2.1 Derivatives and Rates of Change Calculus grew out of four major problems that European mathematicians were working on during the 17th century: • The tangent line problem • The velocity and acceleration problem • The minimum and maximum problem • The area problem If a curve C has equation y = f (x) and we want to find the tangent line to C at the point (a, f (a)), then we consider a nearby point (a + h, f (a + h)) and compute the slope of the secant line

Letting h → 0, the nearby points approach the point (a, f (a)). If the slopes of the secant lines approach a number m, then we define the tangent to be the line through (a, f (a)) with slope m. The P (a, f (a)) is the line through P with slope

to the curve y = f (x) at the point

provided that this limit exists. Example 1. Find an equation of the tangent line to the parabola y = x2 at the point P (1, 1).

The slope of the tangent line to the graph of f at the point (a, f (a)) is also called the at the point. The idea is that if we zoom in far enough toward the point, the curve looks almost like a straight line. 1

There is another expression for the slope of a tangent line:

Example 2. Find the slope of the graph of f (x) = 4x − 3 at the point (2, 1).

Note: Here, the limit definition of the slope of f agrees with the definition of the slope of a line. Example 3. Find an equation of the tangent line to the hyperbola y = 4/x at the point (4, 1).

The derivative can also be used to determine the rate of change of one variable with respect to another. Examples of rates of change in the real world are population growth rates, production rates, water flow rates, velocity and acceleration. A common use of rate of change is to describe the motion of an object. In such problems, it is customary to use either a horizontal or vertical line with a designated origin to represent the line of motion. On such lines, movement to the right (or upward) is considered to be in the positive direction, and movement to the left (or downward) is considered to be in the negative direction.

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Suppose an object moves along a straight line according to an equation of motion s = f (t), where s is the displacement (directed distance) of the object from the origin at time t. The function f that describes of the object. In the the motion is called the time interval t = a to t = a + h the change in position is f (a + h) − f (a). The average velocity over this time interval is

Suppose we compute the average velocities over shorter and shorter time intervals [a, a + h] (h → 0). (or ) Define the v(a) at time t = a to be the limit of these average velocities:

This means that the velocity at time t = a is equal to the slope of the tangent line at P . The speed is given by:

Example 4. A ball is dropped from the top of LEX LUTHOR: Drop of Doom, 415 feet above the ground, and its height H above the ground t seconds after being dropped is given by H(t) = 415 − 16t2 . What is the velocity of the ball after 5 seconds?

Note: The velocity is

, thus the object is moving 3

.

We have seen that the same type of limit arises in finding the slope of a tangent line or the velocity of an object. In fact, limits of the form

arise whenever we calculuate a rate of change in any of the sciences or engineering. The by f 0 (a), is

, denoted

if this limit exists.

An equivalent definition of the derivative:

Example 5. Find the derivative of the function f (x) =

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√ x at the number a.

The tangent line to y = f (x) at (a, f (a)) is the line through (a, f (a)) whose slope is equal to f 0 (a).

If we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve y = f (x) at the point (a, f (a)) :

Example 6. Find an equation of the tangent line to the curve f (x) =

√

x at the point (4, 2).

Suppose y is a quantity that depends on another quantity x. Thus y is a function of x and we write y = f (x). If x changes from x1 to x2 , then the change in x (also called the of x) is

and the corresponding change in y is

The difference quotient

is called the the interval [x1 , x2 ] and can be interpreted as the slope of the secant line P Q.

over

Consider the average rate of change over smaller and smaller intervals by letting x2 approach x1 and therefore letting ∆x approach 0. The limit of these average rates of change is called the at x = x1 , which is interpreted as the slope of the tangent to the curve y = f (x) at P (x1 , f (x1 )):

We recognize this limit as being the derivative f 0 (x1 ).

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We now have two interpretations of the derivative: (1) The slope of the tangent line to the curve y = f (x) when x = a. (2) The instantaneous rate of change of y = f (x) with respect to x when x = a. If we sketch the curve y = f (x), then the instantaneous rate of change is the slope of the tangent to this curve at the point where x = a. This means that when the derivative is large (and therefore the curve is steep), the y-values change rapidly. When the derivative is small, the curve is relatively flat and the y-values change slowly.

If s = f (t) is the position function of a particle that moves along a straight line, then f 0 (a) is the rate of change of the displacement s with respect to the time t. In other words, . The of the particle is the absolute value of the velocity, that is, . Example 7. The number of bacteria after t hours in a controlled laboratory experiment is n = f (t). a) What is the meaning of the derivative f 0 (5)? What are its units?

b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, f 0 (5) or f 0 (10)? If the supply of nutrients is limited, would that affect your conclusion? Explain.

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