Calculus I (AB): Implicit Differentiation

Given , find .

Difficulty:

Solution:

In order to derive with respect to x, you need to be careful. All x terms are differentiated normally. However, y terms are done a bit differently. In essence, you derive them normally but add a to the end of each term. For example, the derivative of 2y is not just 2. It's 2. Be careful when differentiating the 4xy term. Since there are two variables, you must use the product rule:

Now, move all of the non- terms to the right side of the equation, factor out the and solve for it:

Given the differential equation , and the fact that y(2) = 1, use Euler's Method to estimate y(3) with four steps of size .

Difficulty:

Solution:

I'll walk you through the first step. You'll have to repeat this three more times once we're done, since the problem asks you to use four steps. We start at the point (2,1) and want to proceed units to the right, ending up at a new x value of . The question is this: What is the new y that matches this new x? We need a complete point. In order to find that y value, we first calculate at the point (2,1):

Almost there. Multiply this value by the step size given in the problem: . This tells us that height changes by 0 units from the original y value of 1 (from our initial point). Therefore, our new point is:

You now repeat the process using the new point as you used (2,1). Here's the next three steps:

This gives you an approximate solution to the differential equation of 62/55 when x = 3. This is approximately equal to 1.12727.