A student studies a curve whose slope at the point (x, y) is dy/dx = (2x + y + 1)/(4x + 2y + 3)…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A student studies a curve whose slope at the point is . The numbers and are plain numbers. The two lines that make the top and the bottom zero do not cross. The curve passes through the point . She expects the slope of the curve to settle towards a fixed number far out. The curve can be written as . Find the constant to four decimal places, the slope at , and the limiting value of the slope when becomes very large.
Need a reminder? The key ideaShow
A substitution replaces a combination of and by a new unknown . For put . A Bernoulli equation becomes linear with .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Given. and the point .
Plan. Put , separate, integrate, and use the point.
Step 1. Substitute. The bottom is . So .
Step 2. Separate. . Write . Integrating gives .
Step 3. Return to and . Multiply by : . This simplifies to .
Step 4. The constant. At : .
Step 5. The slopes. At the slope is . In terms of the slope is , which tends to as becomes very large.
Answer: , slope , and limiting slope .
Watch out. Shifting the origin does not work here, because the lines are parallel.
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