![]() Later, in connection with Taylor series.) We will study more careful ways to judge the accuracy of such approximations When f is a power function, this will be true when What we really need for a linear approximationīe almost a straight line when viewed on the small scale set by the (How do we know that dr is "rather small"? ![]() ![]() We expect the difference between dA and DA Since dr is rather small (compared to r),Īnd all the quantities are uncertain anyway, Is the estimated error or uncertainty in A. Then dA = 2r dr (which is approximately 3 cm 2) We say that r = 10.00 cm and dr, the error, is 0.05 This means that r may be as large as 10.05, or as small as 9.95, or Radius to be 10.00 plus-or-minus 0.05 cm. Suppose that a competent experimenter or field engineer has measured the The curve, or the width of the pencil mark used to draw it.) (In fact, any real circle is never a "perfect" circle, so the radius is notĮven defined to a precision beyond the scale of the irregularities in In practice the radius of a circle can be measured only to within a certainĪ better measuring instrument may reduce the uncertainty in the radius, but it Let's go back once more to the circle problem. This example, that we don't know the formula for the derivative of Note: We are going to pretend, just for the sake of Here is a simple model of this type of argument:Įxample: Find a formula for the rate of change of the area of a The basic formulas and equations that are applied to concrete problems in the These are the sections of your science and engineering textbooks that derive Point of view of the discipline making the application. "applied" from the point of view of mathematics, but "theoretical" from the The most important use of differentials is in discussions that are Is still very helpful in applying calculus.ĭifferentials in theoretical arguments in applications Today we avoid this kind of talk in discussing mathematical fundamentals weīut thinking of f ' = dy/dx as a ratio of small changes Shrank to " d" to indicate this smallness). Numbers - numbers so small that the approximate equation Mathematicians thought of dx and dy as "infinitesimal" In fact, Leibnitz (the co-inventor of calculus) and other early Here dx is any (sufficiently small) change in x,Īmd dy is the corresponding change in y when you adopt the tangent-line approximation. Now we see that it can also be broken up literally as a fraction, That is, it was to be broken up into d/dx, the operation ofĭifferentiation, acting on y, some function or physical quantity. Heretofore we have thought of the notation "dy/dx" as just another name for ![]() Then, in that approximation, Dy equals dy. (This means that it vanishes even after being divided by Dx.įor instance, (Dx) 3/Dx = (Dx) 2 -> 0 in that More precisely, you will ignore anything that vanishes faster ![]() (positive integral) powers of Dx that might arise. Often means doing a calculation in which you will throw away any In other words, " dx" is what Dx is called when you areĪnnouncing your intention to use a linear approximation. The linear approximation to Dy, and also to write " dx" In this context it is traditional to write " dy" for ( f approx is the function whose graph is the tangent line Is that it tells us how to build a linear approximation to Using the definition of the increments Dy and Dx, we canįrom this point of view, the importance of the number f '(a) Then we can solve the approximate equation for Dy: Today we are going to turn the equation around and think of the left side as Taking x closer to a gives a better numerical estimate for We thought of the difference quotient on the right,Īs an approximation to the derivative on the left: The even more important subject of Taylor series, which will occupy ourĪttention for several weeks toward the end of this semester. Thought about and talked about in applications of calculus. Today's topic is very important for understanding how derivatives are Note: Since the Web does not yet speak Greek easily, we will use theĬapital D for "delta", and the grade-school division sign, Fulling) 1998 Class 17.T Differentials and Linear Approximations Reading assignment for Tuesday, January 27 Differentials (c) copyright Foundation Coalition (S. ![]()
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