MAT 137Y: Calculus with proofs Test 1 - Part B - Version 1.(a)Let gbe a function with domain R. Write the formal de nition of lim x!1 g(x) = 1. (b)Prove that lim x!1 3 p x 5 = 1: Write a proof directly from the formal de nition of limit. Do not use any of the limit laws or any other theorems. Solutions: (a)The standard de nition is

Text. This course is based on Applied Calculus, by Shana Calaway, Dale Hoffman, and David Lippman. The full textbook can be downloaded for free here.Students will learn to apply calculus in economic and business settings, like maximizing profit or minimizing average cost, finding elasticity of demand, or finding the present value of a continuous income stream.

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MAT 137Y: Calculus with proofs Test 1 - Part B - Version 1.(a)Let gbe a function with domain R. Write the formal de nition of lim x!1 g(x) = 1. (b)Prove that lim x!1 3 p x 5 = 1: Write a proof directly from the formal de nition of limit. Do not use any of the limit laws or any other theorems. Solutions: (a)The standard de nition is Some of these solutions omit some details that you will be required to show on a homework assignment, a quiz, and an exam. These details were omitted in some of these solutions for sake of brevity. You should consult the homework solutions in your respective courses to get a sense of the amount of detail you should provide in your solutions.

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    (1.02)≈2.7+1∙0.02=2.72. 3. Here we use the fact that the antiderivative of sec tan is sec and apply the fundamental theorem of calculs: ∫sec tan 𝜋 4 0 =[sec ] 0 𝜋 4=sec 𝜋 4 −sec0 The question asks you to simplify as much as possible. From trigonometry, we know that

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    Calculus II Math 142 Fall 2008 Professor Ben Richert Exam 2 Solutions A few integrals: Z sec3(u) du= 1 2 tan(u)sec(u) + 1 2 (1) lnjsec(u) + tan(u)j+ C Z p 2au u2 u du= p 2au u2 + acos 1 a u a (2) + C Z p a+ bu u du= 2 p a+ bu+ p aln p ... Comparison Test that Z 1 1 1 + sin2(ecosx) p x dxdiverges. Problem 5. (15pts) Use the trapezoidal rule to ...

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    Calculus II Math 142 Fall 2008 Professor Ben Richert Exam 2 Solutions A few integrals: Z ... Comparison Test that Z 1 1 1 + sin2(ecosx) p x dxdiverges. Problem 5 ... Chapter 0 Revision 0.1 Exponents Definition (1) Let n be a positive integer and let a be a real number. We define an to be the real number given by an = a| a{z}a n factors (2) Let n be a negative integer n, that is, n = k where k is a positive integer, and let a be a real number

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