微積分英文版4課件

1、CHAPTER 3APPLICATIONS OF THE DERIVATIVE三、其他未定式 二、 型未定式一、 型未定式機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 洛必達(dá)法則 第三章 一、存在 (或?yàn)?)定理 1.型未定式(洛必達(dá)法則) 機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 ( 在 x , a 之間)證:無妨假設(shè)在指出的鄰域內(nèi)任取則在以 x, a 為端點(diǎn)的區(qū)間上滿足柯故定理?xiàng)l件: 西定理?xiàng)l件,機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 存在 (或?yàn)?)推論1.定理 1 中換為之一,推論 2.若理1條件, 則條件 2) 作相應(yīng)的修改 , 定理 1 仍然成立.洛必達(dá)法則定理1 目錄 上頁 下頁 返回 結(jié)束 例1.

2、 求解:原式注意: 不是未定式不能用洛必達(dá)法則 !機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 例2. 求解:原式 機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 二、型未定式存在 (或?yàn)?定理 2.證: 僅就極限存在的情形加以證明 .(洛必達(dá)法則)機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 1)的情形從而機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 2)的情形.取常數(shù)可用 1) 中結(jié)論機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 3)時(shí), 結(jié)論仍然成立. ( 證明略 )說明: 定理中換為之一,條件 2) 作相應(yīng)的修改 , 定理仍然成立.定理2 目錄 上頁 下頁 返回 結(jié)束 例3. 求解:原式機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 例3. 求解:

3、原式機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 例3. 例4.說明:例如,而用洛必達(dá)法則在滿足定理?xiàng)l件的某些情況下洛必達(dá)法則不能解決 計(jì)算問題 . 機(jī)動(dòng) 目錄 上頁 下頁 返回 結(jié)束 3.1Maxima & MinimaMaxima: point whose function value is greater than or equal to function value of any other point in the intervalMinima: point whose function value is less than or equal to function value of any

4、other point in the intervalExtrema: Either a maxima or a minimaWhere do extrema occur?Peaks or valleys (either on a smooth curve, or at a cusp or corner)f(c)=0 or f(c) is undefinedDiscontinutiesEndpoints of an intervalThese are known as the critical points of the functionOnce you know you have a cri

5、tical point, you can test a point on either side to determine if its a max or min (or maybe neitherjust a leveling off point)3.2Monotonicity and ConcavityLet f be defined on an interval I (open, closed, or neither). Then f isINCREASING on I if, DECREASING on I if, MONTONIC on I if it is ether increa

6、sing or decreasingMonotonicity TheoremLet f be continuous on an interval I and differentiable at every interior point of I.If f(x)0 for all x interior to I, then f is increasing on IIf f(x)0 for all x in (1,c) and f(x)0 for all x in (c,b), then f(c) is a local max. value.If f(x)0 for all x in (c,b),

7、 then f(c) is a local min. value.If f(x) has the same sign on both sides of c, then f(c) is not a local extreme value.Second Derivative TestLet f and f exist at every point in an open interval (a,b) containing c, and suppose that f(c)=0.a)If f(c)0, then f is a local min. value of f.3.4Practical Prob

8、lemsOptimization problems finding the “best” or “l(fā)east” of “most cost effective”, etc. often involves finding the extrema of the functionUse either 1st or 2nd derivative testExampleA fence is to be constructed using three lengths of fence (the 4th side of the enclosure will be the side of the barn).

9、I have 120 yd. of fencing and the barn is 150 long. In order to enclose the largest possible area, what dimensions of fence should be used?(continued on next slide)Example continuedArea is to be optimized: A = l x wPerimeter = 120 yd = 360=2l + ww = 360 2lSo, 2 lengths of 90 and a width of 180. HOWE

10、VER, the barn is only 150 wide, so in order to enclose the greatest area, we wont use a critical point of the function, rather we will evaluate the area using the endpoints of the interval, with w=150. The length = 55 and the area enclosed = 8250 sq. ft.3.5Graphing Functions Using CalculusCritical p

11、oints & Inflections pointsIf f(x) = 0, function levels off at that point (check on either side or use 2nd deriv. test to see if max. or min.)If f(x) is undefined: cusp, corner, discontinuity, or vertical asymptote (look at behavior and limits of function on either side)If f(x) = 0: inflection point,

12、 curvature changes3.6Mean Value Theorem for DerivativesIf f is continuous on a closed interval a,b and differentiable on the open interval (a,b), then there is at least one number c in (a,b) where Example: Find a point within the interval (2,5) where the instantaneous velocity is the same as the ave

13、rage velocity between t=2 and t=5.If functions have the same derivatives, they differ by a constant.If F(x) = G(x) for all x in (a,b), then there is a constant C such that F(x) = G(x) + C for all x in (a,b).3.7Solving Equations NumericallyBisection MethodNewtons MethodFixed-Point AlgorithmBisection

14、MethodLet f(x) be a continuous function, and let a and b be numbers satisfying ab and f(a) x f(b) 0. Let E denote the desired bound for the error (difference between the actual root and the average of a and b).Repeat steps until the solution is within the desired bound for error.Continue next slide.

15、Bisection MethodNewtons MethodLet f(x) be a differentiable function and let x(1) be an initial approximation to the root r of f(x) = 0. Let E denote a bound for the error. Repeat the following step for n = 1,2, until the difference between successive error terms is within the error.Fixed-Point Algor

16、ithmLet g(x) be a continuous function and let x(1) be an initial approximation to the root ro of x = g(x). Let E denote a bound for the error (difference between r and the approximation). Repeat the following step for n 1,2, until the difference between succesive approximations are within the error.

17、3.8 AntiderivativesDefinition: We call F an antiderivative of f on an interval if F(x) = f(x) for all x in the interval.Power RuleIntegrate = AntidifferentiateIndefinite integral = AntiderivativeConstants can be moved out of the integralIntegral of a sum is the sum of the integralsIntegral of a difference is the difference of the integrals3.9 Introduction to Differential EquationsAn equation in which the unknown is a function and that involves derivates (or di