Math%20119%202005-2006%20Fall%20MidTerm1

# Math%20119%202005-2006%20Fall%20MidTerm1 - Compute(Show...

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Group List No. Last Name Name Student No. Department Section Signature : : : : : : : : : : : : : Code Acad. Year Semester Coordinator Date Time Duration M E T U Department of Mathematics CALCULUS I Mid Term 1 Math 119 2005-2006 Fall Muhiddin Uguz November.12.2005 09:00 100 minutes 6 QUESTIONS ON 4 PAGES TOTAL 60 POINTS 1 2 3 4 5 6 Question 1 (8 points) Let f ( x ) = ± x 2 if x 6 = 1 1 if x = 1 a) Does lim x 1 f ( x ) exits? Explain your answer. b) Is f ( x ) continuous at x = 1? Explain your answer. c) Is f ( x ) diﬀerentiable at x = 1? Explain your answer. d) Using ± - δ technique, prove that lim x 0 f ( x ) = 0 . Muhiddin UĞUZ

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Question 2 (4+4+4=12 points) Compute the limits below without using the L’Hopital’s Rule. If you use a theorem or a well known limit, state it explicitly. a) lim x 0 | 2 x - 3 | - | x - 3 | x b) lim x →∞ sin( x ) e x c) lim x 3 x 3 - 27 x 2 - 9 Question 3 (4+4+4=12 points)

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Unformatted text preview: Compute : (Show your work) a) lim h → 3 h-1 h ( do not use the L’Hopital’s Rule) Muhiddin UĞUZ b) d dx tan ± cos( x ) x ² c) d dx h x sin( x ) + ln( x √ x ) i Question 4 (5+3=8 points) a) Show that the equation f ( x ) = x 5 + 15 x + 1 = 0 has a solution. b) Show that there is no more than one solution. (Hint: you may use derivative) Muhiddin UĞUZ Question 5 (10 points) Find an equation of the tangent line to the curve e xy + y 2 sin( πx )-e = 0 at the point (1 , 1) . Question 6 (10 points) Using diﬀerentiation prove that 2 arctan( r 1 + x 1-x ) + arccos( x ) = π holds for all 0 ≤ x < 1 . Muhiddin UĞUZ...
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## This note was uploaded on 11/01/2011 for the course CHEMICAL E CHEM-107 taught by Professor Bilinmiyor during the Spring '11 term at Middle East Technical University.

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Math%20119%202005-2006%20Fall%20MidTerm1 - Compute(Show...

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