131F08.Assignments

131F08.Assignments - Assignment 1 for MTH 131: Fall 2008...

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Unformatted text preview: Assignment 1 for MTH 131: Fall 2008 Due date: Wednesday September 17. Reading: Sections 1.1, 1.2 in Logan; also review one- and two-variable calculus from any undergraduate text. 1). a) Verify that the functions e- 3 x , e x are solutions of the ODE y 00 + 2 y- 3 y = 0. b) Find values of r so e rx is a solution of the ODE y 000- 3 y 00 + 2 y = 0 c) Find values of r so x r is a solution of the ODE x 2 y 00 + 4 xy + 2 y = 0 2). Find the partial derivatives f x and f y for the functions a) f = xy 2 b) f = x 2 e y c) f = sin 2 ( x- y ) d) f = p x 2- 3 y 3). Use lHopitals rule to evaluate the limits: lim x sin x x , lim x x tan 3 x sin 2 x , lim x x ( e 2 /x- 1) 4). Review the statement of the Mean Value Theorem (MVT). Use the MVT to prove the following: if f ( x ) is constant for all x in the closed interval [ a, b ] then f is a linear function on [ a, b ]. 5). Show that the function f ( x ) = sin(1 /x ) is continuous and bounded on the open interval (0 , ). [Hint: the composition of two continuous functions is continuous]. Show that there is no number c such that the function g is continuous on [0 , ) where g ( x ) = n sin(1 /x ) for x > c for x = 0 6). 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? 7). 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? 8). Logan p.11: Problems #4,#7. Assignment 2 for MTH G131: Fall 2008 Due date: Wednesday September 24. Reading: Chapter 1, Sections 2.12.2 in Logan; also review one-variable calculus and Taylor polynomials from any undergraduate calculus text. 1). Anti-derivatives Compute indefinite integrals of these functions: (i) x 5 / 2 (ii) x + 2 (iii) x x 2 + 4 (iv) cos(3 x ) (v) cos( x ) 1+sin( x ) 2). ODEs analytical solutions a) Logan, p.18, #3 b) Logan, p.18, #4 c) Logan, p.57, #1 d) Logan, p.58, #4 e) Logan, p.67, #3 3). Taylor polynomials Compute Taylor polynomials of these functions to the stated order at the given point: (i) ln(1 + x ), fourth order at x = 0 (ii) x , third order at x = 1 (iii) ( x- 3)- 2 , third order at x = 1 4). Remainder estimates a) Derive the fifth order Taylor polynomial P 5 ( x ) for the function sin( x ) at x = 0....
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This note was uploaded on 10/19/2011 for the course MATH 3354 taught by Professor Drager during the Fall '08 term at Texas Tech.

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131F08.Assignments - Assignment 1 for MTH 131: Fall 2008...

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