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MA 105 D4 T5 Homework Assignment Deadline: Oct-14-2013 Max. Marks: 20 Answers should be clear and neat. Zero marks will be given to unclear answers. Please try to highlight the imp. and final points, deductions and theorem state- ments, if any. Answer the questions in order. Answers out of order will not be corrected and no exception will be made in this regard. 1. Let a, b be real numbers such that 0 <a<b. Consider the sequence {x 1 } defined recursively by x 1 = a and x n+1 = q ab 2 +x 2 n a+1 for n ∈ N Show that {x n } is bounded above and monotonically increasing. Also, find lim x→+∞ x n . [4] 2. Let f : R → R be such that |f (x) - f (y)|≤ (x - y) 2 for all x, y ∈ R. If f (0) = 5 find f (1). [4] 3. Consider the function f : (0, 1) → R defined by f (x)= x 1 x . (a) Find the intervals in (0, 1) where f is increasing/decreasing and Anal- yse convexity/concavity (b) find the points in (0, 1), if there are any, where (i) f has a local minimum, and (ii) f has a local maximum. Justification required! [6] 4. Consider the planar region D bounded by the curves y = e x and y = ln x and vertical lines x = 1 and x = e 2 . (a) Find area of D. (b) Find the volume of the solid generated by revolving the planar region D about x-axis. [6] Submission details : Answers are to be hand written and the order of ques- tions must be maintained. If you are not answering a question, leave space for that. Front page must have roll number and name. Submission venue : In monday’s tutorial Oct 14th. In any case, any one misses the tutorial, it is the students responsibility to handover the copy before Tuesday night by meeting me at H8 R207. No further answer books will be taken. 1
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MA 105 D4 T5 Homework Assignment
Deadline: Oct-14-2013 Max. Marks: 20Answers should be clear and neat. Zero marks will be given to unclear answers.Please try to highlight the imp. and final points, deductions and theorem state-ments, if any. Answer the questions in order. Answers out of order will not becorrected and no exception will be made in this regard.
1. Let a, b be real numbers such that 0 < a < b. Consider the sequence {x1}defined recursively by
x1 = a and xn+1 =√
ab2+x2n
a+1 for n ∈ NShow that {xn} is bounded above and monotonically increasing. Also,find limx→+∞ xn. [4]
2. Let f : R → R be such that |f(x) − f(y)| ≤ (x − y)2 for all x, y ∈ R. Iff(0) = 5 find f(1). [4]
3. Consider the function f : (0, 1)→ R defined by f(x) = x1x .
(a) Find the intervals in (0, 1) where f is increasing/decreasing and Anal-yse convexity/concavity
(b) find the points in (0, 1), if there are any, where (i) f has a localminimum, and (ii) f has a local maximum. Justification required!
[6]
4. Consider the planar region D bounded by the curves y = ex and y = lnxand vertical lines x = 1 and x = e2.
(a) Find area of D.
(b) Find the volume of the solid generated by revolving the planar regionD about x-axis.
[6]
Submission details : Answers are to be hand written and the order of ques-tions must be maintained. If you are not answering a question, leave space forthat. Front page must have roll number and name.Submission venue : In monday’s tutorial Oct 14th. In any case, any onemisses the tutorial, it is the students responsibility to handover the copy beforeTuesday night by meeting me at H8 R207. No further answer books will betaken.
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