Learning Objectives
A student will be able to:
- Summarize the properties of function including intercepts, domain, range, continuity, asymptotes, relative extreme, concavity, points of inflection, limits at infinity.
- Apply the First and Second Derivative Tests to sketch graphs.
Introduction
In this lesson we summarize what we have learned about using derivatives to analyze the graphs of functions. We will demonstrate how these various methods can be applied to help us examine a function’s behavior and sketch its graph. Since we have already discussed the various techniques, this lesson will provide examples of using the techniques to analyze the examples of representative functions we introduced in the Lesson on Relations and Functions, particularly rational, polynomial, radical, and trigonometric functions. Before we begin our work on these examples, it may be useful to summarize the kind of information about functions we now can generate based on our previous discussions. Let's summarize our results in a table like the one shown because it provides a useful template with which to organize our findings.
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | |
Asymptotes and limits at infinity | |
Differentiability | |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema | |
Concavity | |
Inflection points |
Example 1: Analyzing Rational Functions
Consider the function
General Properties: The function appears to have zeros at However, once we factor the expression we see
Hence, the function has a zero at there is a hole in the graph at the domain is and the intercept is at
Asymptotes and Limits at Infinity
Given the domain, we note that there is a vertical asymptote at To determine other asymptotes, we examine the limit of as and . We have
Similarly, we see that . We also note that since
Hence we have a horizontal asymptote at
Differentiability
. Hence the function is differentiable at every point of its domain, and since on its domain, then is decreasing on its domain, .
in the domain of Hence there are no relative extrema and no inflection points.
So when Hence the graph is concave up for
Similarly, when Hence the graph is concave down for
Let’s summarize our results in the table before we sketch the graph.
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zero at intercept at |
Asymptotes and limits at infinity | VA at HA at hole in the graph at |
Differentiability | differentiable at every point of its domain |
Intervals where is increasing | nowhere |
Intervals where is decreasing | |
Relative extrema | none |
Concavity | concave up in concave down in |
Inflection points | none |
Finally, we sketch the graph as follows:
Let’s look at examples of the other representative functions we introduced in Lesson 1.2.
Example 2:
Analyzing Polynomial Functions
Consider the function
General Properties
The domain of is and the intercept at
The function can be factored
and thus has zeros at
Asymptotes and limits at infinity
Given the domain, we note that there are no vertical asymptotes. We note that and
Differentiability
if . These are the critical values. We note that the function is differentiable at every point of its domain.
on and ; hence the function is increasing in these intervals.
Similarly, on and thus is decreasing there.
if where there is an inflection point.
In addition, . Hence the graph has a relative maximum at and located at the point
We note that for . The graph is concave down in
And we have ; hence the graph has a relative minimum at and located at the point
We note that for The graph is concave up in
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zeros at intercept at |
Asymptotes and limits at infinity | no asymptotes |
Differentiability | differentiable at every point of it’s domain |
Intervals where is increasing | and |
Intervals where is decreasing | |
Relative extrema |
relative maximum at and located at the point ; relative minimum at and located at the point |
Concavity |
concave up in . concave down in . |
Inflection points | , located at the point |
Here is a sketch of the graph:
Example 3: Analyzing Radical Functions
Consider the function
General Properties
The domain of is , and it has a zero at
Asymptotes and Limits at Infinity
Given the domain, we note that there are no vertical asymptotes. We note that .
Differentiability
for the entire domain of Hence is increasing everywhere in its domain. is not defined at , so is a critical value.
everywhere in . Hence is concave down in is not defined at , so is an absolute minimum.
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zeros at , no intercept |
Asymptotes and limits at infinity | no asymptotes |
Differentiability | differentiable in |
Intervals where is increasing | everywhere in |
Intervals where is decreasing | nowhere |
Relative extrema |
none absolute minimum at , located at |
Concavity | concave down in |
Inflection points | none |
Here is a sketch of the graph:
Example 4: Analyzing Trigonometric Functions
We will see that while trigonometric functions can be analyzed using what we know about derivatives, they will provide some interesting challenges that we will need to address. Consider the function on the interval
General Properties
We note that is a continuous function and thus attains an absolute maximum and minimum in Its domain is and its range is
Differentiability
at .
Note that on and ; therefore the function is increasing in and .
Note that on ; therefore the function is decreasing in .
if Hence the critical values are at
hence there is a relative minimum at
; hence there is a relative maximum at
on and on Hence the graph is concave up and decreasing on and concave down on There is an inflection point at located at the point
Finally, there is absolute minimum at located at and an absolute maximum at located at
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | |
Asymptotes and limits at infinity | no asymptotes |
Differentiability | differentiable in |
Intervals where is increasing | and |
Intervals where is decreasing | |
Relative extrema |
relative maximum at relative minimum at absolute maximum at , located at absolute minimum at , located at |
Concavity | concave up in |
Inflection points | located at the point |
Lesson Summary
- We summarized the properties of functions, including intercepts, domain, range, continuity, asymptotes, relative extreme, concavity, points of inflection, and limits at infinity.
- We applied the First and Second Derivative Tests to sketch graphs.
Multimedia Links
Each of the problems above started with a function and then we analyzed its zeros, derivative, and concavity. Even without the function definition it is possible to sketch the graph if you know some key pieces of information. In the following video the narrator illustrates how to use information about the derivative of a function and given points on the function graph to sketch the function. Khan Academy Graphing with Calculus (9:44).
Another approach to this analysis is to look at a function, its derivative, and its second derivative on the same set of axes. This interactive applet called Curve Analysis allows you to trace function points on a graph and its first and second derivative. You can also enter new functions (including the ones from the examples above) to analyze the functions and their derivatives.
For more information about computing derivatives of higher orders (7.0), see Math Video Tutorials by James Sousa, Higher-Order Derivatives: Part 1 of 2 (7:34)
and Math Video Tutorials by James Sousa, Higher-Order Derivatives: Part 2 of 2 (5:21).
Review Questions
Summarize each of the following functions by filling out the table. Use the information to sketch a graph of the function.
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | |
Asymptotes and limits at infinity | |
Differentiability | |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema | |
Concavity | |
Inflection points |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | |
Asymptotes and limits at infinity | |
Differentiability | |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema | |
Concavity | |
Inflection points |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | |
Asymptotes and limits at infinity | |
Differentiability | |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema | |
Concavity | |
Inflection points |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | |
Asymptotes and limits at infinity | |
Differentiability | |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema | |
Concavity | |
Inflection points |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | |
Asymptotes and limits at infinity | |
Differentiability | |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema | |
Concavity | |
Inflection points |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | |
Asymptotes and limits at infinity | |
Differentiability | |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema | |
Concavity | |
Inflection points |
- on the interval
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | |
Asymptotes and limits at infinity | |
Differentiability | |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema | |
Concavity | |
Inflection points |
Review Answers
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zeros at intercept at |
Asymptotes and limits at infinity | no asymptotes |
Differentiability | differentiable at every point of its domain |
Intervals where is increasing | and |
Intervals where is decreasing | |
Relative extrema |
relative maximum at located at the point ; relative minimum at located at the point |
Concavity |
concave up in concave down in |
Inflection points | located at the point |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zeros at intercept at |
Asymptotes and limits at infinity | no asymptotes |
Differentiability | differentiable at every point of its domain |
Intervals where is increasing | and |
Intervals where is decreasing | and |
Relative extrema |
relative maximum at , located at the point ; and at at located at the point relative minimum at , located at the point |
Concavity |
concave up in concave down in and |
Inflection points | , located at the points and |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zeros at , no intercept |
Asymptotes and limits at infinity | HA |
Differentiability | differentiable at every point of its domain |
Intervals where is increasing | |
Intervals where is decreasing | and |
Relative extrema | relative maximum at located at the point |
Concavity |
concave up in concave down in and |
Inflection points | located at the point |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zeros at intercept at |
Asymptotes and limits at infinity | no asymptotes |
Differentiability | differentiable in |
Intervals where is increasing | and |
Intervals where is decreasing | |
Relative extrema |
relative maximum at located at the point relative minimum at located at the point |
Concavity |
concave up in concave down in |
Inflection points | located at the point |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zero at no intercept |
Asymptotes and limits at infinity | no asymptotes |
Differentiability | differentiable in |
Intervals where is increasing | nowhere |
Intervals where is decreasing | everywhere in |
Relative extrema | none absolute maximum at located at |
Concavity | concave up in |
Inflection points | none |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zeros at intercept at |
Asymptotes and limits at infinity | no asymptotes |
Differentiability | differentiable in |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema | relative minimum at located at the point |
Concavity | concave up in |
Inflection points | none |
Analysis | |
---|---|
Domain and Range | |
Intercepts and Zeros | zeros at intercept at |
Asymptotes and limits at infinity | no asymptotes; does not exist |
Differentiability | differentiable at every point of its domain |
Intervals where is increasing | |
Intervals where is decreasing | |
Relative extrema |
absolute max at located at the point absolute minimums at located at the points and |
Concavity | concave down in , concave up in and |
Inflection points | , located at the points and |