned with growing plants in containers that are placed above the ground. Many hobbies involve performances by the hobbyist, such as singing , acting, juggling , magic , dancing, playing a musical instrument , martial arts , and other performing arts . Some hobbies may result in an end product. Examples of this would be woodworking , photography , moviemaking , jewelry making , software projects such as Photoshopping and home music or video production , making bracelets , artistic projects such as drawing , painting , Cosplay (design, creation, and wearing

4270-1002: Is f ′ ( x ) = 4 x ( 4 − 1 ) + d ( x 2 ) d x cos ⁡ ( x 2 ) − d ( ln ⁡ x ) d x e x − ln ⁡ ( x ) d ( e x ) d x + 0 = 4 x 3 + 2 x cos ⁡ ( x 2 ) − 1 x e x − ln ⁡ ( x ) e x . {\displaystyle {\begin{aligned}f'(x)&=4x^{(4-1)}+{\frac {d\left(x^{2}\right)}{dx}}\cos \left(x^{2}\right)-{\frac {d\left(\ln {x}\right)}{dx}}e^{x}-\ln(x){\frac {d\left(e^{x}\right)}{dx}}+0\\&=4x^{3}+2x\cos \left(x^{2}\right)-{\frac {1}{x}}e^{x}-\ln(x)e^{x}.\end{aligned}}} Here

4392-464: Is differentiable at ⁠ a {\displaystyle a} ⁠ , then f {\displaystyle f} must also be continuous at a {\displaystyle a} . As an example, choose a point a {\displaystyle a} and let f {\displaystyle f} be the step function that returns the value 1 for all x {\displaystyle x} less than ⁠

4514-482: Is a demanding hobby that requires a multitude of large and expensive machine tools , such as lathes and mills . This hobby originated in the United Kingdom in the late 19th century, later spreading and flourishing in the mid-20th century. Due to the expense and space required, it is becoming rare. 3D Printing is a relatively new technology and already a major hobby as the cost of printers has fallen sharply. It

4636-571: Is a function from an open subset of R n {\displaystyle \mathbb {R} ^{n}} to ⁠ R m {\displaystyle \mathbb {R} ^{m}} ⁠ , then the directional derivative of f {\displaystyle f} in a chosen direction is the best linear approximation to f {\displaystyle f} at that point and in that direction. However, when ⁠ n > 1 {\displaystyle n>1} ⁠ , no single directional derivative can give

4758-460: Is a good example of how hobbyists quickly engage with new technologies, communicate with one another and become producers related to their former hobby. 3D modeling is the process of making mathematical representations of three dimensional items and is an aspect of 3D printing. Dressmaking has been a major hobby up until the late 20th century, in order to make cheap clothes, but also as a creative design and craft challenge. It has been reduced by

4880-439: Is a healthy trade in them at auctions . Hobby Hobbyists may be identified under three sub-categories: casual leisure which is intrinsically rewarding, short-lived, pleasurable activity requiring little or no preparation, serious leisure which is the systematic pursuit of an amateur, hobbyist, or volunteer that is substantial, rewarding and results in a sense of accomplishment, and finally project-based leisure which

5002-422: Is a short-term, often one-off, project that is rewarding. In the 16th century, the term "hobby" had the meaning of "small horse and pony". The term " hobby horse " was documented in a 1557 payment confirmation for a "Hobbyhorse" from Reading, England. The item, originally called a "Tourney Horse", was made of a wooden or basketwork frame with an artificial tail and head. It was designed for a child to mimic riding

Meccano Magazine - Misplaced Pages Continue

5124-485: Is a vector in ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ , so the norm in the denominator is the standard length on R n {\displaystyle \mathbb {R} ^{n}} . However, f ′ ( a ) h {\displaystyle f'(\mathbf {a} )\mathbf {h} } is a vector in ⁠ R m {\displaystyle \mathbb {R} ^{m}} ⁠ , and

5246-660: Is also called a pastime , derived from the use of hobbies to pass the time . A hobby became an activity that is practiced regularly and usually with some worthwhile purpose. Hobbies are usually, but not always, practiced primarily for interest and enjoyment, rather than financial reward. Prior to the mid-19th century, hobbies were generally considered as an obsession, childish or trivial, with negative connotations. However, as early as 1676 Sir Matthew Hale, in Contemplations Moral and Divine , wrote "Almost every person hath some hobby horse or other wherein he prides himself." He

5368-524: Is an important role in being in touch with fellow hobbyists. Some hobbies are of communal nature, like choral singing and volunteering. People who engage in hobbies have an interest in and time to pursue them. Children have been an important group of hobbyists because they are enthusiastic for collecting, making and exploring, in addition to this they have the leisure time that allows them to pursue those hobbies. The growth in hobbies occurred during industrialization which gave workers set time for leisure. During

5490-492: Is as small as possible. The total derivative of f {\displaystyle f} at a {\displaystyle \mathbf {a} } is the unique linear transformation f ′ ( a ) : R n → R m {\displaystyle f'(\mathbf {a} )\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}} such that lim h → 0 ‖ f (

5612-655: Is by using the prime mark in the symbol of a function ⁠ f ( x ) {\displaystyle f(x)} ⁠ . This is known as prime notation , due to Joseph-Louis Lagrange . The first derivative is written as ⁠ f ′ ( x ) {\displaystyle f'(x)} ⁠ , read as " ⁠ f {\displaystyle f} ⁠ prime of ⁠ x {\displaystyle x} ⁠ , or ⁠ y ′ {\displaystyle y'} ⁠ , read as " ⁠ y {\displaystyle y} ⁠ prime". Similarly,

5734-432: Is called k {\displaystyle k} times differentiable . If the k {\displaystyle k} - th derivative is continuous, then the function is said to be of differentiability class ⁠ C k {\displaystyle C^{k}} ⁠ . A function that has infinitely many derivatives is called infinitely differentiable or smooth . Any polynomial function

5856-414: Is fundamental for the study of the functions of several real variables . Let f ( x 1 , … , x n ) {\displaystyle f(x_{1},\dots ,x_{n})} be such a real-valued function . If all partial derivatives f {\displaystyle f} with respect to x j {\displaystyle x_{j}} are defined at

5978-422: Is infinitely differentiable; taking derivatives repeatedly will eventually result in a constant function , and all subsequent derivatives of that function are zero. One application of higher-order derivatives is in physics . Suppose that a function represents the position of an object at the time. The first derivative of that function is the velocity of an object with respect to time, the second derivative of

6100-403: Is on the high part of the step, so the secant line from a {\displaystyle a} to a + h {\displaystyle a+h} has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. For example,

6222-418: Is one; if h {\displaystyle h} is negative, then the slope of the secant line from 0 {\displaystyle 0} to h {\displaystyle h} is ⁠ − 1 {\displaystyle -1} ⁠ . This can be seen graphically as a "kink" or a "cusp" in the graph at x = 0 {\displaystyle x=0} . Even

SECTION 50

#1732858812770

6344-469: Is represented as the ratio of two differentials , whereas prime notation is written by adding a prime mark . Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. The higher order derivatives can be applied in physics; for example, while the first derivative of

6466-505: Is socially valorized precisely because it produces feelings of satisfaction with something that looks very much like work but that is done of its own sake." "Hobbies are a contradiction: they take work and turn it into leisure, and take leisure and turn it into work." A 2018 study using survey results identified the term "hobby" to most accurately describe activities associated with making or collecting objects, especially when done alone. Cultural trends related to hobbies change with time. In

6588-426: Is still a function, but its domain may be smaller than the domain of f {\displaystyle f} . For example, let f {\displaystyle f} be the squaring function: f ( x ) = x 2 {\displaystyle f(x)=x^{2}} . Then the quotient in the definition of the derivative is f ( a + h ) − f (

6710-446: Is the second derivative , denoted as ⁠ f ″ {\displaystyle f''} ⁠ , and the derivative of f ″ {\displaystyle f''} is the third derivative , denoted as ⁠ f ‴ {\displaystyle f'''} ⁠ . By continuing this process, if it exists, the ⁠ n {\displaystyle n} ⁠ th derivative

6832-548: Is the derivative of the ⁠ ( n − 1 ) {\displaystyle (n-1)} ⁠ th derivative or the derivative of order ⁠ n {\displaystyle n} ⁠ . As has been discussed above , the generalization of derivative of a function f {\displaystyle f} may be denoted as ⁠ f ( n ) {\displaystyle f^{(n)}} ⁠ . A function that has k {\displaystyle k} successive derivatives

6954-414: Is typically used in differential equations in physics and differential geometry . However, the dot notation becomes unmanageable for high-order derivatives (of order 4 or more) and cannot deal with multiple independent variables. Another notation is D-notation , which represents the differential operator by the symbol ⁠ D {\displaystyle D} ⁠ . The first derivative

7076-393: Is valid as long as h ≠ 0 {\displaystyle h\neq 0} . The closer h {\displaystyle h} is to ⁠ 0 {\displaystyle 0} ⁠ , the closer this expression becomes to the value 2 a {\displaystyle 2a} . The limit exists, and for every input a {\displaystyle a}

7198-444: Is viewed as a functional relationship between dependent and independent variables . The first derivative is denoted by ⁠ d y d x {\displaystyle \textstyle {\frac {dy}{dx}}} ⁠ , read as "the derivative of y {\displaystyle y} with respect to ⁠ x {\displaystyle x} ⁠ ". This derivative can alternately be treated as

7320-421: Is written D f ( x ) {\displaystyle Df(x)} and higher derivatives are written with a superscript, so the n {\displaystyle n} -th derivative is ⁠ D n f ( x ) {\displaystyle D^{n}f(x)} ⁠ . This notation is sometimes called Euler notation , although it seems that Leonhard Euler did not use it, and

7442-600: The n {\displaystyle n} -th derivative of y = f ( x ) {\displaystyle y=f(x)} . These are abbreviations for multiple applications of the derivative operator; for example, d 2 y d x 2 = d d x ( d d x f ( x ) ) . {\textstyle {\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}{\Bigl (}{\frac {d}{dx}}f(x){\Bigr )}.} Unlike some alternatives, Leibniz notation involves explicit specification of

SECTION 60

#1732858812770

7564-586: The x {\displaystyle x} and y {\displaystyle y} direction. However, they do not directly measure the variation of f {\displaystyle f} in any other direction, such as along the diagonal line ⁠ y = x {\displaystyle y=x} ⁠ . These are measured using directional derivatives. Given a vector ⁠ v = ( v 1 , … , v n ) {\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})} ⁠ , then

7686-406: The absolute value function given by f ( x ) = | x | {\displaystyle f(x)=|x|} is continuous at ⁠ x = 0 {\displaystyle x=0} ⁠ , but it is not differentiable there. If h {\displaystyle h} is positive, then the slope of the secant line from 0 to h {\displaystyle h}

7808-595: The directional derivative of f {\displaystyle f} in the direction of v {\displaystyle \mathbf {v} } at the point x {\displaystyle \mathbf {x} } is: D v f ( x ) = lim h → 0 f ( x + h v ) − f ( x ) h . {\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\rightarrow 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.} If all

7930-399: The gradient of f {\displaystyle f} at a {\displaystyle a} . If f {\displaystyle f} is differentiable at every point in some domain, then the gradient is a vector-valued function ∇ f {\displaystyle \nabla f} that maps the point ( a 1 , … ,

8052-417: The gradient vector . A function of a real variable f ( x ) {\displaystyle f(x)} is differentiable at a point a {\displaystyle a} of its domain , if its domain contains an open interval containing ⁠ a {\displaystyle a} ⁠ , and the limit L = lim h → 0 f (

8174-567: The pushforward of v {\displaystyle \mathbf {v} } by f {\displaystyle f} . If the total derivative exists at ⁠ a {\displaystyle \mathbf {a} } ⁠ , then all the partial derivatives and directional derivatives of f {\displaystyle f} exist at ⁠ a {\displaystyle \mathbf {a} } ⁠ , and for all ⁠ v {\displaystyle \mathbf {v} } ⁠ , f ′ (

8296-1343: The standard part function , which "rounds off" each finite hyperreal to the nearest real. Taking the squaring function f ( x ) = x 2 {\displaystyle f(x)=x^{2}} as an example again, f ′ ( x ) = st ⁡ ( x 2 + 2 x ⋅ d x + ( d x ) 2 − x 2 d x ) = st ⁡ ( 2 x ⋅ d x + ( d x ) 2 d x ) = st ⁡ ( 2 x ⋅ d x d x + ( d x ) 2 d x ) = st ⁡ ( 2 x + d x ) = 2 x . {\displaystyle {\begin{aligned}f'(x)&=\operatorname {st} \left({\frac {x^{2}+2x\cdot dx+(dx)^{2}-x^{2}}{dx}}\right)\\&=\operatorname {st} \left({\frac {2x\cdot dx+(dx)^{2}}{dx}}\right)\\&=\operatorname {st} \left({\frac {2x\cdot dx}{dx}}+{\frac {(dx)^{2}}{dx}}\right)\\&=\operatorname {st} \left(2x+dx\right)\\&=2x.\end{aligned}}} If f {\displaystyle f}

8418-447: The 20th century there was extensive research into the important role that play has in human development. While most evident in childhood, play continues throughout life for many adults in the form of games, hobbies, and sport. Moreover, studies of aging and society support the value of hobbies in healthy aging. There have been many instances where hobbyists and amateurs have achieved significant discoveries and developments. These are

8540-458: The 21st century, the video game industry has been popular as a hobby involving millions of children and adults. Stamp collecting declined along with the importance of the postal system. Woodwork and knitting declined as hobbies, because manufactured goods provide cheap alternatives for handmade goods. Through the internet, an online community has become a hobby for many people; sharing advice, information and support, and in some cases, allowing

8662-452: The Depression there was an increase in the participation in hobbies because the unemployed had the time and a desire to be purposefully occupied. Hobbies are often pursued with an increased interest by retired people because they have the time and seek the intellectual and physical stimulation a hobby provides. Hobbies are a diverse set of activities and it is difficult to categorize them in

8784-846: The Meccano Guild in 1919, the magazine carried regular Guild news to keep Meccano clubs informed of each other's activities. But over time Meccano Magazine became a general hobby magazine aimed at "boys of all ages". Aside from Meccano related articles, it also featured Hornby trains, Dinky Toys and other products of Meccano Ltd , plus a wide variety of general interest articles, including, engineering , aircraft, trains, modelling, camping, photography and philately . Commonwealth countries always featured strongly in articles as Meccano Ltd exported its products to these countries. The magazines today are an excellent source of historic information and an invaluable aid to collectors of toys from those years. They have also become collector's items and there

8906-521: The Serious leisure category. Casual leisure is intrinsically rewarding, short-lived, pleasurable activity requiring little or no preparation. Serious leisure is the systematic pursuit of an amateur, hobbyist, or volunteer that is substantial, rewarding and results in a sense of accomplishment. Finally, project-based leisure is a short-term often a one-off project that is rewarding. The terms amateur and hobbyist are often used interchangeably. Stebbins has

9028-512: The United States as "Meccano Engineer", and was a month ahead of the UK issue. The first copies were given away free but in 1918 readers had to pay two pence for postage for four issues. In 1917 and again in 1932 Hornby published a history of Meccano , its manufacture, and marketing in the magazine. In 1919 it doubled its size to eight pages and now cost one penny. New Meccano parts were advertised for

9150-581: The above definition of derivative applies to them. The derivative of y ( t ) {\displaystyle \mathbf {y} (t)} is defined to be the vector , called the tangent vector , whose coordinates are the derivatives of the coordinate functions. That is, y ′ ( t ) = lim h → 0 y ( t + h ) − y ( t ) h , {\displaystyle \mathbf {y} '(t)=\lim _{h\to 0}{\frac {\mathbf {y} (t+h)-\mathbf {y} (t)}{h}},} if

9272-660: The advent of modern plastics, the amount of skill required to get the basic shape accurately shown for any given subject was lessened, making it easier for people of all ages to begin assembling replicas in varying scales. Superheroes, aero planes, boats, cars, tanks, artillery, and even figures of soldiers became quite popular subjects to build, paint and display. Although almost any subject can be found in almost any scale, there are common scales for such miniatures which remain constant today. Model engineering refers to building functioning machinery in metal, such as internal combustion motors and live steam models or locomotives. This

9394-455: The amateur from the hobbyist is because the amateur has the ethos of the professional practitioner as a guide to practice. An amateur clarinetist is conscious of the role and procedures of a professional clarinetist. A large proportion of hobbies are mainly solitary in nature. However, individual pursuit of a hobby often includes club memberships, organized sharing of products and regular communication between participants. For many hobbies there

9516-405: The application of a differential operator to a function, d y d x = d d x f ( x ) . {\textstyle {\frac {dy}{dx}}={\frac {d}{dx}}f(x).} Higher derivatives are expressed using the notation d n y d x n {\textstyle {\frac {d^{n}y}{dx^{n}}}} for

9638-472: 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

9760-501: The constant 7 {\displaystyle 7} , were also used. Higher order derivatives are the result of differentiating a function repeatedly. Given that f {\displaystyle f} is a differentiable function, the derivative of f {\displaystyle f} is the first derivative, denoted as ⁠ f ′ {\displaystyle f'} ⁠ . The derivative of f ′ {\displaystyle f'}

9882-514: The culture that is most truly native centers round things which even when they are communal are not official—the pub, the football match, the back garden, the fireside and the 'nice cup of tea'." Deciding what to include in a list of hobbies provokes debate because it is difficult to decide which pleasurable pass-times can also be described as hobbies. During the 20th century the term hobby suggested activities, such as stamp collecting, embroidery, knitting, painting, woodwork, and photography. Typically

10004-424: The derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones. This process of finding a derivative is known as differentiation . The following are

10126-416: The derivative of a function is Leibniz notation , introduced by Gottfried Wilhelm Leibniz in 1675, which denotes a derivative as the quotient of two differentials , such as d y {\displaystyle dy} and ⁠ d x {\displaystyle dx} ⁠ . It is still commonly used when the equation y = f ( x ) {\displaystyle y=f(x)}

10248-405: The description did not include activities like listening to music, watching television, or reading. These latter activities bring pleasure, but lack the sense of achievement usually associated with a hobby. They are usually not structured, organized pursuits, as most hobbies are. The pleasure of a hobby is usually associated with making something of value or achieving something of value. "Such leisure

10370-492: The end of 1972 publication stopped due to a decline in popularity. In April 1973 it was resurrected again, this time as a quarterly magazine, and from 1977 it incorporated Meccano Engineer and the Meccanoman's Journal. Meccano Magazine' s last issue was Spring 1981. Meccano Magazine was originally aimed at Meccano builders and featured articles on Meccano construction and new Meccano developments. When Frank Hornby launched

10492-578: The first time in 1920 and in 1922 the magazine became a monthly publication. From 1921, the magazine was edited by Ellison Hawks who worked at Meccano as an advertising manager. From the May 1924 issue, the magazine had full-colour covers, and the December 1924 issue was 96 pages, costing six pence. By the 1930s Meccano Magazine had a circulation of 70,000. During the Second World War the content and quality of

10614-844: The foundations of calculus is called nonstandard analysis . This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the d {\displaystyle d} in the Leibniz notation. Thus, the derivative of f ( x ) {\displaystyle f(x)} becomes f ′ ( x ) = st ⁡ ( f ( x + d x ) − f ( x ) d x ) {\displaystyle f'(x)=\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)} for an arbitrary infinitesimal ⁠ d x {\displaystyle dx} ⁠ , where st {\displaystyle \operatorname {st} } denotes

10736-472: The function f {\displaystyle f} is differentiable at ⁠ a {\displaystyle a} ⁠ , that is if the limit L {\displaystyle L} exists, then this limit is called the derivative of f {\displaystyle f} at a {\displaystyle a} . Multiple notations for the derivative exist. The derivative of f {\displaystyle f} at

10858-1143: The function is the acceleration of an object with respect to time, and the third derivative is the jerk . A vector-valued function y {\displaystyle \mathbf {y} } of a real variable sends real numbers to vectors in some vector space R n {\displaystyle \mathbb {R} ^{n}} . A vector-valued function can be split up into its coordinate functions y 1 ( t ) , y 2 ( t ) , … , y n ( t ) {\displaystyle y_{1}(t),y_{2}(t),\dots ,y_{n}(t)} , meaning that y = ( y 1 ( t ) , y 2 ( t ) , … , y n ( t ) ) {\displaystyle \mathbf {y} =(y_{1}(t),y_{2}(t),\dots ,y_{n}(t))} . This includes, for example, parametric curves in R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} . The coordinate functions are real-valued functions, so

10980-488: The function near that input value. For this reason, the derivative is often described as the instantaneous rate of change , the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation . There are multiple different notations for differentiation, two of the most commonly used being Leibniz notation and prime notation. Leibniz notation, named after Gottfried Wilhelm Leibniz ,

11102-518: The graph of f {\displaystyle f} at a {\displaystyle a} . In other words, the derivative is the slope of the tangent. One way to think of the derivative d f d x ( a ) {\textstyle {\frac {df}{dx}}(a)} is as the ratio of an infinitesimal change in the output of the function f {\displaystyle f} to an infinitesimal change in its input. In order to make this intuition rigorous,

11224-445: The group of activities which occur outdoors. These hobbies include gardening, hill walking , hiking , backpacking , cycling , canoeing , climbing , caving , fishing, hunting , target shooting (informal or formal), wildlife viewing (as birdwatching ) and engaging in watersports and snowsports . One large subset of outdoor pursuits is gardening. Residential gardening most often takes place in or about one's own residence, in

11346-414: The internet is a common hobby, and one that can trace its origins back hundreds of years. A love of literature, later in life, may be sparked by an interest in reading children's literature as a child. Many of these fall under the category literary arts . Knitting or Crocheting is a calming and productive hobby. It allows for creativity while making cozy items like scarves, blankets, or hats. It's easy on

11468-485: The items being collected. An alternative to collecting physical objects is collecting records of events of a particular kind. Examples include train spotting , bird-watching , aircraft spotting , and any other form of systematic recording a particular phenomenon. The recording form can be written, photographic, online, etc. Making and tinkering includes working on self-motivated projects for fulfillment. These projects may be progressive, irregular tasks performed over

11590-811: The joints and can be done at a leisurely pace, making it perfect for staying engaged and creating thoughtful gifts. Stebbins distinguishes an amateur sports person and a hobbyist by suggesting a hobbyist plays in less formal sports, or games that are rule bound and have no professional equivalent. While an amateur sports individual plays a sport with a professional equivalent, such as football or tennis. Amateur sport may range from informal play to highly competitive practice, such as deck tennis or long distance trekking. The Department for Culture, Media, and Support in England suggests that playing sports benefits physical and mental health. A positive relationship appeared between engaging in sports and improving overall health. During

11712-464: The limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of y {\displaystyle \mathbf {y} } exists for every value of ⁠ t {\displaystyle t} ⁠ , then y ′ {\displaystyle \mathbf {y} '} is another vector-valued function. Functions can depend upon more than one variable . A partial derivative of

11834-448: The limit is 2 a {\displaystyle 2a} . So, the derivative of the squaring function is the doubling function: ⁠ f ′ ( x ) = 2 x {\displaystyle f'(x)=2x} ⁠ . The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function ⁠ f {\displaystyle f} ⁠ , specifically

11956-473: The low cost of manufactured clothes. Cooking is for some people an interest, a hobby, a challenge and a source of significant satisfaction. For many other people it is a job, a chore, a duty, like cleaning. In the early 21st century the importance of cooking as a hobby was demonstrated by the high popularity of competitive television cooking programs. Activity participation includes partaking in "non-competitive, rule-based pursuits." Outdoor pursuits are

12078-467: The magazine was reduced greatly, and the size decreased from approximately A4 to A5 . The A5 pocket-size format remained until 1961 when it was increased again to A4. In 1963 the magazine started reporting a loss and Meccano Ltd handed its publishing over to its printers, Thomas Skinner. In August 1967 Thomas Skinner terminated production of the magazine and passed the magazine on to Model & Allied Publications (MAP) of Hemel Hempstead in 1968. At

12200-404: The making process, often applied to the hobby of tinkering with car repairs, and various kinds of restoration: of furniture, antique cars , etc. It also applies to household tinkering: repairing a wall, laying a pathway, etc. Examples of Making and Tinkering hobbies include Scale modeling , model engineering , 3D printing , dressmaking , and cooking . Scale modeling is making a replica of

12322-396: The most basic rules for deducing the derivative of functions from derivatives of basic functions. The derivative of the function given by f ( x ) = x 4 + sin ⁡ ( x 2 ) − ln ⁡ ( x ) e x + 7 {\displaystyle f(x)=x^{4}+\sin \left(x^{2}\right)-\ln(x)e^{x}+7}

12444-399: The norm in the numerator is the standard length on R m {\displaystyle \mathbb {R} ^{m}} . If v {\displaystyle v} is a vector starting at ⁠ a {\displaystyle a} ⁠ , then f ′ ( a ) v {\displaystyle f'(\mathbf {a} )\mathbf {v} } is called

12566-399: The notation f ( n ) {\displaystyle f^{(n)}} for the ⁠ n {\displaystyle n} ⁠ th derivative of ⁠ f {\displaystyle f} ⁠ . In Newton's notation or the dot notation, a dot is placed over a symbol to represent a time derivative. If y {\displaystyle y} is

12688-1410: The notation was introduced by Louis François Antoine Arbogast . To indicate a partial derivative, the variable differentiated by is indicated with a subscript, for example given the function ⁠ u = f ( x , y ) {\displaystyle u=f(x,y)} ⁠ , its partial derivative with respect to x {\displaystyle x} can be written D x u {\displaystyle D_{x}u} or ⁠ D x f ( x , y ) {\displaystyle D_{x}f(x,y)} ⁠ . Higher partial derivatives can be indicated by superscripts or multiple subscripts, e.g. D x y f ( x , y ) = ∂ ∂ y ( ∂ ∂ x f ( x , y ) ) {\textstyle D_{xy}f(x,y)={\frac {\partial }{\partial y}}{\Bigl (}{\frac {\partial }{\partial x}}f(x,y){\Bigr )}} and ⁠ D x 2 f ( x , y ) = ∂ ∂ x ( ∂ ∂ x f ( x , y ) ) {\displaystyle \textstyle D_{x}^{2}f(x,y)={\frac {\partial }{\partial x}}{\Bigl (}{\frac {\partial }{\partial x}}f(x,y){\Bigr )}} ⁠ . In principle,

12810-489: The partial derivative of a function f ( x 1 , … , x n ) {\displaystyle f(x_{1},\dots ,x_{n})} in the direction x i {\displaystyle x_{i}} at the point ( a 1 , … , a n ) {\displaystyle (a_{1},\dots ,a_{n})} is defined to be: ∂ f ∂ x i (

12932-526: The partial derivative of function f {\displaystyle f} with respect to both variables x {\displaystyle x} and y {\displaystyle y} are, respectively: ∂ f ∂ x = 2 x + y , ∂ f ∂ y = x + 2 y . {\displaystyle {\frac {\partial f}{\partial x}}=2x+y,\qquad {\frac {\partial f}{\partial y}}=x+2y.} In general,

13054-732: The partial derivatives of f {\displaystyle f} exist and are continuous at ⁠ x {\displaystyle \mathbf {x} } ⁠ , then they determine the directional derivative of f {\displaystyle f} in the direction v {\displaystyle \mathbf {v} } by the formula: D v f ( x ) = ∑ j = 1 n v j ∂ f ∂ x j . {\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\sum _{j=1}^{n}v_{j}{\frac {\partial f}{\partial x_{j}}}.} When f {\displaystyle f}

13176-430: The partial derivatives of f {\displaystyle f} measure its variation in the direction of the coordinate axes. For example, if f {\displaystyle f} is a function of x {\displaystyle x} and ⁠ y {\displaystyle y} ⁠ , then its partial derivatives measure the variation in f {\displaystyle f} in

13298-418: The point ⁠ ( a 1 , … , a n ) {\displaystyle (a_{1},\dots ,a_{n})} ⁠ , these partial derivatives define the vector ∇ f ( a 1 , … , a n ) = ( ∂ f ∂ x 1 ( a 1 , … ,

13420-404: The points ( a , f ( a ) ) {\displaystyle (a,f(a))} and ( a + h , f ( a + h ) ) {\displaystyle (a+h,f(a+h))} . As h {\displaystyle h} is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the tangent to

13542-432: The position of a moving object with respect to time is the object's velocity , how the position changes as time advances, the second derivative is the object's acceleration , how the velocity changes as time advances. Derivatives can be generalized to functions of several real variables . In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation)

13664-462: The provision of counter-attractions came to the fore in the 1830s, and has rarely waned since. Initially, the bad habits were perceived to be of a sensual and physical nature, and the counter attractions, or perhaps more accurately alternatives, deliberately cultivated rationality and the intellect." The book and magazine trade of the day encouraged worthwhile hobbies and pursuits. The burgeoning manufacturing trade made materials used in hobbies cheap and

13786-410: The rules for the derivatives of the most common basic functions. Here, a {\displaystyle a} is a real number, and e {\displaystyle e} is the base of the natural logarithm, approximately 2.71828 . Given that the f {\displaystyle f} and g {\displaystyle g} are the functions. The following are some of

13908-611: The second and the third derivatives can be written as f ″ {\displaystyle f''} and ⁠ f ‴ {\displaystyle f'''} ⁠ , respectively. For denoting the number of higher derivatives beyond this point, some authors use Roman numerals in superscript , whereas others place the number in parentheses, such as f i v {\displaystyle f^{\mathrm {iv} }} or ⁠ f ( 4 ) {\displaystyle f^{(4)}} ⁠ . The latter notation generalizes to yield

14030-565: The second term was computed using the chain rule and the third term using the product rule . The known derivatives of the elementary functions x 2 {\displaystyle x^{2}} , x 4 {\displaystyle x^{4}} , sin ⁡ ( x ) {\displaystyle \sin(x)} , ln ⁡ ( x ) {\displaystyle \ln(x)} , and exp ⁡ ( x ) = e x {\displaystyle \exp(x)=e^{x}} , as well as

14152-648: The total derivative can be expressed using the partial derivatives as a matrix . This matrix is called the Jacobian matrix of f {\displaystyle f} at a {\displaystyle \mathbf {a} } : f ′ ( a ) = Jac a = ( ∂ f i ∂ x j ) i j . {\displaystyle f'(\mathbf {a} )=\operatorname {Jac} _{\mathbf {a} }=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{ij}.} The concept of

14274-543: The turn of the Industrial Age and through the 1920s, some families could afford things such as electric trains , wind-up toys (typically boats or cars) and the increasingly valuable tin toy soldiers. Scale modeling as we know it today became popular shortly after World War II . Before 1946, children as well as adults were content in carving and shaping wooden replicas from block wood kits, often depicting enemy aircraft to help with identification in case of an invasion. With

14396-605: The variable x {\displaystyle x} is variously denoted by among other possibilities. It can be thought of as the rate of change of the function in the x {\displaystyle x} -direction. Here ∂ is a rounded d called the partial derivative symbol . To distinguish it from the letter d , ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee". For example, let ⁠ f ( x , y ) = x 2 + x y + y 2 {\displaystyle f(x,y)=x^{2}+xy+y^{2}} ⁠ , then

14518-663: The variable for differentiation, in the denominator, which removes ambiguity when working with multiple interrelated quantities. The derivative of a composed function can be expressed using the chain rule : if u = g ( x ) {\displaystyle u=g(x)} and y = f ( g ( x ) ) {\displaystyle y=f(g(x))} then d y d x = d y d u ⋅ d u d x . {\textstyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.} Another common notation for differentiation

14640-415: The world. Others focus on a subtopic within their area of interest, perhaps 19th century postage stamps, milk bottle labels from Sussex, or Mongolian harnesses and tack, Firearms (both modern and vintage). Collecting is an ancient hobby, with the list of coin collectors showing Caesar Augustus as one. Sometimes collectors have turned their hobby into a business, becoming commercial dealers that trade in

14762-483: Was acknowledging that a "hobby horse" produces a legitimate sense of pride. The cultural shift towards acceptance of hobbies was thought to begin during the mid 18th century as working people had more regular hours of work and greater leisure time, spending more time to pursue interests that brought them satisfaction. However, there was concern that these working people might not use their leisure time in worthwhile pursuits. "The hope of weaning people away from bad habits by

14884-492: Was responsive to the changing interests of hobbyists. In 1941, George Orwell identified hobbies as central to English culture at the time: "Another English characteristic which is so much a part of us that we barely notice it … is the addiction to hobbies and spare-time occupations, the privateness of English life. We are a nation of flower-lovers, but also a nation of stamp-collectors, pigeon-fanciers, amateur carpenters, coupon-snippers, darts-players, crossword-puzzle fans. All

#769230