Review Blog

So this week, like sarah stated, we've gone over AB stuff (AKA stuff that cam back to haunt us). Although I thought it was the hardest thing last year, it is relatively easy this time around. Here are some tips that you may need to remember when practicing AB for the AP exam. :P

RELATED RATES:
your steps are:
1. Identify what you have (dr/dt etc.)
2. Determine which formula you are dealing with (a lot of times its volume or area...)
3. You usually take the derivative somewhere around this step (like of the equation...)
4. Plug in
5. Solve

EXAMPLE Air is being pumped into a spherical balloon at a rate of 5 cm3/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm.



Solution

The first thing that we’ll need to do here is to identify what information that we’ve been given and what we want to find. Before we do that let’s notice that both the volume of the balloon and the radius of the balloon will vary with time and so are really functions of time.

We know that air is being pumped into the balloon at a rate of 5 cm3/min. This is the rate at which the volume is increasing. Recall that rates of change are nothing more than derivatives and so we know that, dv/dt=5.

We want to determine the rate at which the radius is changing. Again, rates are derivatives and so it looks like we want to determine,
dr/dt= ?
r= diameter/2=10 cm

Note that we needed to convert the diameter to a radius.



Now that we’ve identified what we have been given and what we want to find we need to relate these two quantities to each other. In this case we can relate the volume and the radius with the formula for the volume of a sphere.

V= 4/3 pi r^3

Now we don’t really want a relationship between the volume and the radius. What we really want is a relationship between their derivatives. We can do this by differentiating both sides with respect to t. In other words, we will need to do implicit differentiation on the above formula. Doing this gives,

dV/dt=4 pi r^2 dr/dt

Now I would just plug in giving me:

dr/dt = 1/80pi cm/min.