|
所在平台: Udemy |
课程主页: https://www.udemy.com/course/applied-math-limit-derivative-applications-of-derivative/
课程评论：没有评论
Coursera应用数学课程：“极限、导数及其应用” 本课程旨在深入讲解微积分中的核心概念——极限与导数，并探讨导数在实际问题中的广泛应用。 **课程内容概述：** * **极限（Limits）** * 课程将从直观的理解入手，介绍极限的概念，包括左极限和右极限。 * 学习如何计算多项式、有理函数、三角函数、指数函数和对数函数的极限。 * 理解极限的定义，并认识到极限值与函数在某点的值可能不同，甚至其中一个可能未定义。 * **导数（Derivatives）** * 导数被引入为变化率，既可以描述距离函数的变化率，也可以从几何上理解为函数图像切线的斜率。 * 深入学习导数的定义，并掌握导数的基本计算法则，包括和、差、积、商的求导法则。 * 学习多项式函数和三角函数的导数。 * **导数的应用（Applications of Derivatives）** * **变化率：** 将导数应用于描述物体运动、数量变化等实际场景。 * **单调性：** 利用导数判断函数的增减区间。 * **切线与法线：** 掌握如何利用导数求解函数图像的切线和法线方程。 * **近似计算：** 探索导数在数值近似计算中的应用。 * **极值（最大值与最小值）：** * **一阶导数判定法：** 从几何角度直观理解如何利用导数符号的变化来判断局部极值。 * **二阶导数判定法：** 学习并掌握作为严格数学工具的二阶导数判别法。 * **极值点的定义：** 学习识别临界点（导数为零或导数不存在的点）。 * **增量与微分：** 理解 $\Delta y$（y的增量）与 $dy$（y的微分）的概念。 **总结要点：** * **极限定义：** 左极限、右极限以及当两者相等时，该共同值即为该点的极限。 * **导数法则：** $(u \pm v)' = u' \pm v'$，$(uv)' = u'v + uv'$。 * **基本导数：** $\frac{d}{dx}(\sin x) = \cos x$，$\frac{d}{dx}(\cos x) = -\sin x$。 * **一阶导数判别局部极值：** 通过导数符号变化分析。 * **二阶导数判别局部极值：** $f'(c)=0$ 且 $f''(c)<0$ 时为局部极大值；$f'(c)=0$ 且 $f''(c)>0$ 时为局部极小值。若 $f'(c)=0$ 且 $f''(c)=0$，则需回退到一阶导数判别法。 * **工作规则求绝对极值：** 寻找所有临界点和区间端点，计算函数值并比较。 * **临界点：** 导数为零或导数不存在的点。 本课程将通过一系列简单的问题，帮助学习者巩固基础知识，理解核心原理，并能将所学知识应用于解决现实生活中的问题。
Limits and DerivativesDerivative introduced as rate of change both as that of distance function and geometrically.Intuitive idea of limitLimits of −Polynomials and rational functions.Trigonometric, exponential and logarithmic functions.Definition of derivative, relate it to slope of tangent of a curve, derivative of sum, difference, product and quotient of functions.The derivative of polynomial and trigonometric functions.Applications of DerivativesApplications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normal, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool)Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)SUMMARYLimits and Derivatives1. We say lim x→a- f(x) is the expected value of f at x = a given the values of f near x to the left of a. This value is called the left hand limit of f at a. 2. We say lim x→a+ f(x) + is the expected value of f at x = a given the values of f near x to the right of a. This value is called the right hand limit of f(x) at a. 3. If the right and left hand limits coincide, we call that common value as the limit of f(x) at x = a and denote it by lim x→a f(x).4. The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit. 5. Limit of a function at a point is the common value of the left and right hand limits, if they coincide.6. For a function f and a real number a, lim x→a f(x) and f (a) may not be same (In fact, one may be defined and not the other one).7. For functions u and v the following holds: (u ± v)' = u' ± v' (uv)' = u'v + uv'.8. Following are some of the standard derivative -d/dx (sin x) = cos xd/dx (cos x) = -sin xApplications of Derivatives1. First Derivative Test Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then (i) If f ′(x) changes sign from positive to negative as x increases through c, i.e., if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima. (ii) If f ′(x) changes sign from negative to positive as x increases through c, i.e., if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima. (iii) If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Infact, such a point is called point of inflexion. 2. Second Derivative Test Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then (i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0 The values f (c) is local maximum value of f. (ii) x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0 In this case, f (c) is local minimum value of f. (iii) The test fails if f ′(c) = 0 and f ″(c) = 0. In this case, we go back to the first derivative test and find whether c is a point of maxima, minima or a point of inflexion.3. Working rule for finding absolute maxima and/or absolute minima Step 1: Find all critical points of f in the interval, i.e., find points x where either f ′(x) = 0 or f is not differentiable. Step 2:Take the end points of the interval. Step 3: At all these points (listed in Step 1 and 2), calculate the values of f. Step 4: Identify the maximum and minimum values of f out of the values calculated in Step 3. This maximum value will be the absolute maximum value of f and the minimum value will be the absolute minimum value of f.4. A point c in the domain of a function f at which either f ′(c) = 0 or f is not differentiable is called a critical point of f.5. Let y = f(x), ∆x be a small increment in x and ∆y be the increment in y corresponding to the increment in x, i.e., ∆y = f(x + ∆x) - f(x).
