Page 73 - Calc for the Clueless

But if y = u 100 and we did not know what u was (only a function of x),

And if y = u 100 and u was a function of t (which we did not know),

That's all there is to it.

2 +
3
Let's get back to z = x y . Differentiating with respect to time, we get

How many variables are in this equation? Five!! x, y, dz/dt, dx/dt, and dy/dt. In order to solve this equation,
four of the variables must be given or found from the problem.

Let's try one more.

Example 17—

Now we are ready for related-rates problems. In general these problems are easier than max and min problems
since the most difficult part of the problem, the original equation, is either given or much easier to find than
before.

Example 18—

The radius of a circle is increasing at 2 feet per minute. Find the rate at which the area is increasing when the
radius is 10 feet.

We are given that dr/dr = 2 and we are looking for dA/dt. All we need is to know the area of a circle,
differentiate it with respect to time, and substitute to get the answer.

2
A = πr dA/dt = 2πr dr/dt = 2π(10)(2) = 40π square feet per minute.
Are all the problems this easy? Well, no, but they are not too much more difficult.

Example 19—