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Note To follow this tutorial, you should be familiar with the chain rule for derivatives.

Rates of change

We start by recalling some facts about the rate of change of a quantity:
Warmup: Rate of change of Q

If $Q$ is a quantity that changes with time, then the derivative $dQ/dt$ measures how fast $Q$ is increasing or decreasing at the instant $t:$
Rate of change of $Q = \frac{dQ}{dt}$
Example

The volume of rocket fuel in a space ship booster is given by
    $V = %12$   m3
$t$ seconds after its launch.

Related rates problem

In a related rates problem, we are given the rate of change of one or more quantities, and are required to find the rate of change of one or more related quantities. For instance (as in the first example in %4) we may be given the rate at which the radius of a circle is growing, and want to know how fast the area is growing. %9:

The %21 of %23 is %33 at a rate of %34 %24/sec. How fast is its %22 %33 at the instant when its radius is %35 cm?
Step-by-step solution

A. The problem
    1. Identify the changing quantities.
    2. Restate the problem in terms of rates of change.
    The given problem is "The %21 of %23 is %33 at a rate of %34 %24/sec. How fast is its %22 %33 at the instant when its radius is %35 cm?"
    3. Rewrite the problem using mathematical notation.

B. The relationship
    1. Draw a diagram, if appropriate, showing the changing quantities.
    Sketch of changing quantities (Click on the correct sketch.)
         Note Changing quantities are represented by letters; non-changing quantities are represented by numbers.
    2. Write down an equation that relates the changing quantities.
    You have already identified the two changing quantities above. An equation that relates them is

    3. Take the derivative with respect to time of the equation relating the quantities.
    This gives the derived equation, which relates the rates of change of the quantities.

C. The solution
    1. Substitute into the derived equation the given values of the quantities and their derivatives.

    2. Solve for the derivative required. This gives you the solution to the problem..
    The required rate of change of the %22 is therefore

The falling ladder

Variants of "the falling ladder" problem are found in practically every calculus textbook (see, for instance, Example 2 in %4). Here is one of them:

A carelessly placed %40 ft ladder is sliding down a wall in such a way that %55 at a rate of %45 ft/sec. Your siamese cat Papanutski is sitting %56 directly in line with the approaching base of the ladder%57. How fast is %58 when Papnutski is hit?
A. The problem
    1. Identify the changing quantities.
    2. Restate the problem in terms of rates of change.
    The given problem is: A carelessly placed %40 ft ladder is sliding down a wall in such a way that %55 at a rate of %45 ft/sec. Your siamese cat Papanutski is sitting %56 directly in line with the approaching base of the ladder%57. How fast is %58 when Papnutski is hit?
    3. Rewrite the problem using mathematical notation.

B. The relationship
    1. Draw a diagram, if appropriate, showing the changing quantities.
    Sketch of changing quantities (Click on the correct sketch.)
         Note Changing quantities are represented by letters; non-changing quantities are represented by numbers.
    An equation that relates the changing quantities is

    3. Write down the derived equation..

C. The solution
    1. Substitute into the derived equation the given values of the quantities and their derivatives.

    2. Solve for the derivative required..
    To solve for $%61$, we first need to know the value of $%63$. For this, use the equation that relates the changing quantitites.

    The required rate of change is therefore

Now try the exercises in %4, some the %8, or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.

Last Updated: April, 2016
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