Calculus - guide for tricky questions

The concept of limits is the foundation of calculus and serves as the crucial bridge connecting differentiation and integration. This unit will focus on the computation of limits and how to use limits to find derivatives, differential functions, and asymptotes. Table of Contents Precise Definition of Limits Limits of Composite Functions Squeeze Theorem Definition of Derivatives Finding Asymptotes Precise Definition of Limits Do not underestimate this concept. In high school, we learn an intuitive definition of limits, but the college definition is different and often a focal point for professors. Here’s a review of the precise definition of limits: We write \[\lim\limits_{x\to a}f\left(x\right)=L\] if for every number \(\delta \gt 0\) there is a number \(\varepsilon \gt 0\) such that \(\bbox[#e0ebeb,5px,border:3px solid #b3f0ff]{\text{if}~0<\left|x-a\right|<\delta~~~ \text{then}~~  \left|f(x)-L\right|<\

微分公式可以為我們省下大量用極限求導數或導函數的時間。本單元將著重於探討鏈鎖律、隱函數的微分和反函數的導數。 目錄 鏈鎖律 隱函數微分 反函數的導數 鏈鎖律 鏈鎖律非常實用，使用羅必達時很常會搭配鏈鎖律，但因為其實就是按部就班套公式，頂多有點複雜，所以通常會搭配別的觀念出題，本身只是配角。看一題範例： \( \text{Find the derivative of } ~y = \left( \tan^{-1} x \right)^{\sin x},  x > 0 .\) 【台大105】 看到 指數有變數 或 高次因式相乘除 ，不論是取極限還是求導，一般可以換成以 \(e\) 為底或取 \(\ln\) 。 \[ \frac{dy}{dx} = \frac{d}{dx} e^{\sin x \cdot \ln(\tan^{-1} x)} \] \[ = e^{\sin x \cdot \ln(\tan^{-1} x)} \cdot \frac{d}{dx} \left( \sin x \cdot \ln(\tan^{-1} x) \right) \] \[ = e^{\sin x \cdot \ln(\tan^{-1} x)} \cdot \left( \cos x \cdot \ln(\tan^{-1} x) + \sin x \cdot \frac{d}{dx} \left( \ln(\tan^{-1} x) \right) \right) \] \[ = e^{\sin x \cdot \ln(\tan^{-1} x)} \cdot \left( \cos x \cdot \ln(\tan^{-1} x) + \sin x \cdot \frac{1}{\tan^{-1} x} \cdot \frac{1}{1 + x^2} \right) \] \[ = \left( \tan^{-1} x \right)^{\sin x} \c

English version 極限的概念是微積分的基礎，也是連接微分和積分的重要橋梁。本單元將聚焦於極限的計算，以及如何利用極限求出導數、導函數和漸進線。 目錄 極限的精確定義 合成函數的極限 夾擠定理 導數定義 求漸進線 極限的精確定義 不要小看這個觀念，在高中時我們學的是極限的直觀定義，但到了大學端的定義有所不同，反而會是教授命題的重點。這裡為大家複習一下極限的精確定義： We write \[\lim\limits_{x\to a}f\left(x\right)=L\] if for every number \(\delta \gt 0\) there is a number \(\varepsilon \gt 0\) such that \(\bbox[#e0ebeb,5px,border:3px solid #b3f0ff]{\text{if}~0<\left|x-a\right|<\delta~~~ \text{then}~~  \left|f(x)-L\right|<\varepsilon}\) 廢話不多說，直接上一道例題： Let \(A=\{0.6,0.7,0.8,0.9\},\) Find the largest number, \(\delta\) , in \(A\) such that \(\left|\sqrt{4x+5}-3\right|\lt0.6\), whenever \(\left|x-1\right|\lt\delta\). (A) 0.6 ,      (B) 0.7 ,      (C) 0.8 ,      (D) 0.9 . 【交大100】 由左式可以得到： \begin{aligned} &−0.6\lt\sqrt{4x+5}−3\lt0.6\\ &\Rightarrow2.4\lt\sqrt{4x+5}\lt3.6\\ &\Rightarrow5.76\lt\sqrt{4x+5}\lt12