ws24 - WORKSHEET 24 Fall 1995 1 a Find three distinct...

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WORKSHEET 24 - Fall 1995 1. a) Find three distinct functions f ( x ) such that f 0 ( x ) = 0 for all x . b) Find three distinct functions g ( x ) such that g 0 ( x )=4 x for all x . c) Find three distinct functions h ( x ) such that h 0 ( x )=3 x x 2 for all x . 2. Let f 1 ( x ) be a function such that f 0 1 ( x )=s in( x 2 ). (We do not yet know that there is such a function, but we will see that there is in about a week. We will then describe how to compute it, but we will never be able to write it down as a combination of any functions that you already know.) In the following problems, your answer will be in terms of f 1 ( x ). a) Find f 2 ( x ) such that f 0 2 ( x x 2 )and f 2 ( p π/ 2) = 10 . b) Find f 3 ( x ) such that f 0 3 ( x x 2 f 3 ( p 4) = 18 . c) Draw the graph of f 2 ( x ) as best you can from the information known. (Hint: consider the interval [ kπ, p ( k +1) π ] for each integer k .) 3. Solve the following di±erential equations subject to the prescribed initial conditions. a) dy dx =4( x 7) 3 ,x =8 ,y =10 b) dy dx = x 2 +1 x 2 =1 c) dy dx = x 2 y 2 =0 d) dy dx = 4 p (1 + y 2 ) 3 y . (Hint: For c) and d), move anything involving the variable
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