More Differential Equation Problems

Yeees....the day started off with a quiz just like old times.



So this was a familiar question, a related rates problem. All i remembered while writing this quiz was the steps in the book. Find your equation, which is the pythagorean theorem and differentiate all the variables with respect to time. They gave you the rate of change of x so all that is left is to...solve for dy/dx.



This question was a shell question. Revolving around the y-axis will give you a cylinder of some sort. You can take a sample piece of the cylinder and you'll end up with a rectangle. Then integrate the area of the rectangle. The length would be 2pir since the length is the part of the cylinders circumference. The width is the little change so it'll be dx. The height is equal to the difference between top and bottom (2 and e^x).



This question was straight application of the chain rule, which is just the derivative of the outer function multiplied by the derivative of the inner.



The answer for this one was none. To find the answer quickly we can use process of elimination. Or you can do the long way and derive each of the options...



So back to differential equation problems. What made this problem hard for me was figuring out the initial value. So you set it up the usual way by first finding your differential equation. It said that population of bacteria increased proportionate to the present population. That means the rate is just a constant multiplid by the present. Then all thats left to do is the algebra. The initial value will be 100% and will be 300% when the original population triples and 400% when it quadruples.

K homework is bee question also 2004 ap examp i believe. yeah yinan next scribe you.