131F06.Assignments

131F06.Assignments - Assignment 1 for MTH 131: Fall 2006...

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Unformatted text preview: Assignment 1 for MTH 131: Fall 2006 Due date: Wednesday September 13. Reading: Sections 1.1, 1.2 in Logan; also review one-variable calculus from any undergraduate text. 1). Limits (i) lim x 1 x 2- 1 1- x (ii) lim x cos( x ) (iii) lim x sin( x ) x (iv) lim x sin( x )- x x 2 2). Continuity a) Show that the function f ( x ) = 1 + x 2 is continuous at x = 0 as follows: for any fixed positive number a , find a positive number b so that the interval [- b, b ] around x = 0 gets mapped inside the interval J = [1- a, 1+ a ] around y = 1 by the function f . If you can do this for any value of a (no matter how small) then you are done! b) Try to repeat the argument above for the function g ( x ) which is equal to f ( x ) on the right side (that is for x > 0) but is identically zero on the left side (that is for x < 0). Can you choose a value for g (0) which makes it continuous at x = 0? If not, why? c) Make up a definition of one-sided continuity (from the right side) for a function f ( x ) at the point x = 0 (use s and s). By applying your definition to the function g ( x ) in (b) above, show that you can choose a value for g (0) so that g ( x ) is continuous from the right side at x = 0. 3). Derivatives Compute derivatives of these functions: (i) ( x 2 + 3) 4 (ii) exp( x 2 + 5 x ) (iii) cos( x 2 + 1) (iv) ( x- 1)- 5 (v) log(cos( x 1 / 2 + 4)) 4). ODEs I a) Show that the function y = sin(1 /x- C ) is a solution of the following ODE for any number C : x 2 dy dx + p 1- y 2 = 0 b) Note that the solution in (a) is not continuous at x = 0 for any value C . Question: can you find a different solution which is continuous at x = 0? 5). ODEs II Verify that the second order ODE d 2 y dx 2 + y = 1 has the general solution y = 1 + A sin( x ) + B cos( x ) where A, B are any numbers. a) Find the values of A, B which solve this equation with boundary values y (0) = 0 and y ( 2 ) = 0. b) Try the same for boundary values y (0) = 0 and y ( ) = 0. What do you find? c) Try the same for boundary values y (0) = 0 and y ( ) = 2. What do you find? 6). ODEs III Logan p.11: Problems #4,#7. Assignment 1 for MTH 131: Fall 2006 Due date: Wednesday September 13. Problems assigned from Logans text 6). ODEs III Logan p.11: Problem #4 Find a solution u = u ( t ) of u + 2 u = t 2 + 4 t + 7 in the form of a quadratic function of t . Logan p.11: Problem #7 Show that the one-parameter family of straight lines u = Ct + f ( C ) is a solution to the differential equation tu- u + f ( u ) = 0 for any value of the constant C . Assignment 2 for MTH G131: Fall 2006 Due date: Wednesday September 20. Reading: Chapter 1, Sections 2.12.2 in Logan; also review one-variable calculus from any undergraduate text....
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131F06.Assignments - Assignment 1 for MTH 131: Fall 2006...

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