Related rates

Definition:

In differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time.

In related rates type of problems all worked in a very different manner, but the process will be essentially the same in each. In each problem we will identify what we were given and what we wanted to find. We next write down a relationship between all the various quantities and used implicit differentiation to arrive at a relationship between the various derivatives in the problem. Finally, we plugged into the equation to find the value we were after.

So, in a general sense each problem was worked in pretty much the same manner. The only real difference between them was coming up with the relationship between the known and unknown quantities. This is often the hardest part of the problem. In many problems the best way to come up with the relationship is to sketch a diagram that shows the situation. This often seems like a silly step, but can make all the difference in whether we can find the relationship or not.

In some problem we can see an important idea in dealing with related rates. In order to find the asked for rate all we need is an equations that relates the rate we're looking for to a rate that we already know. Sometimes there are multiple equations that we can use and sometimes one will be easier than another.

Also, in some problems there we will often have an equation that contains more variables that we have information about and so, in these cases, we will need to eliminate one (or more) of the variables. we can eliminate the extra variable using the idea of similar triangles. This will not always be how we do this, but many of these problems do use similar triangles so make sure you can use that idea.

If Q is a quantity that is varying with time, we know that the derivative measures how fast Q is increasing or decreasing. Specifically, if we let t stand for time, then we know the following.

Rate of change of Q = dQ/dt

In a related rates problem, we are given the rate of change of certain quantities, and are required to find the rate of change of related quantities.

Solving a Related Rates Problem

Step 1: Identify the changing quantities, possibly with the aid of a sketch.

Step 2: Write down an equation that relates the changing quantities.

Step 3: Differentiate both sides of the equation with respect to t.

Step 4: Go through the whole problem and restate it in terms of the quantities and their rates of change. Rephrase all statements regarding changing quantities using the phrase the rate of change of . . . .

Last Step: Substitute the given values in the derived equation you obtained above, and solve for the required quantity.

One of the most fundamental concepts of calculus is the fact that the derivative is used to model change. In related rate applications one tries to find the rate at which one quantity is changing by relating it to other quantities with known rates of change. In the preceding applet we applied related rates to the determination of the rate at which the distance between two ships changes. However, this example is just one of a variety of applications that pertain to related rates