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.
Now we will find an exact value for f(3), and see which approximation is closer to being correct. We will do this by finding a solution to the Initial Value Problem: f′(x)=x+12​,f(0)=1 3. (a) First, find the general antiderivative of f′(x)=x+12​. [Hint: This answer should have an unknown +C in it. ] (b) Now, find the specific antiderivative which satisfies the initial value f(0)=1. [ Hint: Find which value for C makes f(0)=1.] (c) Using your answer to Part (b), find an exact value for f(3). Then, use a calculator to estimate f(3) to 3 decimal places. (d) Compare the value of f(3) to the approximations you made in Problems 1 and 2 . Which approximation was closer, L0​(3) or E(3) ?
4. On the axes below, graph the following functions: (a) The linear approximation function L0​(x), (b) the Euler approximation function E(x), and (c) the actual function f(x) Make sure you label each of the three functions you graph, so that the grader can tell them apart. [Hint: Find decimal approximations for E(1) and E(2) to help make an accurate graph]