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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 Î” (the uppercase form of the Greek letter Delta) is an abbreviation for "change in". It follows that Î”y = mÎ”x.

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 âˆ‚y/âˆ‚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.

Graphing calculator

A graphing calculator (also graphic calculator) typically refers to a class of handheld calculators that are capable of plotting graphs, solving simultaneous equations, and performing numerous other tasks with variables. Most popular graphing calculators are also programmable, allowing the user to create customized programs, typically for scientific/engineering and education applications. Due to their large displays intended for graphing, they can also accommodate several lines of text and calculations at a time. Some graphing calculators also have colour displays, and others even include 3D graphing.

Many graphing calculators can be attached to devices like electronic thermometers, pH gauges, weather instruments, decibel and light meters, accelerometers, and other sensors and therefore function as data loggers.

Since graphing calculators are readily user-programmable, such calculators are also widely used for gaming purposes, with a sizable body of user-created game software on most popular platforms.

There is also computer software available to emulate or perform the functions of a graphing calculator. One such example is Grapher for Mac OS X and is a basic software graphic calculator.

## History

Casio produced the world's first graphic calculator, the fx-7000G, in 1985. After Casio, Hewlett Packard followed shortly in the form of the HP-28C. This was followed by the HP-28S (1988), HP-48SX (1990), HP-48S (1991), and many other models. Recent models like the HP 50g (2006), feature a computer algebra system (CAS) capable of manipulating symbolic expressions and analytic solving. The HP-28 and -48 range were primarily meant for the professional science/engineering markets; the HP-38/39/40 were sold in the high school/college educational market; while the HP-49 series cater to both educational and professional customers of all levels. The HP series of graphing calculators is best known for its Reverse Polish notation interface, although the HP-49 introduced a standard expression entry interface as well.

Texas Instruments has produced graphing calculators since 1990, the oldest of which was the TI-81. Some of the newer calculators are similar, with the addition of more memory, faster processors, and USB connection such as the TI-82, TI-83 series, and TI-84 series. Other models, designed to be appropriate for students 10&ndash;14 years of age, are the TI-80 and TI-73. Other TI graphing calculators have been designed to be appropriate for calculus, namely the TI-85, TI-86, TI-89 series, and TI-92 series (TI-92, TI-92 Plus, and Voyage 200). TI offers a CAS on the TI-89, TI-Nspire CAS and TI-92 series models with the TI-92 series featuring a QWERTY keypad. TI calculators are targeted specifically to the educational market, but are also widely available to the general public.

Graphing calculators are also manufactured by Sharp but they do not have the online communities, user-websites and collections of programs like the other brands.

## Graphing calculators in schools

• In the Canadian and American educational systems, many high school mathematics teachers allow and even encourage their students to use graphing calculators in class. In some cases (especially in calculus courses) they are required. Some of them are banned in certain classes such as chemistry or physics due to their capacity to contain full periodic tables.
• In the Accelerated Christian Education (A.C.E) Curriculum, 4th year high school students are encouraged to use graphing calculators for the Trigonometry subject.

Also, some high school courses offered in these countries require a graphing calculator to fulfill.

• In the United Kingdom, a graphic calculator is required for most A-level maths courses, the use of such devices is both taught and tested. However, for GCSE maths exams, a limited number of calculator models are allowed, none of which are capable of graphic operations (although they are capable of scientific and statistical operations).
• In Finland, Slovenia and certain other countries, it is forbidden to use calculators with symbolic calculation (CAS) or 3D graphics features in the matriculation exam.
• In Norway, calculators with wireless communication capabilities, such as IR links, have been banned at some technical universities.
• The College Board of the United States permits the use of most graphing or CAS calculators that do not have a QWERTY-style keyboard for parts of its AP and SAT exams, but IB schools do not permit the use of calculators with computer algebra systems on its exams.
• In Australia, policies vary from state to state.

Question:okay...i dont have a calculator, and im sick lol..and im doing a math project that requires calculating 3d figures,their surface areas and volumes...and i need a calculator that can do problems like these 2*pi(3.5)^2 (the little square button) because i found calculators online that do this 2*pi(3.5) but dont have the square button...and instead of getting the answer or 24.5, i get like 75.XX...im just using an example from the text book...if anyone can post a good calculator,ill be thankful...thanks well,, whenever i do an equation on the calculators for a cylinder(surface area), i do this 2*pi(3.5)(11.5) and instead of getting 80.5 like my text book says,i get 227....so i need a calculator that can actually do equations like that because i dont have a calculator... i tried using my calculator from my computer,and instead of getting 80.5, i got 227....thats why im confused

Answers:Start-->All Programs-->Accessories-->Calculator Once the calculator is open it is in standard view by default. Go to the top and click view-->scientific and now your should have a pi button and exponent button along with others you may want to use You may be getting wrong answers because: 1. You are not following the order of operations.... (Parentheses, Exponent, Multiplication*, Division*, Addition**, Subtraction**) *-order does not matter between the two **-order does not matter between the two 2.Your book has the wrong answer which is unlikely...

Question:Suppose a surface has a patch with first fundamental form ds^2 = du^2 +W^2(u, v)* dv^2 where lim u -->0 W(u, v) = 0, defined for the u values where u > 0 and W(u, v) > 0. Show that the curves v = constant are geodesics, with u an affine parameter. (The coordinates are called geodesic polars, in comparison to plane polars.)

Answers:calculate the cristofell's symbols, and put everything in the equations of geodetics and you 'll get the result.

Question:Continuous compounding means that, at any times, interest is being accrued at a rate that is a fixed percentage of the balance at that moment. A bank account earns 10% annual interest, compounded continuously. Money is deposited in a continuous cash flow at a rate of $1200 per year into the account. a. write a differential equation that describes the rate at which the balance B= f(t) is changing b. solve the differential equation given an initial balance B(0) = f(0) =0 c. find the balance after 5 years, f(5) d. now suppose the money is deposited once a month (instead of continuously) but still at a rate of$1200 per year. i. write down the sum that gives the balance after 5 years, assuming the first deposit is made one month from today, and today is t=0 ii. the sum you wrote in part (i) is a Riemann sum approximation to the integral 1200e^(0.1t)dt. Determine whether it is a left sum or a right sum, and determine what t and n are. Then use your calculator to evaluate sum iii. compare your answer in part (ii) to your answer in part (c) I know this is a lengthy question, but I am really confused. thanks for the help

Answers:Okay, so maybe it is best to start with /daily/ compounded interest. An interest rate of "36.5% annual interest, compounded daily" literally means that every day, they would add exactly 0.1% interest to your balance. It is very important to understand that this is how the actual number is gotten: divide the "nominal rate" 36.5% by the number of evaluations of that rade (365 days in one year). This is the difference between a "nominal rate" (nominal means "in name only") and the /actual/ rates that you will be experiencing, when the interest is added to your debt. Okay, so let's be totally clear. You've got a nominal rate R (like 10% = 0.1) which you pay over some big time-scale T (like 1 year). N times per year, you have a little change in your balance B, which looks like this: B = (R/N) * B. If we define that the time period t = T/N, and if we explicitly include the time in the nominal rate r = R/T (e.g. "r = 0.365 per year") we can write: B = r B t In the continuum case: dB/dt = r B B(t) = B e^(r (t t )) B(t ) = B B'(t ) = r B . We can also ask what actually happens after one year -- how much do you really make with continuous interest? B(t + T) = B e^R So the actual annual earnings per T is not R but is instead e^R 1. If you are earning r = 0.05 per year, or 5% annual interest at a bank, this works out to a 5.1% per year actual earnings. But when your credit card tries to charge you 30% interest compounded continuously, this works out to 35% if you let it "compound" for a year. The "compounding" happens when the interest earns interest, you see -- that is what ultimately makes compound interest "the most powerful force in the universe," to quote Einstein. (Everything else in the universe goes like t^n, but compound interest goes like e^t. If you owe the credit card $100, and don't make any payments for 10 years at 30% interest, they will expect you to pay back a full$100 * e^3 = $2000 back at the end of that. This is one of the reasons that they don't care too much if you pay off your credit card bills with other credit cards, within some limits of reason.) Okay, so we have the /discrete/ case B = r B t, useful for the Riemann sum in part d, and the /continuous/ case: dB/dt = r B Now, what is happening in the situation described? A /continuous cash flow/, which we might write as k =$1200/year, is being deposited. Thus: dB/dt = r B + k We have added an inhomogeneous term. To solve it, we find a particular solution: B(t) = -k / r, B'(t) = 0. Then we add that to the general solution: B(t) = C e^(r t) k / r Solving C for B(0) = B gives: C = B + k/r B(t) = [B + k/r] e^(r t) k/r B(t) = B e^(r t) + (k/r) [e^(r t) 1] So, given an initial balance B = 0, we have: B(t) = (k/r) [e^(r t) 1]. B(5 years) = ($1200/year)/(0.1/year) [e^(0.1/year * 5 years) 1] B(5 years) = ($12 000) [e^(0.5) 1] = $7 785. Pretty good, considering the deposits only amounted to$6 000. Do you see how to complete the discrete case for the rest of it?

Question:One that shows the steps of the equations would be nice...

Answers:I think that this website is a pretty good one...Try it out!