Answer Happy

We are given the following information about a function f : f′(x)=x+12​,f(0)=1. We want to use this information to estimate f(3). In Problem 1, we will use a linear approximation. In Problem 2, we will break up the interval [0,3] into smaller parts to come up with a better approximation. This process is called Euler's Method. In Problem 3, we will find an exact values for f(3) and see which approximation was better. And in Problem 4 we will graph all of the relevant functions to see what is going on.
The problem with this approximation is that x=3 is relatively far from x=0. Over the interval [0,3], the function f(x) bends away from the linear approximation function L0​(x). We can get a better estimate by using a technique called Euler's Method. In this method, we will define a piecewise function E(x)=⎩⎨⎧​m0​x+b0​m1​x+b1​m2​x+b2​​ if 0≤x≤1 if 1<x≤2 if 2<x≤3​ Note that the function has 3 different pieces, and each piece is a straight line. We will want this function to satisfy the following properties: - E(x) is continuous on the interval [0,3]. In particular, this includes the splitting points x=1 and x=2, and - The slope of the line over the interval (n,n+1) should be f′(n). This requirement means that the derivative f′(x) will be used to direct E(x). Once we have constructed E(x), we can use it to estimate f(3)≈E(3). To construct E(x), we need to find appropriate values for the constants m0​,m1​,m2​,b0​,b1​,b2​. Complete the following steps to find these values. (a) The constants m0​,m1​,m2​ are the slopes of the three different pieces of E(x). The slope over the interval (n,n+1) should be f′(n). Use this relation for n=0,n=1, and n=2 to find the values for m0​,m1​, and m2​. (b) The constant b0​ should be chosen so that E(0)=f(0). Use the given value of f(0) to find the correct value for b0​.
(c) We have now found values for b0​,m0​, and m1​. Use these to find a value for b1​ which makes E(x) continuous at x=1. Make sure you use the definition of continuity. (d) We have now found values for b1​,m1​, and m2​. Use these to find a value for b2​ which makes E(x) continuous at x=2. Make sure you use the definition of continuity. (e) We've now found all of the necessary constants. Summarize your findings by filling in the following for E(x) : E(x)=⎩⎨⎧​​ if 0≤x≤1 if 1<x≤2 if 2<x≤3​​ (f) Use the function E(x) to approximate f(3). [ Hint: f(3)≈E(3).]