Notice this time the formula involves both variables, t and y in this case. So we can rephrase the question with y ‘ = t(2 – y). Just substitute y for f( t) to make it easier to work with. Here, the step size is h = 0.5, so there will only be two new time values before landing on the goal time t = 1: Use Euler’s Method with step size 0.5 to estimate the value of f(1). nįinally, we see that our final answer is: y(1) = y 4 = 6.586. Now remember, the derivative is given you you: y ‘ = 3 y – 1. Note, here we are using x instead of t because sometimes the AP Calculus question will be in terms of x and y rather than t and y.īecause the step size is 0.25, we use x n = x 0 + nh to get: The equation y(0) = 1 means that the initial time is x 0 = 0 and initial value is y 0 = 1. The first clue is in the given information. Use Euler’s Method with h = 0.25 to estimate y(1) if y ‘ = 3 y – 1 and y(0) = 1.
Let’s get to the examples! Example 1 - Getting the Basics Down Instead, if you have to solve a differential equation or initial value problem, then try using methods like separation of variables or in the simplest cases, integration (for a refresher, take a look at: AP Calculus Exam Review: Integrals).
Unless the directions specify to use Euler’s Method, do not use it! It’s important to realize that this method does not give exact answers - just good estimates. One clue ( y-value) leads you to the next. Using Euler’s Method is like detective work. On the other hand, why stop at just one step? Once you know the value of y 1, then you can use that value in a similar way to find out y 2. If you’re only going from t 0 to t 1 (a single step), then you get: That means that you can replace y ‘ by its current value as determined by the function f. The final piece of the puzzle is to use the given information that y ‘ = f( t, y). Letting the difference in time be h = t 1 – t 0, then the formula becomes: Now if y changes from y 0 to y 1 in the time interval from t 0 to t 1, then we can express this fact using the derivative: So even if you don’t know what y is, if you can tell how fast it’s changing, then that info can help you to build up the values of y. The key is that the y ‘ always measures the rate of change of the function y.
In fact, a good grasp of the theory helps you in other areas of calculus as well. It’s helpful to understand a little of the theory behind Euler’s Method. Then the solution at later times t 1, t 2, t 3, … can be found using the following algorithm. The given time t 0 is the initial time, and the corresponding y 0 is the initial value.įirst, you must choose a small step size h (which is almost always given in the problem statement on the AP exam). Euler’s MethodĮuler’s Method is a step-based method for approximating the solution to an initial value problem of the following type. What is Euler’s Method? In this post we review this technique for approximating solutions to certain kinds of differential equations and work out a few examples based on what you may see on the AP Calculus BC exam.
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