Precalculus Mathematics for Calculus Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

1 Fundamentals

Fundamentals When scientists talk about a mathematical model for a real-world phenomenon, they often mean an equation that describes the relationship between two quantities. For instance, the model may describe how the population of an animal species varies with time or how the pressure of a gas varies as its temperature changes.

In this section, we study a kind of modeling called variation. Fundamentals In this section, we study a kind of modeling called variation.

1.11 Modeling Variation

Modeling Variation Two types of mathematical models occur so often that they are given special names.

Direct Variation

Direct Variation Direct variation occurs when one quantity is a constant multiple of the other. So, we use an equation of the form y = kx to model this dependence.

The constant k is called the constant of proportionality. Direct Variation If the quantities x and y are related by an equation y = kx for some constant k ≠ 0, we say that: y varies directly as x. y is directly proportional to x. y is proportional to x. The constant k is called the constant of proportionality.

Direct Variation Recall that the graph of an equation of the form y = mx + b is a line with: Slope m y-intercept b

Direct Variation So, the graph of an equation y = kx that describes direct variation is a line with: Slope k y-intercept 0

E.g. 1—Direct Variation During a thunderstorm, you see the lightning before you hear the thunder because light travels much faster than sound. The distance between you and the storm varies directly as the time interval between the lightning and the thunder.

E.g. 1—Direct Variation Suppose that the thunder from a storm 5,400 ft away takes 5 s to reach you. Determine the constant of proportionality and write the equation for the variation.

(b) Sketch the graph of this equation. E.g. 1—Direct Variation (b) Sketch the graph of this equation. What does the constant of proportionality represent? (c) If the time interval between the lightning and thunder is now 8 s, how far away is the storm?

E.g. 1—Direct Variation Example (a) Let d be the distance from you to the storm and let t be the length of the time interval. We are given that d varies directly as t. So, d = kt where k is a constant.

To find k, we use the fact that t = 5 when d = 5400. E.g. 1—Direct Variation Example (a) To find k, we use the fact that t = 5 when d = 5400. Substituting these values in the equation, we get: 5400 = k(5)

E.g. 1—Direct Variation Example (a) Substituting this value of k in the equation for d, we obtain: d = 1080t as the equation for d as a function of t.

E.g. 1—Direct Variation Example (b) The graph of the equation d = 1080t is a line through the origin with slope 1080. The constant k = 1080 is the approximate speed of sound (in ft/s).

E.g. 1—Direct Variation Example (c) When t = 8, we have: d = 1080 ∙ 8 = 8640 So, the storm is 8640 ft ≈ 1.6 mi away.

Inverse Variation

Inverse Variation Another equation that is frequently used in mathematical modeling is y = k/x where k is a constant.

Inverse Variation If the quantities x and y are related by the equation for some constant k ≠ 0, we say that: y is inversely proportional to x. y varies inversely as x.

The graph of y = k/x for x > 0 is shown for the case k > 0. Inverse Variation The graph of y = k/x for x > 0 is shown for the case k > 0. It gives a picture of what happens when y is inversely proportional to x.

Boyle’s Law states that: E.g. 2—Inverse Variation Boyle’s Law states that: When a sample of gas is compressed at a constant temperature, the pressure of the gas is inversely proportional to the volume of the gas.

E.g. 2—Inverse Variation Suppose the pressure of a sample of air that occupies 0.106 m3 at 25°C is 50 kPa. Find the constant of proportionality. Write the equation that expresses the inverse proportionality.

If the sample expands to a volume of 0.3 m3, find the new pressure. E.g. 2—Inverse Variation If the sample expands to a volume of 0.3 m3, find the new pressure.

Let P be the pressure of the sample of gas and let V be its volume. E.g. 2—Inverse Variation Example (a) Let P be the pressure of the sample of gas and let V be its volume. Then, by the definition of inverse proportionality, we have: where k is a constant.

To find k, we use the fact that P = 50 when V = 0.106. E.g. 2—Inverse Variation Example (a) To find k, we use the fact that P = 50 when V = 0.106. Substituting these values in the equation, we get: k = (50)(0.106) = 5.3

Putting this value of k in the equation for P, we have: E.g. 2—Inverse Variation Example (a) Putting this value of k in the equation for P, we have:

When V = 0.3, we have: So, the new pressure is about 17.7 kPa. E.g. 2—Inverse Variation Example (b) When V = 0.3, we have: So, the new pressure is about 17.7 kPa.

Joint Variation

A physical quantity often depends on more than one other quantity. Joint Variation A physical quantity often depends on more than one other quantity. If one quantity is proportional to two or more other quantities, we call this relationship joint variation.

Joint Variation If the quantities x, y, and z are related by the equation z = kxy where k is a nonzero constant, we say that: z varies jointly as x and y. z is jointly proportional to x and y.

Joint Variation In the sciences, relationships between three or more variables are common. Any combination of the different types of proportionality that we have discussed is possible. For example, if we say that z is proportional to x and inversely proportional to y.

E.g. 3—Newton’s Law of Gravitation Newton’s Law of Gravitation says that: Two objects with masses m1 and m2 attract each other with a force F that is jointly proportional to their masses and inversely proportional to the square of the distance r between the objects. Express the law as an equation.

E.g. 3—Newton’s Law of Gravitation Using the definitions of joint and inverse variation, and the traditional notation G for the gravitational constant of proportionality, we have:

Gravitational Force If m1 and m2 are fixed masses, then the gravitational force between them is: F = C/r2 where C = Gm1m2 is a constant.

The figure shows the graph of this equation for r > 0 with C = 1. Gravitational Force The figure shows the graph of this equation for r > 0 with C = 1. Observe how the gravitational attraction decreases with increasing distance.