Delete End rage Down Enter Hon B M 4 Shit 4 |WRITTEN HOMEWORK 3 (WH3) SP19.nb (III) Solve the Initial Value Problem (IVP) : f' (x) = x2+1 -; f (0)=1. (a) First, solve the differential equation (DE) : f' (x) = x2 x2+1 Hint : - x2 x2+1 -1 x2+1 1 x2+1 x2+1 x241 x2+1 =1- 1 x2 +1 Guess a function whose derivative is ( 1 - yaz ) i.e., find an antiderivative of (1 - ya z ) - ) flow (b) Find the solution of the Initial Value Problem (IVP). fw) =1 V () = c , C-( fix= > _ ant and + ANSWER : f (x)= x- arctanal + (The graph of f is given in the figure above.)End PDn) WRITTEN HOMEWORK 3 (WH3) SP19.nb | 3 SH but x2 MS (II) The figure shows the graph of f, the particular solution of the IVP : f' (x) = x2 +1 f (0) =1 Du #21 (6 (a) In the same figure, sketch the graph of the function g, as defined in part I (f). (I ON) (a) Or y =f (x ) Fi (b) 2.23 1.t (b) Do you think that g is a good approximation of f ? Why do you think that this method (of constructing a piece - wise linear approximation of f) can be useful ?2 | WRITTEN HOMEWORK 3 (WH3) SP19.nb (c) On the interval [2, 3], g is a linear function and its graph is a line whose slope is f' (2). Find an expression for g (x), 2 s x s 3. g (x) = 2sxS3. (d) On the interval [3, 4], g is a linear function and its graph is a line whose slope is f' (3). Find an expression for g (x), 3 sxs4 g (x) = , 3SX$4. (e) On the interval [4, 5], g is a linear function and its graph is a line whose slope is f' (4). Find an expression for g (x), 4 s x s 5. 4SX$5 = (x) 6 hut (f) Summarize your work in parts (a) - (e) and write an expression for g (x) on the interval [0, 5] : 0sxs1 , 1