How to Solve Word Problems in Calculus

Chapter 1

Extracting Functions from Word Problems

Calculus is the study of the behavior of functions. The ability to solve real life problems using calculus hinges upon the ability to extract a function from a given description or physical situation.

Students usually find that a word problem is easily solved once the underlying mathematical function is determined. In this chapter we discuss techniques that will form the basis for solving a variety of word problems encountered in calculus courses.

One definition of a function found in calculus texts reads:

A function is a rule that assigns to each number x ε A, a unique number y ε B.

In calculus, A and B are sets of real numbers. A is called the domain and B the range. It is important to understand that a function is not a number, but a correspondence between two sets of numbers. In a practical sense, one may think of a function as a relationship between y and x. The important thing is that there be one, and only one, value of y corresponding to a given value of x.

EXAMPLE 1

If , then y is a function of x. For each value of x there is clearly one and only one value of y. However, if the equation defines the correspondence between x and y, then y is not a function of x. If , for example, then y could be 4 or −4.

Functions are usually represented symbolically by a letter such as f or g. For convenience, function notation is often used in calculus. In terms of the definition above, if f is a function and x ε A, then f(x) is the unique number in B corresponding to x.

It is not uncommon to use a letter that reminds us of what a function represents. Thus for example, A(x) may be used to represent the area of a square whose side is x or V(r) may represent the volume of a sphere whose radius is r.

EXAMPLE 2

Suppose f represents the squaring function, i.e., the function that squares x. We write

To compute the value of this function for a particular value of x, simply replace x by that value wherever it appears in the definition of the function.

The domain of a function is the set of numbers for which the function is defined. While polynomials have the set of all real numbers as their domain, many functions must, by their very definition, have restricted domains. For example, has, as its domain, the set of nonnegative real numbers. (The square root of a negative number is undefined.) The domain of is the set of all real numbers except 0.

Because of the geometric or physical nature of a problem, many word problems arising from everyday situations involve functions with restricted domains. For example, the squaring function discussed in Example 2 allows all real x, but if f(x) represents the area of a square of side x, then negative values make no sense. The domain would be the set of all nonnegative real numbers. (We shall see later that it is sometimes desirable to allow 0 as the dimension of a geometric figure, even though a square or rectangle whose side is 0 is difficult to visualize).

Finally, please note that when dealing with problems in elementary calculus, such as those discussed in this book, only functions of a single variable are considered. We may write to represent the area of a rectangle of width x and length y, but A is not a function of a single variable unless it is expressed in terms of only one variable. Techniques for accomplishing this are discussed in the pages that follow.

Strategy for Extracting Functions

The most important part of obtaining the function is to read and understand the problem. Once the problem is understood, and it is clear what is to be found, there are three steps to determining the function.

Step 1

Draw a diagram (if appropriate). Label all quantities, known and unknown, that are relevant.

Step 2

Write an equation representing the quantity to be expressed as a function. This quantity will usually be represented in terms of two or more variables.

Step 3

Use any constraints specified in the problem to eliminate all but one independent variable. A constraint defines a relationship between variables in the problem. The procedure is not complete until only one independent variable remains.

Number Problems

Although number problems are relatively simple, they illustrate the above steps quite clearly.

EXAMPLE 3

The sum of two numbers is 40. Express their product as a function of one of the numbers.

Solution

Step 1

In most number problems a diagram is not called for. We label the numbers using the variables x and y.

Let x be the first number

y be the second number

Step 2

We wish to express the product P as a function, so .

Step 3

Since , . We substitute into the equation involving P obtained in step 2 and express using function notation.

EXAMPLE 4

The product of two numbers is 32. Find a function that represents the sum of their squares.

Solution

Step 1

Let x be the first number and y the second.

Step 2

Step 3

Since , It follows that

Two–Dimensional Geometry Problems

Most geometry problems are composed of rectangles, triangles, and circles. It is therefore useful to review the formulas for the perimeter and area of these standard geometric shapes.

A rectangle of length l and width w has a perimeter equal to the sum of the lengths of its four sides. Its area is the product of its length and width.

A special case arises when l and w are equal. The resulting figure is a square, whose side is s.

A triangle with base b and altitude h has an area of ½bh. Special cases include right triangles and equilateral triangles. One often encounters triangles where two sides and their included angle are known.

The perimeter of a triangle is the sum of the lengths of its 3 sides. The perimeter of an equilateral triangle of side s is simply 3s.

A circle is measured by its radius r. The perimeter of a circle is known as its circumference. Occasionally the diameter d will be given in place of the radius. Since the area and circumference may also be expressed in terms of d.

EXAMPLE 5

A farmer has 1500 feet of fencing in his barn. He wishes to enclose a rectangular pen, subdivided into two regions by a section of fence down the middle, parallel to one side of the rectangle. Express the area enclosed by the pen as a function of its width x. What is the domain of the function?

Solution

Step 1

We draw a simple diagram, labeling the dimensions of the rectangle.

Step 2

We express the area of the rectangle in terms of the variables x and y. Observe that the area of the pen is determined by its outer dimensions only; the inner section has no affect on the area.

Step 3

We use the constraint of 1500 feet of fence to obtain a relationship between x and y.

Next we solve for y in terms of x.

Finally, we substitute this expression for y into the area equation obtained in step 2.

Mathematically, the domain of A(x) is the set of all real numbers. However, in this problem, as with all geometry problems, negative dimensions are unrealistic. Although may appear to be unrealistic as well, we generally allow a rectangle of zero width or length with the understanding that its area is 0. Such a rectangle is called a degenerate rectangle. Since the perimeter is fixed, y gets smaller as x gets larger so the largest value of x occurs when .

The function describing the area of the farmer’s pen is

EXAMPLE 6

A piece of wire 12 inches long is to be used to form a square and/or a circle. Determine a function that expresses the combined area of both figures. What is its domain?

Solution

Step 1

Let x be the side of the square and r the radius of the circle. We shall express the area as a function of x.

Step 2

Step 3

Since the combined perimeter of the two figures must be 12 inches, we have

It follows that

Replacing r in terms of x in step 2 gives the area as a function of x.

If all the wire