[PDF] Calculus II, Worksheet 1 Name: Please answer the following





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[PDF] CALCULUS II

CALCULUS II MATH 2414 Calculus I Review Worksheet-Answers 1 1 2 p 558 – Basic Integration Formulas (Thomas' CALCULUS Media Upgrade 11th edition)




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[PDF] CALCULUS II

Calculus I Review Worksheet Find the derivative ( ) dy y or dx ′ for each function 1 y = (x 2 - 4x + 5)2/3 2 y = tan(2πx) 3 cos(5 ) y x = 4 y = sin2 (3x)

[PDF] Calculus II, Worksheet 1 Name: Please answer the following

Philosophical Introduction: In Calculus II, we have a number of ideas The goals of this worksheet are to: 1) build familiarity with basic series, and to 2) look at

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[PDF] Calculus II, Worksheet 1 Name: Please answer the following 40130_6ws1.pdf

Calculus II, Worksheet 1

Name: Please answer the following questions in the spaces provided, or on your own paper. You may use your textbook, but do not consult any other sources or with each other. This worksheet is due onJuly 21st. As you have plenty of time for this, you will not receive credit for illegible or excessively disorganized work. Philosophical Introduction:In Calculus II, we have a number of ideas (Squeezing Theo- rem, Comparison Test, etc.) which rely on intuition in deciding what comparisons to make. The goals of this worksheet are to: 1) build familiarity with basic series, and to 2) look at the properties of series abstractly, to help you see which details are and are not important.

1. (10 pts) Find examples of divergent series

1X k=1a kand1X k=1b kwith limk!1ak= 0 and lim k!1bk= 0 such that: (a) 1X k=1(ak+bk) converges. (b) 1X k=1(ak+bk) diverges. (c) 1X k=1(akbk) converges. (d) 1X k=1(akbk) diverges.

2. (10 pts)

(a) Suppose that ak 1 k=1andbk 1 k=1are sequences such thatak>0 andbk>0 for allkand limk!1bkconverges to a limitB >0.

Show that

1X k=1a kconverges if and only if1X k=1a kbkconverges. (b) Suppose that ak 1 k=1is a sequence such that 0< ak<1 for allkand that the series 1X k=1a kconverges. Determine whether1X k=1a

2kconverges.

(c) Suppose that ak 1 k=1is a sequence bounded below byM1>0 and bounded above byM2. (Note that we do not require thatak 1 k=1converges!)

Compute lim

k!1a kk and determine for what values ofpthe series1X k=1a kk pconverges. 2

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