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# Calculus MATH 1A

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This 2 page Study Guide was uploaded by Kavon Feest on Thursday October 22, 2015. The Study Guide belongs to MATH 1A at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/226594/math-1a-university-of-california-berkeley in Mathematics (M) at University of California - Berkeley.

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Date Created: 10/22/15

MATH 1A FALL 2008 FINAL STUDY AIDE MARTIN OLSSON Disclaimer You are responsible for all the material indicated on the syllabus The questions below are intended only as a way to get more practice7 and to give an indication of the type of problems that could appear on the exam A few general comments 1 The nal will cover all the material for the course Dont forget to study the early material 2 It is a good idea to go over carefully the midterm exams 3 Since the material after the second midterm has not been tested yet7 at least some of it is sure to show up on the nal 1 a Let f be a function de ned on all real numbers Explain what is meant by limmace f 17 b Show using only the de nition in a that lim 17 395 714953 17 2 Let f Explain what happens if you try to use Newton7s method to solve f 0 with a starting guess of 0 1 3 Compute each of the following integrals a 2 H 111m 6 dx 1 12 0 71 7r2 sin3xdx 0 fczdx 4 A car approaches a large hotel at night7 driving along a semicircular driveway which becomes tangent to the front wall of the hotel at the entrance The headlights of the car illuminate a spot on the front wall If the radius of the driveway is 10 meters7 and if the car is moving at a speed of 10 kmhr7 with what speed is the spot moving when the angle between the entrance7 the center of the circle7 and the car is 7r3 radians 1 b 2 MARTIN OLSSON 5 Prove using mathematical induction the formula 2 3 2 5 BMW ltn gtltn6 gtlt n gt75 i1 Conclude that for any integer n 2 1 the rational number n 2n 32n 5 6 is in fact an integer for the curious prove this another way 6 a Give an example of a function f on 01 with an absolute maximum at 14 and absolute minimum at 34 b Give an example of a function on 01 which is not integrable c Give an example of a continuous function on 01 with fol fzd 2 7 Compute the following limits 1 lim Jr 171 we lixlim 39 b lim M zaO x2 0 lim lnz 7 lnz 8 Differentiate the following functions a tan2m39 b i c log22x2 sin 9 Let S be the solid obtained by rotating the region 2 4y2 3 4 about the x axis a Find the volume of S using discs slicing b Find the volume of S using cylindrical shells c Check that the answers in a and b are the same 10 a State the Mean Value Theorem b True or False Let f be a function which is continous and differentiable everywhere lf fx 0 has two distinct solutions then f 0 also has a solution If true explain why if false give a counterexample

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