Related rates problems and optimization problems, both of which are explained in Stewart's chapter 4, are examples of mathematical modeling problems. A model is a mathematical representation of a physical system. Models can be extremely complicated or very simple. Most of the models you'll encounter in math courses are simple. Some problems are already stated in mathematical language. ("Find the dimensions of the rectangle of largest area which can be inscribed in a circle of radius 5.") Others involve descriptions of physical objects. ("A man 6 feet tall is walking away from a 15-foot high lamppost at the rate of 5 feet per second. How fast is the length of his shadow changing when he's 5 feet from the base of the lamppost?")
 

There are usually four phases to these problems:

1.  Make a reasonable guess at the answer.
2.  Change the physical problem into a mathematical problem.
3.  Solve the mathematical problem.
4.  Determine what the mathematical solution says about the original problem, and compare that with your original guess.

    1.  Draw a picture.
        Since derivatives measure rate of change, it's better to draw two pictures.

    2.  Label everything you can in the pictures.
        Be careful not to use numbers to label quantities that are changing.
        (Here's one of the ways drawing two pictures helps.)
        Use letters that mean something to you.

    3.  Find some relationships among the quantities you've labeled. The best sources are:

        Geometry:    similar triangles
                            distance formulas
                            Pythagorean theorem
                            areas of familiar plane regions
                            volumes of familiar solids
                            surface areas of familiar solids
        Calculus:    rates of change are derivatives
        Formulas from physics (velocity, momentum, energy, etc.)
        Formulas from biology, chemistry, and other sciences

    4. Try to reduce your collection of relationships to one equation.
       Solve one equation for one variable, and substitute your solution for that variable in the other equations.
        (There are perfectly good techniques that work directly on systems of equations, but you don't normally learn them in your first year of calculus.)

    5. If the quantities in an equation vary over time but the relationship given by the equation is always valid,
       differentiating both sides of the equation with respect to time should give another valid equation.

    6. Pay attention to the units.
        Equal quantities should be measured in the same units.
        Your calculator has an excellent unit conversion utility if your original units aren't satisfactory.
 

Exercises Follow as many of these steps as you can for the two problems in parentheses in the first paragraph above. Then apply the same steps to some of the problems in the textbook.

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