Test 3 - , , x n } , an arbitrary partition of [0 , 2] ,...

This preview shows pages 1–7. Sign up to view the full content.

Test 3, Version A Math 1501, Fall 02. Prof. W. Gangbo * October 28, 2002 Student Name: Student Section: Teaching Assistant Name: Instructions. You are to work independently these exercises for the next fourty ﬁve minutes (45 mn.). You may not use any textbook or your class notes during the text. Read carefully each exercise and show all your work. * School of Mathematics, Georgia Institute of technology. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Exercise 1 ( 25 points ) Assume that f ( x )= 2 - 2 x - x 2 if - 2 x 0 | x - 2 | if 0 <x< 3 1 3 ( x - 2) 3 if 3 x 4 . (a) ( 10 points ) Find the critical points of f. (b) ( 10 points ) Find the local extreme values of f. (c) ( 5 points ) Classify the local extreme values of f. 2
Exercise 2 ( 25 points )Set f ( x )=2cos 2 x - x 2 ,x [0 ] . a) ( 5 points )Eva luate f 0 and f . b) ( 15 points ) Find the points of inﬂection of f. c) ( 5 points ) Describe the concavity of f. 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Exercise 3 ( 20 points ) Determine whether or not the graph of x f ( x )=2 x 3 5 - x 6 5 has a vertical tangent or a vertical cusp at c =0 . Justify your answer. 4
Exercise 4 ( 30 points ) Assume that f ( x )=3 x 2 ,x [0 , 5 1 3 ] . a) ( 5 points )For P = { x 0 ,x 1 ,

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , , x n } , an arbitrary partition of [0 , 2] , write L f ( P ) and U f ( P ) , the lower sum and the upper sum of f. b) ( 5 points ) For each i = 1 , 2 , , n consider the three numbers x 2 i , x 2 i-1 , and 1 3 ( x 2 i-1 + x i-1 x i + x 2 i ) . Which one of them is the largest and which one is the smaller? 5 c) ( 10 points ) Denote the following sum by M f ( P ) : ( x 1-x ) x 2 + x x 1 + x 2 1 3 +( x 2-x 1 ) x 2 1 + x 1 x 2 + x 2 2 3 + +( x n-x n-1 ) x 2 n-1 + x n-1 x n + x 2 n 3 . Use b) to deduce which one of the numbers L f ( P ) , U f ( P ) , and M f ( P ) is the largest, and which one is the smaller one. d) ( 5 points ) For each i = 1 , 2 , , n say which one of the two numbers ( x i-x i-1 ) x 2 i-1 + x i-1 x i + x 2 i 3 , x 3 i-x 3 i-1 3 is the smaller. 6 e) ( 5 points ) Use a), c) and d) to evaluate R 5 1 3 f ( x ) dx. 7...
View Full Document

This note was uploaded on 09/22/2009 for the course MATH 1501 taught by Professor Gangbo during the Spring '09 term at University of Georgia Athens.

Page1 / 7

Test 3 - , , x n } , an arbitrary partition of [0 , 2] ,...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online