The First Derivative and the Shape of a Function f(x)
Read this section to learn how the first derivative is used to determine the shape of functions. Work through practice problems 1-9.
The First Derivative and the Shape of a Function f
This section examines some of the interplay between the shape of the graph ofand the behavior of . If we have a graph of , we will see what we can conclude about the values of . If we know values of , we will see what we can conclude about the graph of .
These same ideas make sense if we consider
Example 1: List the intervals on which the function given in Fig. 1 is increasing or decreasing.
Practice 1: List the intervals on which the function given in Fig. 2 is increasing or decreasing.
If we have an accurate graph of a function, then it is relatively easy to determine where
First Shape Theorem
For a function
(i) if is increasing on , then for all in
(ii) if is decreasing on , then for all in
(iii) if is constant on , then for all in .
Proof: Most people find a picture such as Fig. 3 to be a convincing justification of this theorem: if the graph of
(i) Assume that
(ii) Assume thatis decreasing on : The proof of this part is very similar to part (i). If , then since is decreasing on . Then the numerator of the limit, , will be negative and the denominator, , will still be positive, so the limiting value, , must be negative or zero: .
(iii) The derivative of a constant is zero, so if
The previous theorem is easy to understand, but you need to pay attention to exactly what it says and what it does not say. It is possible for a differentiable function which is increasing on an interval to have horizontal tangent lines at some places in the interval (Fig 4).
It is also possible for a continuous function which is increasing on an interval to have an undefined derivative at some places in the interval (Fig. 4).
Finally, it is possible for a function which is increasing on an interval to fail to be continuous at some places in the interval (Fig. 5).
The First Shape Theorem has a natural interpretation in terms of the height
Example 2: Fig. 6 shows the height of a helicopter during a period of time. Sketch the graph of the upward velocity of the helicopter,
Solution: The graph of
Practice 2: Fig. 8 shows the population of rabbits on an island during 6 years. Sketch the graph of the rate of population change,
Example 3: The graph ofis shown in Fig. 9. Sketch the graph of .
Solution: It is a good idea to look first for the points where
ifthen is decreasing so is negative,
ifthen is increasing so is positive,
ifthen is decreasing so is negative, and
The graph of
Practice 3: The graph of
The next theorem is almost the converse of the First Shape Theorem and explains the relationship between the values of the derivative and the graph of a function from a different perspective. It says that if we know something about the values of
Second Shape Theorem
For a function
(i) if for all in the interval , then is increasing on ,
(ii) if for all in the interval , then is decreasing on ,
(iii) if for all in the interval , then is constant on .
Proof: This theorem follows directly from the Mean Value Theorem, and part (c) is just a restatement of the First Corollary of the Mean Value Theorem.
(a) Assume that
(b) Assume that
Practice 4: Rewrite the Second Shape Theorem as a statement about the height
The value of the function at a number
Practice 5: Graph a continuous function which satisfies the conditions on
The Second Shape Theorem is particularly useful if we need to graph a function
Example 4: Use information about the values of
If , then (positive number)(negative number) so is decreasing.
If , then (positive number)(positive number) so is increasing.
Even though we don't know the value of
To plot the graph of
Practice 6: Use information about the values of
Example 5: Use the graph of
Solution: Several functions which have the derivative we want are given in Fig. 16 , and each of them is a correct answer. By the Second Corollary to the Mean Value Theorem, we know there is a whole family of parallel functions which have the derivative
we want, and each of these functions is a correct answer. If we had additional information about the function such as a point it went through, then only one member of the family would satisfy the extra condition and that function would be the only
Practice 7: Use the graph of
Practice 8: A weather balloon is released from the ground and sends back its upward velocity measurements (Fig. 18). Sketch a graph of the height of the balloon. When was the balloon highest?
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.3-First-Derivative.pdf
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