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From Wikipedia

Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.

Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.

Differentiation has applications to all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. By applying game theory, differentiation can provide best strategies for competing corporations.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.

The derivative

Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, we can determine the value of y. This relationship is written as: y = f(x). Where f(x) is the equation for a straight line, y = mx + b, where m and b are real numbers that determine the locus of the line in Cartesian coordinates. m is called the slope and is given by:

m={\mbox{change in } y \over \mbox{change in } x} = {\Delta y \over{\Delta x}},

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in". It follows that Δy = mΔx.

In linear functions the derivative of f at the point x is the best possible approximation to the idea of the slope of f at the point x. It is usually denoted f'(x) or dy/dx. Together with the value of f at x, the derivative of f determines the best linear approximation, orlinearization, of f near the point x. This latter property is usually taken as the definition of the derivative. Derivatives cannot be calculated in nonlinear functions because they do not have a well-defined slope.

A closely related notion is the differential of a function.

When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f' at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, which is usually denoted ∂y/∂x. The linearization of f in all directions at once is called thetotal derivative. It is alinear transformation, and it determines the hyperplane that most closely approximates the graph of f. This hyperplane is called the osculating hyperplane; it is conceptually the same idea as taking tangent lines in all directions at once.


Definition of Sound

Definition of Sound was a London based dance musicgroup, consisting of Kevin Clark and Don Weekes, working with musicians Rex Brough (aka The Red King) and latterly, Mike Spencer. Their second and fourth singles, "Wear Your Love Like Heaven" (1991) and "Moira Jane's Café" (1992) were in the Top 40 in the UK Singles Chart. They also had several songs enter the U.S.BillboardHot Dance Club Playchart, including "Moira Jane's Café", which hit #1 in 1992.

Career

Weekes, who had recorded with Coldcut's Matt Black, and was briefly a member of X Posse, was impressed with Clark's skills and soon the two were working together on material. They recorded a demo and, under the name Top Billin', released two underground hits, "Naturally" and "Straight From the Soul" on the Dance Yard record label. This led to a recording contract with Cardiac Records and their first album, Love and Life: A Journey With the Chameleons and single, "Now Is Tomorrow". Love and Life: A Journey With the Chameleons was named rap album of the year by Record Mirrorand had glowing reviews in Billboard,The Source, and otherStateside publications. Their second album The Lick got buried under record label takeovers, and Clark and Weeks signed a deal with Mercury and started making their third album, Experience (1996). According to the NMEit was "like the delayed hit of a powerful drug".

Although they had no Billboard Hot 100 entries, the song "Now Is Tomorrow" (a #10 dance hit) climbed to #68 on the Hot 100 Airplay chart in 1991. Vocal duties on this single and some album tracks were handled by singer, Elaine Vassel.

After recording their fourth and final album for MCA/Universal, which was ultimately never released, they split up. Initially the final members Clark, Weekes and Spencer continued as a songwritingremix and production team. Clark went on to a career as a A&R manager for Parlophone and eventually Universal. He has worked with Beverley Knight, Jamelia, Tracie Spencer, Freestylers, Betty Boo and Beats International. His later career saw a move into music publishing with Clarkmusic. Mike Spencer went onto have successful a career producing, amongst others, Jamiroquai, Kylie MinogueAlphabeat and Newton Faulkner . Weekes, after the release of a solo album, left the music industry.

Discography

Albums

  • Love and Life: A Journey With the Chameleons (1991)
  • The Lick (1992)
  • Experience (1996)

Singles

  • "Dream Girl (1991)
  • "Wear Your Love Like Heaven" (1991) - UK #12, U.S. Dance #28
  • "Now is Tomorrow" (1991) UK #39, U.S. Dance #10
  • "Moira Jane's Cafe" (1992) UK #27, U.S. Dance #1
  • "What Are You Under" (1992) UK #68, U.S. Dance #4
  • "Can I Get Over" (1992) #61
  • "Boom Boom" (1995) UK #33
  • "Pass the Vibes" (1995) UK #17
  • "Child" (1996) UK #48

Don worked with Jamelia,beverly knight and the free stylers not kevin.

Music video

  • "Child" Directed by Dani Jacobs

Differential (mechanical device)

A differential is a device, usually but not necessarily employing gears, capable of transmitting torque and rotation through three shafts, almost always used in one of two ways: in one way, it receives one input and provides two outputs—this is found in most automobiles—and in the other way, it combines two inputs to create an output that is the sum, difference, or average, of the inputs.

In automobiles and other wheeled vehicles, the differential allows each of the driving roadwheels to rotate at different speeds, while for most vehicles supplying equal torque to each of them.

Purpose

A vehicle's wheels rotate at different speeds, mainly when turning corners. The differential is designed to drive a pair of wheels with equal torque while allowing them to rotate at different speeds. In vehicles without a differential, such as karts, both driving wheels are forced to rotate at the same speed, usually on a common axle driven by a simple chain-drive mechanism. When cornering, the inner wheel needs to travel a shorter distance than the outer wheel, so with no differential, the result is the inner wheel spinning and/or the outer wheel dragging, and this results in difficult and unpredictable handling, damage to tires and roads, and strain on (or possible failure of) the entire drivetrain.

History

There are many claims to the invention of the differential gear but it is likely that it was known, at least in some places, in ancient times. Some historical milestones of the differential include:

Note: TheAntikythera mechanism (150 BC–100 BC), discovered on an ancient shipwreck near the Greek island of Antikythera, was once suggested to have employed a differential gear. This has since been disproved.

Functional description

The following description of a differential applies to a "traditional" rear-wheel-drive car or truck with an "open" or limited slip differential:

Torque is supplied from the engine, via the transmission, to a drive shaft (British term: 'propeller shaft', commonly and informally abbreviated to 'prop-shaft'), which runs to the final drive unit that contains the differential. A spiral bevelpinion gear takes its drive from the end of the propeller shaft, and is encased within

Homogeneous differential equation

The term homogeneous differential equation has several distinct meanings.

One meaning is that a first-order ordinary differential equation is homogeneous (of degree 0) if it has the form

\frac{dy}{dx} = F(x,y)

where F(x,y) is a homogeneous function of degree zero; that is to say, such that F(tx,ty) = F(x,y).

In a related, but distinct, usage, the term linear homogeneous differential equation is used to describe differential equations of the form

Ly = 0 \,

where the differential operatorL is a linear operator, and y is the unknown function.

The remainder of this article is about homogeneous differential equations in the first sense defined above.

Solving homogeneous differential equations

By the definition above, it can be seen that F(tx,ty) = F(x,y) for all t, so t can be arbitrarily chosen to simplify the form of the equation. One can solve this equation by making a simple change of variables y = ux, and then using the product rule on the left hand side as follows,

\frac{d(ux)}{dx} = x\frac{du}{dx} + u\frac{dx}{dx} = x\frac{du}{dx} + u.

and then using the identity F(tx,ty) = F(x,y) to simplify the right hand side by choosing to set t to be 1/x, transforming the original problem into the separable differential equation

x\frac{du}{dx} + u = F(1,u)

which can then be integrated by the usual methods.



From Yahoo Answers

Question:Let f(x)= 2x -3x-5. show that the slope of the secant line through (2,f(2)) and (2+h, f(2+h)) is 2h+5. Then use this formula to compute the slope of: a) the secant line through (2, f(2)) and (3, f(3)) b) the tangent line at x=2 (by taking a limit) I have no idea how to do this stuff. please help

Answers:h represents a very small distance away from the x value of 2. The theory is that we can get a tangent line by drawing a secant line then making h so small that it's essentially the tangent line. f(2) means you put 2 in for every x you see. f(2) = 2(2) - 3(2) - 5 = -3 f(3) means you put 3 in for every x you see. f(3) = 2(3) - 3(3) - 5 = 4 So, slope is the difference in y values (f(x)) over the difference of x values. [f(3) - f(2)]/(3-2) = [4 - (-3)]/1 = 7 f(2 + h) means you put 2 + h in for every x you see. f(2 + h) = 2(2 + h) - 3(2 + h) - 5 = 2h + 8h + 8 - 6 -3h - 5 = 2h + 5h - 3 You use the same slope formula here. [f(2 + h) - f(2)]/((2 + h) -2) = [2h + 5h - 3 - (-3)]/h = (2h + 5h)/h So, that's 2h + 5 Next, take the limit of that to minimize h and get the tangent line. lim[h=>0] 2h + 5 = 5 So the slope of the tangent line at that point is 5.

Question:The definition of asexual reproduction in angiosperms by leaves and shoots ? What is the asexual reproduction in angiosperms? define the artificial and natural? This homework is for my little brother, please cite any websites. Thank you

Answers:. ASEXUAL REPRODUCTION involves NO FERTILIZATION AND PRODUCES OFFSPRING THAT ARE GENETICALLY IDENTICAL TO THE PARENTS -CLONES. 2. Most plants reproduce Asexually at least some of the time, while other plants reproduce Asexually most of the time. 3. In a sable environment with abundant resources, asexually reproduction is FASTER, and produces offspring that are well adapted to the existing environment. 4. ASEXUAL REPRODUCTION THAT OCCURS NATRUALLY IN PLANTS IS CALLED VEGETATIVE REPRODUCTION. Reproduction occurs from Non-Reproductive Parts, such as Leaves, Stems, and Roots. (Figure 32-15) 5. WHEN WE USE ASEXUAL METHODS TO GROW PLANTS WE CALL IT VEGETATIVE (ARTIFICIAL) PROPAGATION. 6. VEGETATIVE PROPAGATION IS A BY-PRODUCT OF A PLANT'S ABILITY TO REGENERATE LOST PARTS. 7. Many species of plants are Vegetative Propagated from Specialized Structures such as Runners, Rhizomes, Bulbs, and Tubers. (Table 32-2) 8. METHODS OF VEGETATIVE PROPAGATION INCLUDE CUTTINGS, GRAFTING, TISSUE CULTURING AND LAYERING. A. CUTTING - Taking a piece of Stem or Leaf and planting it in soil to grow a new plant. B. GRAFTING - A way to make TWO Different plants grow as one by fusing their cut ends. C. TISSUE CULTURING - Growing a new plant from individual cells, or from small pieces of Leaf, Stem or Roots. (Figure 32-16) D. LAYERING - Roots form on Stems where they make Contact with the Soil. People Stake the Branch Tips to the Soil or Cover the Base of Stems with Soil to Propagate the Plants.

Question:PLEASE HELP!!!

Answers:How many have you got?

Question:1. Match the following terms with their definitions. words. Asexual Reproduction,Sexual Reproduction,Mitosis,Meiosis Definition. A) The formation of sex cells. B) Two parents are needed for this type of reproduction. C) The process of duplicating the nucleus. D) Only one parent it needed for this type of reproduction. 2. There are two types of cell division. In reproduction, sperm and egg cells are formed by ____________________ . 3. During meiosis the chromosome number is reduced by _____________(half/quarter). 4. There are many diseases that affect or change a cell. _____________ is one of the most common forms of disease affecting cells. 5. 1. Do you agree or disagree with the production of cloned embryos for the purpose of medical research? Please be specific and give an explanation. 2. Do you agree with the use of stem cells for the production of transplant organs? Please be specific and give an explanation YES OR NO 6. Mitosis is reproduction by... one parent or twoi parents?

Answers:so i'm doing your homework for you? you wont learn anything that way... but i'm bored so... A) Meiosis B) Sexual Reproduction C) Mitosis D) Asexual Reproduction 2) Meiosis 3) half 4) cancer 5 is YOUR opinion. 6) two parents

From Youtube

Integration by Parametric Differentiation :Here is a method of evaluating definite integrals that is not often taught to undergraduate engineers and physicists. It probably should be. NOTE: IN THE FINAL EQUATION where I wrote alpha, it should be 3. That should be clear, as mentioned at the time that I was letting alpha equal 3, I just got nervous at the camera. If you have any questions, feel free to ask.

Structure of Gear Differential for SAKURA ZERO :New from 3 Racing is Gear Differential 37 T & 38T for SAKURA ZERO, it contains 6 plastic gears inside. Mechanical operation makes it much more durable than ball differential and roller differential. Drivers possessing this new part no more need to change worn and eroded differential frequently, which definitely suits competition grade drivers who always crave for rebuilding their chassis promptly in the pit. Two outer differential joints are made of 7075 aluminium alloy with hard coating, which is extraordinary strong and sturdy. Appended 3mm differential cushion effectively hinders joints get damaged. For adjusting the smoothness and the speed of gear differential, silicon oil with varied viscocity can be applied. Gear Differential 37T and 38T are workable with 3.5 T to 4.5T modified motors, it fits all car models if some right adjustments of the chassis are made. The new part is light and it weighs 22grams only, a must-have item for every advanced driver this season. US$25, available for sale at the end of June. For our complete listing of option parts, please visit our website:www.3racing.hk