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Unformatted text preview: Calculus Problems Math 504 505 Jerry L. Kazdan 1. Sketch the points ( x , y ) in the plane R 2 that satisfy | y x | 2. 2. A certain function f ( x ) has the property that Z x f ( t ) dt = e x cos x + C . Find both f and the constant C . 3. Compute lim x parenleftBig cos x cos2 x parenrightBig 1 / x 2 . 4. Sketch the curve that is defined implicitly by x 3 + y 3 3 xy = 0. Calculate y ( ) . 5. Calculate n = 1 1 4 n 2 1 . 6. Determine the indefinite integral Z log ( 1 + x 2 ) dx . 7. Let f ( x ) be a smooth function for 0 x 1. If f ( x ) = 0 for all 0 x 1, what can you conclude? Prove all your assertions. 8. Solve the initial value problem ( 1 + e x ) yy = e x with y ( 1 ) = 1 . 9. Let the continuous function f ( ) , 0 2 represent the temperature along the equator at a certain moment, say measured from the longitude at Greenwich.. Show there are antipodal points with the same temperature. 10. a) Let g ( x ) : = x 3 ( 1 x ) . Without computation, show that g ( c ) = 0 for some 0 < c < 1. b) Let h ( x ) : = x 3 ( 1 x ) 3 . Show that h ( x ) has exactly three distinct roots in the interval 0 < x < 1. c) Let p ( x ) : = parenleftbigg d dx parenrightbigg 4 ( 1 x 2 ) 4 . Show that p is a polynomial of degree 4 and that it has 4 real distinct zeroes, all lying in the interval 1 < x < 1. 1 11. In R 2 , let Q 1 = ( x 1 , y 1 ) , Q 2 = ( x 2 , y 2 ) , and Q 3 = ( x 3 , y 3 ) , where x 1 < x 2 < x 3 . a) Show there is a unique quadratic polynomial p ( x ) that passes through these points: p ( x j ) = y j , j = 1 , 2 , 3. b) If y 1 > y 2 and y 3 > y 2 and f ( x ) is any smooth function that passes through these three points, show there is some point c ( x 1 , x 3 ) where f ( c ) > 0. Even better, for some c , f ( c ) p , so p is the optimal constant. [Remark: It is enough to consider the special case where x 2 = 0 and y 2 = 0. Then write x 1 = a < 0, x 3 = b > 0]. 12. a) If f ( x ) > 0 is continuous for all x 0 and the improper integral R f ( x ) dx exists, then lim x f ( x ) = 0. Proof or counterexample. b) If f ( x ) > 0 is continuous for all x 0 and the improper integral R f ( x ) dx exists, then f ( x ) is bounded. Proof or counterexample. 13. Find explicit rational functions f ( x ) and g ( x ) with the following Taylor series: f ( x ) = 1 nx n , g ( x ) = 1 n 2 x n . 14. a) Let x = ( x 1 , x 2 ) be a point in R 2 and consider Z R 2 1 ( 1 + bardbl x bardbl 2 ) p dx . For which p does this improper integral converge? b) This integral can be computed explicitly. Do so. c) Repeat part a) where x R 3 and the integral is over R 3 instead of R 2 ....
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.
- Fall '10