多伦多大学大一微积分作业题汇总

　　多伦多大学的微积分课程学习众多数学主题，不少同学总是觉得课程很难，考试也不知道怎么备考。下面我们提供一些这门课程经常会出现的一些作业题目，同学们在平时学习中或者考前复习时都可以用这些题目来做提升训练。

　　一、课程内容

　　unit1：逻辑、集合、符号、定义和证明

　　unit2：极限和连续性

　　unit3：导数

　　unit4：超越函数

　　unit5：中值定理及其应用

　　unit6：极限和导数的应用

　　unit7：积分的定义

　　unit8：微积分的基本定理

　　unit9：积分方法

　　unit10：积分的应用

　　unit11：序列

　　unit12：反常积分

　　unit13：系列

　　unit14：幂级数和泰勒级数

　　二、作业题分享

　　1.Negate each of the following statements without using any negative words (‘no’,‘not’, ‘none’, etc):

　　(a) “Every page in this book contains at least one word whose first and last letters both come alphabetically before M.”

　　(b) “I have a friend all of whose former boyfriends had at least two siblings with exactly three different vowels in their name.”

　　(c) “If a student in this class likes the musical Cats then they are not my friend.”

　　2. Consider the following definitions about real numbers x:

　　• x is courageous when ∀a > 0, x < a

　　• x is hard-working when ∀a ≥ 0, x < a

　　• x is intelligent when ∀a > 0, x ≤ a

　　• x is ambitious when ∀a ≥ 0, x ≤ a

　　3.Given a real number x, we defined the floor of x, denoted by b xc , as the largest integer smaller than or equal to x. For example, [π] = 3, [ 7 ] = 7, and [−0.5] = −1.

　　(a) Sketch the graph of this function. At which points is the function f(x) = [x] continuous? Which discontinuities are removable and which ones are nonremovable?

　　(b) Consider the function h(x) = [sin x] . Show that h has exactly one removable and one non-removable discontinuity inside the interval (0, 2π).

　　4.The functions G and H have both domain R. They are continuous everywhere. They satisfy G(0) = 0 and H(0) = 0. Moreover, G0 = g and H0 = h, where g and h are the functions in Question 2. Sketch the graphs of G and H.

　　5.You know what your final answers should be. Just make sure that at each point you are only using things that have already been proven.

　　6.Now we have a bar which lies on the x-axis, from the position x = 1 to the position x = 7 (all values of x are measured in meters). The bar is made of different materials. Between x = 1 and x = 2, the density of the bar is 2kg/m; between x = 2 and x = 5, the density is 3kg/m; between x = 5 and x = 5.5, the density is 10kg/m, and between x = 5.5 and x = 7, the density is 1kg/m. What is the total mass of the bar?

　　三、微积分课程作业基本解题思路

　　审题→绘图→思考方法并选择→套用定理or公式→边界条件→检查计算过程

　　1.审题时需要你明确这道题目的具体要求，找到关键信息，尤其是一些隐含条件。对于有些题目，需要绘图才能更清晰地知道怎么解题。一道题目也可能不止一种解题方式，在解题时，你往往需要决定使用哪种方法，另外可以根据题目特点选择恰当的微积分方法，比如求导、积分、极限等等。

　　解题过程中，一些基本定理和公式是关键，导数、积分、微分中值定理分别是什么，怎么使用，这些在你练题之前都可以先复习一遍。最后，做完题目，一定不能忘记检查，不要犯一些低级错误。

　　以上是多伦多大学的大一微积分课作业相关问题分享，希望可以帮助大家更了解题目出题方式。在学习过程中有任何问题，同学们都可以直接向考而思的专业老师提问!