3: The Derivative

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Where are we going?

In this chapter, you will learn about the rate of change of a function, a concept at the heart of calculus.

Suppose F is a function. The rate of change of F at a point a in its domain is the slope of the tangent to the graph of F at (a, F(a)), if such a tangent exists.

The slope of the tangent at (a, F(a)) is approximated by slopes of lines through (a, F(a)) and points (b, F(b)) when b is close to a.

We initiate the use of rate of change to form models of biological systems.

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Calculus is the study of change and rates of change. It has two primitive concepts, the derivative and the integral. Given a function relating a dependent variable to an independent variable, the derivative is the rate of change of the dependent variable as the independent variable changes. We determine the derivative of a function when we answer questions such as

Given population size as a function of time, at what rate is the population growing?
Given the position of a particle as a function of time, what is its velocity?
At what rate does air density decrease with increasing altitude?
At what rate does the pressure of one mole of $O_2$ at $300^{\circ}K$ change as the volume changes if the temperature is constant?

On the other hand, given the rate of change of a dependent variable as an independent variable changes, the integral is the function that relates the dependent variable to the independent variable.

Given the growth rate of a population at all times in a time interval, how much did the population size change during that time interval?
Given the rate of renal clearance of penicillin during the four hours following an initial injection, what will be the plasma penicillin level at the end of that four hour interval?
Given that a car left Chicago at 1:00 pm traveling west on I80 and given the velocity of the car between 1 and 5 pm, where was the car at 5 pm?

The derivative is the subject of this chapter; the integral is addressed in Chapter 9. The derivative and integral are independently defined. Chapter 9 can be studied before this one and without reference to this one. The two concepts are closely related, however, and the relation between them is The Fundamental Theorem of Calculus, developed in Chapter 10.

Explore 3.0.2 You use both the derivative and integral concepts of calculus when you cross a busy street. You observe the nearest oncoming car and subconsciously estimate its distance from you and its speed (use of the derivative), and you decide whether you have time to cross the street before the car arrives at your position (a simple use of the integral). Might there be a car different from the nearest car that will affect your estimate of the time available to cross the street? You may even observe that the car is slowing down and you may estimate whether it will stop before it gets to your crossing point (involves the integral). It gets really difficult when you are traveling on a two-lane road and want to pass a car in front of you and there is an oncoming vehicle. Teenage drivers learn calculus.

Give an example in which estimates of distances and speeds and times are important for successful performance in a sport.