- 3.2.1 Define the derivative function of a given function.
- 3.2.2 Graph a derivative function from the graph of a given function.
- 3.2.3 State the connection between derivatives and continuity.
- 3.2.4 Describe three conditions for when a function does not have a derivative.
- 3.2.5 Explain the meaning of a higher-order derivative.
As we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the function at that point. If we differentiate a position function at a given time, we obtain the velocity at that time. It seems reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about the behavior of the function. However, the process of finding the derivative at even a handful of values using the techniques of the preceding section would quickly become quite tedious. In this section we define the derivative function and learn a process for finding it.
The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows.
Let be a function. The derivative function, denoted by is the function whose domain consists of those values of such that the following limit exists:
A function is said to be differentiable at if exists. More generally, a function is said to be differentiable on if it is differentiable at every point in an open set and a differentiable function is one in which exists on its domain.
In the next few examples we use Equation 3.9 to find the derivative of a function.
Finding the Derivative of a Square-Root Function
Find the derivative of
Start directly with the definition of the derivative function. Use Equation 3.1.
Finding the Derivative of a Quadratic Function
Find the derivative of the function
Follow the same procedure here, but without having to multiply by the conjugate.
Find the derivative of
We use a variety of different notations to express the derivative of a function. In Example 3.12 we showed that if then If we had expressed this function in the form we could have expressed the derivative as or We could have conveyed the same information by writing Thus, for the function each of the following notations represents the derivative of
In place of we may also use Use of the notation (called Leibniz notation) is quite common in engineering and physics. To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line. The slopes of these secant lines are often expressed in the form where is the difference in the values corresponding to the difference in the values, which are expressed as (Figure 3.11). Thus the derivative, which can be thought of as the instantaneous rate of change of with respect to is expressed as
Graphing a Derivative
We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative function, we could graph it. Given both, we would expect to see a correspondence between the graphs of these two functions, since gives the rate of change of a function (or slope of the tangent line to
In Example 3.11 we found that for If we graph these functions on the same axes, as in Figure 3.12, we can use the graphs to understand the relationship between these two functions. First, we notice that is increasing over its entire domain, which means that the slopes of its tangent lines at all points are positive. Consequently, we expect for all values of in its domain. Furthermore, as increases, the slopes of the tangent lines to are decreasing and we expect to see a corresponding decrease in We also observe that is undefined and that corresponding to a vertical tangent to at
In Example 3.12 we found that for The graphs of these functions are shown in Figure 3.13. Observe that is decreasing for For these same values of For values of is increasing and Also, has a horizontal tangent at and
Sketching a Derivative Using a Function
Use the following graph of to sketch a graph of
The solution is shown in the following graph. Observe that is increasing and on Also, is decreasing and on and on Also note that has horizontal tangents at and and and
Sketch the graph of On what interval is the graph of above the -axis?
Derivatives and Continuity
Now that we can graph a derivative, let’s examine the behavior of the graphs. First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons.
Differentiability Implies Continuity
Let be a function and be in its domain. If is differentiable at then is continuous at
If is differentiable at then exists and
We want to show that is continuous at by showing that Thus,
Therefore, since is defined and we conclude that is continuous at
We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability. To determine an answer to this question, we examine the function This function is continuous everywhere; however, is undefined. This observation leads us to believe that continuity does not imply differentiability. Let’s explore further. For
This limit does not exist because
See Figure 3.14.
Let’s consider some additional situations in which a continuous function fails to be differentiable. Consider the function
Thus does not exist. A quick look at the graph of clarifies the situation. The function has a vertical tangent line at (Figure 3.15).
The function also has a derivative that exhibits interesting behavior at We see that
This limit does not exist, essentially because the slopes of the secant lines continuously change direction as they approach zero (Figure 3.16).
- We observe that if a function is not continuous, it cannot be differentiable, since every differentiable function must be continuous. However, if a function is continuous, it may still fail to be differentiable.
- We saw that failed to be differentiable at because the limit of the slopes of the tangent lines on the left and right were not the same. Visually, this resulted in a sharp corner on the graph of the function at From this we conclude that in order to be differentiable at a point, a function must be “smooth” at that point.
- As we saw in the example of a function fails to be differentiable at a point where there is a vertical tangent line.
- As we saw with a function may fail to be differentiable at a point in more complicated ways as well.
A Piecewise Function that is Continuous and Differentiable
A toy company wants to design a track for a toy car that starts out along a parabolic curve and then converts to a straight line (Figure 3.17). The function that describes the track is to have the form where and are in inches. For the car to move smoothly along the track, the function must be both continuous and differentiable at Find values of and that make both continuous and differentiable.
For the function to be continuous at Thus, since
and we must have Equivalently, we have
For the function to be differentiable at
must exist. Since is defined using different rules on the right and the left, we must evaluate this limit from the right and the left and then set them equal to each other:
We also have
This gives us Thus and
Find values of and that make both continuous and differentiable at
The derivative of a function is itself a function, so we can find the derivative of a derivative. For example, the derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of can be expressed in any of the following forms:
It is interesting to note that the notation for may be viewed as an attempt to express more compactly. Analogously,
Finding a Second Derivative
Next, find by taking the derivative of
The position of a particle along a coordinate axis at time (in seconds) is given by (in meters). Find the function that describes its acceleration at time
Since and we begin by finding the derivative of
Section 3.2 Exercises
For the following exercises, use the definition of a derivative to find
For the following exercises, use the graph of to sketch the graph of its derivative
For the following exercises, the given limit represents the derivative of a function at Find and
For the following functions,
- sketch the graph and
- use the definition of a derivative to show that the function is not differentiable at
For the following graphs,
- determine for which values of the exists but is not continuous at and
- determine for which values of the function is continuous but not differentiable at
Use the graph to evaluate a. b. c. d. and e. if it exists.
For the following functions, use to find
For the following exercises, use a calculator to graph Determine the function then use a calculator to graph
For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.
denotes the population of a city at time in years.
denotes the total amount of money (in thousands of dollars) spent on concessions by customers at an amusement park.
denotes the total cost (in thousands of dollars) of manufacturing clock radios.
denotes the grade (in percentage points) received on a test, given hours of studying.
denotes the cost (in dollars) of a sociology textbook at university bookstores in the United States in years since
denotes atmospheric pressure in Torrs at an altitude of feet.
Sketch the graph of a function with all of the following properties:
- does not exist.
Suppose temperature in degrees Fahrenheit at a height in feet above the ground is given by
- Give a physical interpretation, with units, of
- If we know that explain the physical meaning.
Suppose the total profit of a company is thousand dollars when units of an item are sold.
- What does for measure, and what are the units?
- What does measure, and what are the units?
- Suppose that what is the approximate change in profit if the number of items sold increases from
The graph in the following figure models the number of people who have come down with the flu weeks after its initial outbreak in a town with a population of citizens.
- Describe what represents and how it behaves as increases.
- What does the derivative tell us about how this town is affected by the flu outbreak?
For the following exercises, use the following table, which shows the height of the Saturn rocket for the Apollo mission seconds after launch.
|Time (seconds)||Height (meters)|
What is the physical meaning of What are the units?
[T] Construct a table of values for and graph both and on the same graph. (Hint: for interior points, estimate both the left limit and right limit and average them. An interior point of an interval I is an element of I which is not an endpoint of I.)
[T] The best linear fit to the data is given by where is the height of the rocket (in meters) and is the time elapsed since takeoff. From this equation, determine Graph with the given data and, on a separate coordinate plane, graph
[T] The best quadratic fit to the data is given by where is the height of the rocket (in meters) and is the time elapsed since takeoff. From this equation, determine Graph with the given data and, on a separate coordinate plane, graph
[T] The best cubic fit to the data is given by where is the height of the rocket (in m) and is the time elapsed since take off. From this equation, determine Graph with the given data and, on a separate coordinate plane, graph Does the linear, quadratic, or cubic function fit the data best?
Using the best linear, quadratic, and cubic fits to the data, determine what are. What are the physical meanings of and what are their units?