• To use knowledge of the first derivative, second derivative, and asymptotes to sketch the graph of a function. Supported by learning objectives 1 through 6.
• To recognize the relationships between the graph of a function, the graph of its derivative, and the graph of its second derivative. Supported by learning objectives 4 through 6.
| E.1 |
| E.2 |
| E.3 |
| E.4 |
| E.5 |
| E.6 |
Objective
E.1 Back to Top
To find the second derivative of a function.
To engage in activities that require exploration in order to develop understandings related to derivatives (CCT).
Instructional Notes
Unit E deals with graphical applications of derivatives. It is important to be able to find both the first and second derivative since knowledge of each is used in analyzing the behaviour of the graph of a function.
Begin by showing the students the table below with the left column covered. Ask the students to examine the table and tell you how it was created (CCT).
Point out to the students that if a function is differentiated, another function is created. If the derivative is then differentiated, another function is created. This process could be continued forever, in some situations, while in others you would eventually obtain 0.
The function that is formed after differentiating once is known as the first derivative . We use the symbol 
or 
to represent the first derivative. The function that is formed by differentiating the derivative, that is, after taking the derivative of the derivative, is known as the second derivative . The second derivative is represented by the symbol 
or 
. We read this as “ f double prime” and “the second derivative of y with respect to x ”. The derivative of the derivative of the derivative, in other words the third derivative , is represented by the symbol 
or 
. Derivatives beyond the first are called higher order derivatives. We will limit our work to second derivatives for now.
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Create some examples to give students practice in finding the second derivative. Be sure to include functions that require a product rule, quotient rule, or chain rule. An example follows.
Example: If 
.
Solution: 
Then 
Simplifying further,
Second derivatives can be a lot of work.
Example: If 
, find 
using implicit differentiation.
Solution: 
. Thus 
. Solving for 
, we have 
. Thus 
(1).
Substituting 
in (1) yields 
(2).
But since 
(the original relation), then 
. Substituting in (2) we have 
.
Examples/Activities
Find 
and 
for each of the following functions.
1. 
2. 
3. 
4. 
5. 
6. 
If 
, find 
.
Find 
for each of the following.
1. 
2. 
3. 
4. 
Suggestions/Extensions
Find a formula for the n th derivative of 
.
If 
, find 
.
Prove that 
.
Have the students create functions for which 
, or 
, or 
for all x .
Have the students compare their results for the second derivative of 
with respect to x when done explicitly or implicitly.
Objective
E.2 Back to Top
To determine the absolute extrema and relative extrema of a continuous function on a closed interval.
To make and justify decisions based upon understanding of calculus concepts (CCT).
Instructional Notes
In this section we apply our knowledge of the derivative to find absolute and relative extrema. A considerable amount of vocabulary must be developed in relation to this objective. The wording of the theorems and definitions is extremely important. It is important to understand why each word is included. Introduce the vocabulary through the use of an illustration such as the following.
Consider the graph of the function below.
The domain of 
is 
.
The function has an absolute maximum or global maximum at 
because this is where the function reaches its highest value. Note that 
. We say that 3 is the maximum value of the function.
The function has an absolute minimum or global minimum at 
because this is where the function reaches its lowest value. Note that 
. We say that 
is the minimum value of the function.
The extreme values or extrema of the function are 3 and 
.
The function has a local or relative minimum at 
because if you move a tiny amount to the left or right of 
, the function's values are higher. The function also has a relative minimum at 
for the same reason. There is both an absolute minimum and a relative minimum at 
.
The relative minimum value at 
is 
. The relative minimum value at 
is 
.
The function has a local or relative maximum at 
because if you move a tiny amount to the left or right of 
, the function's values are lower. The relative maximum value at 
is 2.
The relative extrema (all relative maximum and relative minimum points) occur at 
, 
, and 
.
After introducing the terminology with an example, define the terms in the general sense.
The use of a concept attainment activity supports students in developing an intuitive understanding of these concepts (CCT).
If D is the domain of a function 
, then 
has an absolute maximum or global maximum at c if 
for all values of x in D . The number 
is the maximum value of the function in the domain D . Similarly 
has an absolute minimum or global minimum at e if 
for all values of x in D . The number 
is the minimum value of the function in the domain D . The numbers 
and 
are the extreme values or extrema of the function.
If 
for values of x near c , that is in some open interval about c , then the function has a local or relative maximum at 
. If 
for values of x near e , that is in some open interval about e , then the function has a local or relative minimum at 
.
Whether a function has extrema depends on the domain of the function as well as whether the function is continuous. See below.
On the basis of a discussion surrounding the diagrams above, lead the students to the statement of the extreme value theorem.
The Extreme Value Theorem: If 
is a continuous function on a closed interval, then 
has both an absolute maximum and an absolute minimum value on that closed interval.
Derivatives May Not Exist
Students need to realize that 
does not always exist – that is the function may not be differentiable . In other words, there is no value for the slope of the tangent line or it does not exist. Similarly, it implies that an instantaneous rate of change cannot be determined at that point. This can happen in one of three ways. Show the students figures similar to the following.
Ask students to sketch graphs of functions on overhead transparencies that have a relative maximum and/or a relative minimum. Display these graphs to the class one at a time and ask students to observe the slope of the tangent line at these relative extrema. Hopefully, they will realize that at every relative maximum or minimum, one of two things happens. The derivative is either zero or does not exist. This could lead to the definition of critical numbers.
Critical Numbers: 
is a critical number for the function 
if 
is defined and 
or if 
does not exist.
An important conclusion that the students should reach is: if 
has a relative maximum or minimum at 
, then c is a critical number of 
.
It is also important that the students realize that this does not mean there will be a relative maximum or minimum at 
just because 
or because 
does not exist. For example consider the function 
. 
, and thus 
, but 
does not have a relative maximum or minimum at the origin even though the tangent line is horizontal. Similarly, consider the function 
, whose derivative 
. Clearly 
does not exist, but 
is not a relative maximum or minimum point. The tangent line is vertical there.
In light of this discussion, ask the students how one could locate the absolute extrema on a closed interval.
Have the students discuss how they would determine absolute extrema for 
by considering graphs and examples. This discussion work can then be synthesized into a set of steps such as given below.
Suppose 
is continuous on a closed interval 
. To find the absolute extrema on 
:
• find all critical numbers 
, 
, 
, … in 
.
• evaluate 
, 
, 
, … .
• evaluate 
and 
.
• the greatest of 
, 
, 
, 
, 
, … is the absolute maximum.
• the smallest of 
, 
, 
, 
, 
, … is the absolute minimum.
Next, have the students apply their strategy to some example functions.
Example: Find the absolute maximum and minimum values of the function 
on the interval 
.
Solution: To find the critical numbers, we first find 
. Now 
. To determine where 
is zero or does not exist perform a sign analysis on the derivative.
In the interval 
, 
is a critical number because 
.
left endpoint 
critical number 
right endpoint 
0 The absolute maximum is 0. The absolute minimum is 
.
Some of the practice exercises for this objective should involve identifying absolute and relative maximum and minimum values by:
• looking at the graph of the function.
• determining the critical numbers and examining the value of the function at these critical numbers and at the endpoints of the interval (as was done in the above example).
Examples/Activities
Examine the graph of the function below. For the values a , b , c , d , e , f , g , h , determine if the function has a relative maximum, relative minimum, global maximum, global minimum, both a relative and a global maximum, both a relative and a global minimum, or none of these.
Examine the graph of the function below. Give the absolute and local maximum and minimum values of the function.
Sketch the graph of a function on the interval 
that has:
(a) an absolute maximum that is not a local maximum, and an absolute minimum that is a local minimum.
(b) an absolute maximum and an absolute minimum but no local extrema.
(c) an absolute maximum, an absolute minimum, a local maximum, a local minimum, all different.
(d) an absolute minimum but no local minimum.
(e) a local minimum but no absolute minimum.
Examine the graph of the function below. Which of the numbers a , b , c , . . . , h are critical numbers?
Sketch the graph of a function that:
(a) has a local maximum at 
and is differentiable at 
.
(b) has a local maximum at 
but is not differentiable at 
.
(c) has a local maximum at 
but is not continuous at 
.
Find the critical numbers, if any, for each of the following functions.
1. 
2. 
3. 
4. 
(Hint: Is there a corner at 
?)
5. 
6. 
7. 
8. 
Explain why 
is not a critical number for the function 
.
Examples/Activities
Without drawing the graph, determine the absolute extrema for the function on the given interval.
1. 
; 
2. 
; 
3. 
; 
4. 
; 
5. 
; 
6. 
; 
7. 
; 
8. 
; 
9. 
; 
10. 
; 
11. 
; 
The population of a culture of bacteria after t hours is given by the function 
for 
. Find the maximum and minimum number of bacteria in the culture during this time period.
If two numbers have a sum of 20, then they can be represented by 
and 
. What is the maximum and minimum value of the sum of their squares if neither number can be negative?
Suggestions/Extensions
Have the students talk about the points where absolute maxima and minima occur as well as where the relative extrema occur, and what you need to know on the graph to determine where they will be . More specifically, have the students focus on discussing the behaviour of the graph at local extrema and what that implies about the slope of the tangent line.
Have students explore what an open interval refers to and its implication in the context of the relative extrema.
Challenge the students to try to create the graph of a function f(x) that contradicts the Extreme Value Theorem. Have the students discuss their reasoning and why their attempts fail. Discussions such as these help to reinforce and broaden the students' understanding of continuity (COM, CCT).
It is important for the students to remember that points not in the domain of 
cannot be critical points of 
. As a result, students should be noting all non-permissible values for 
before doing any analysis or determination of its derivative.
See Appendix A for additional objective and activities.
Objective
E.3 Back to Top
To determine the interval(s) in which a function is increasing or decreasing and to use the first derivative test to determine local extrema.
To listen, read and view ideas and concepts analytically and critically (CCT).
To further develop their ability to read graphs and functions for information in order to understand and analyze (NUM).
Instructional Notes
Increasing and decreasing functions were defined in B.4. as follows.
A function is increasing on an interval I if 
whenever 
in the interval I . A function is decreasing on an interval I , if 
whenever 
in the interval I .
Ask one-third of the class to sketch the graph of a function that is everywhere continuous, differentiable, and increasing; then have students draw several tangent lines to their graph at various points. Ask another third of the class to do the same for a continuous, differentiable, decreasing function. Ask the final third of the class to sketch the graph of a constant function drawing tangent lines to it at various points. Then ask each group what common characteristic is shared by the slope of the tangent lines for each type of function (CCT). The following test to determine the intervals in which a function is increasing, decreasing, or constant should follow easily from students' work.
Students need to know and understand the following. Suppose 
is continuous on the closed interval 
.
• If 
for all 
, 
is increasing on the interval 
.
• If 
for all 
, 
is decreasing on the interval 
.
• If 
for all 
, 
is constant on the interval 
.
Note that a function is still considered to be increasing (decreasing) on the interval 
even though there may be a finite number of values c in 
for which 
or for which 
does not exist.
Work through an example similar to the one that follows. Point out to the students that solving inequalities in A.4 was meant to prepare students for this section.
Example: Find the open intervals in which the function 
is increasing and/or decreasing.
Solution: First note that 
is not in the domain of the function. To determine these intervals, find where the slope of the tangent line is positive and/or negative: 
. Thus 
. Next we perform a sign analysis on 
to determine where it is positive or negative.
From the sign analysis, we see that 
is increasing for 
because 
and decreasing for 
because 
.
Ask the students to sketch the graph of a function that has both a local maximum and a local minimum. Ask them to draw several tangent lines to the function on either side of the local extrema. Ask them for their observations about the nature of the slope of the tangent lines on either side of the local extrema (CCT, NUM).
In the above graph b , c , d , and e are critical numbers because both 
and 
are 0 while both 
and 
do not exist. Note that local maxima occur at B and at D and the slopes of the tangent lines immediately left of B and D are positive while immediately to the right of B and D , the tangent line slopes are negative. Local minima occur at C and E . Immediately to the left of C and E , the tangent line slopes are negative while immediately to the right of C and E , the tangent line slopes are positive. A discussion such as the one above should lead to a formation of the first derivative test.
and if 
changes from positive to negative at c , then 
has a local maximum at 
. If c is a critical number for a function 
and if 
changes from negative to positive at c , then 
has a local minimum at 
. Emphasize that the first derivative must have a change in sign at the critical number in order for there to be a local maximum or minimum.
Ask students to determine the coordinates of the relative extrema and find the intervals in which the function is increasing or decreasing using the same sign analysis. Going back to the example that was used before, determine the coordinates of the relative extrema of 
by examining the sign analysis obtained for 
shown below.
There are two critical numbers, namely 
and 2 because 
at these locations. Note 
is not a critical number because 
was not in the domain of the function.
Since 
switched from positive to negative at 
, 
, that is 
, is a local maximum point. Since 
switched from negative to positive at 2, 
, that is 
, is a local minimum point.
Stress that in order to find the intervals in which a function is increasing or decreasing, as well as any local extrema, students should determine 
in simplest factored form and then perform a sign analysis on 
. All the work on sign analysis and factoring from previous sections comes into play.
In this section, introduce students to the type of problem that requires them to match the graph of a function to the graph of its derivative. In addition, students should sketch the graph of 
by looking at the graph of 
and vice versa.
Examples/Activities
By performing a sign analysis on 
, determine the open interval(s) in which each of the following functions is increasing or decreasing. Check your answers by examining the graphs provided.
1. 
2. 
3. 
4. 
You can use a graphing calculator to determine the coordinates of relative extrema. Consider the function 
. First draw the graph using an appropriate window.
Press 2nd TRACE and select 4 if you wish to find the maximum.
The following screen appears.
In response to Left Bound?, press the left arrow key until the cursor is to the left of the maximum, but close to it.
Press ENTER and the following screen results.
In response to Right Bound?, press the right arrow key until the cursor is to the right of the maximum but close to it and press ENTER. In response to Guess?, move the cursor very close (or on) the maximum point and press ENTER. The point at the bottom of the screen is a very close approximation of the maximum value.
Press ENTER and the following screen results.
In response to Guess?, move the cursor near to the maximum.
Examples/Activities
Find the intervals in which the function is increasing or decreasing. Find the coordinates of any relative extrema. Use the first derivative test. Verify with a graphing calculator.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
Students should do several questions like the three that follow.
1. Shown below left is the graph of 
. Sketch the graph of 
.
2. Shown below right is the graph of 
. Sketch a possible graph of 
.
3. Shown below is a graph of 
. For what x values does 
have local extrema? What kind of extrema?
In addition to questions like those above, require students to match the graph of a function with the graph of its derivative. You could have a column of the graphs of five functions and a column of the graphs of the five derivatives in scrambled order.
Suggestions/Extensions
Encourage the students to sketch the graphs of 
to get another representation of what the sign analysis of 
is identifying and how it relates to f(x) .
ENTER
The calculator's algorithm for determining the local maximum has done some rounding and thus the exact coordinates 
are not obtained.
Suppose that x hours after taking a prescriptive drug, the concentration of the drug in the body is given by the function 
. Find the time in which the concentration of the drug is increasing, and decreasing. When does the drug reach its maximum concentration? What is the maximum concentration? (do not worry about the units of your answer for this part).
When a person coughs, the windpipe contracts forcing the velocity of the air through the windpipe to increase. It is estimated that the velocity of the air through the windpipe is given by the function 
, 
, where k is a constant, R is a constant (the normal radius of the windpipe), and r is a variable (the radius of the windpipe during coughing). Find 
. What windpipe radius gives the maximum velocity during coughing?
Do a search on the Internet for growth velocity charts. You will likely be able to find a graph showing the growth rate of boys and girls. From this graph, sketch a possible graph of the height of boys and girls at various ages. Those graphs will also appear at the web site so you can confirm your graphs.
Objective
E.4 Back to Top
To determine the concavity and points of inflection of a function and to apply the second derivative test to determine local extrema.
To represent understandings through a variety of communication modes (COM).
To engage in activities that require exploration and manipulation in order to develop understandings of a concept (CCT).
To use language as a tool for learning and communicating (COM).
To make and justify decisions based upon understanding of calculus concepts (CCT).
Instructional Notes
Ask half the class to sketch the graph of a continuous differentiable function for which the slope of the tangent line is ever increasing as x increases. Ask the other half of the class to sketch the graph of a continuous differentiable function for which the slope of the tangent line is ever decreasing as x increases. Have students show their graphs to the class. (COM, CCT) The following two graphs should result.
After the students have drawn the graphs, introduce the terms concave up and concave down. A function is said to be concave up on an interval if the function lies above its tangent lines on that interval. A function is said to be concave down on an interval if the function lies below its tangent lines on that interval.
The following series of questions and comments will help to establish the sign nature of 
in the different regions of concavity.
• True or False? If a function is increasing, is 
positive?
• True or False? 
describes how 
is changing.
• Does the derivative of a function describe how the differentiated function is changing?
• If I differentiate 
to obtain 
, what would 
describe? (Hopefully students will respond with “How 
is changing.”)
• If 
is positive, then what is happening to 
? (Hopefully students will respond with “ 
is increasing”.)
• If 
is increasing, what is the nature of the graph of 
? (Concave up.)
• What would 
describe? (How 
is changing.)
Ask a similar series of questions to establish that if 
is negative, 
must be decreasing and thus the graph must be concave down.
Ask the students to draw and describe the graph of a continuous differentiable function that on some interval is concave up and on another interval is concave down (COM, CCT). Ask them to label the point at which the concavity changes as P and to draw a tangent line to the curve at that point. Next introduce the term inflection point.
If 
changes its concavity at point P , then P is called an inflection point of the function.
Note that the function crosses its tangent line at a point of inflection.
Since 
changes concavity at a point of inflection, then at a point of inflection, 
must change its sign from positive to negative or from negative to positive. What is the value of 
right at the point of inflection? One of two things can happen, 
can be zero or it may not exist. (Note that this is exactly the same thing that happens to 
at a local extremum.) This is because the first derivative (slope of the tangent line) reaches its maximum or minimum value (quits increasing/decreasing) at a point of inflection or the tangent line is vertical at a point of inflection. See below.
Note that 
is an inflection point for 
. 
is concave down for 
and concave up for 
. Note that 
, and thus 
.
Note that 
is an inflection point for 
. 
is concave up for 
and concave down for 
. Note that 
and thus does not exist for 
.
Just because 
does not guarantee that 
is an inflection point. Consider the function 
for which 
. Clearly 
, yet 
is not an inflection point as can be seen by examining the graph. Moreover 
does not change its sign at 0. Thus 
is always concave up. The message that is conveyed by 
is that at 
, the tangent line is not changing.
.Following a discussion similar to the one on page E-22, establish the test for concavity.
Test For Concavity
Suppose 
is continuous on the closed interval 
.
• If 
for all 
, 
is concave up on the interval 
.
• If 
for all 
, 
is concave down on the interval 
.
Note that a function is still considered to be concave up (down) on the interval 
even though there are a finite number of values c in 
for which 
or does not exist. For example 
and 
are concave up for 
, even though 
and 
does not exist.
To lead the students to the second derivative test, you could divide the class into groups and ask each group to sketch the graphs of six functions, near 
, that satisfy a given set of conditions (CCT).
Graph 1: 
and 
Graph 2: 
and 
Graph 3: 
and 
Graph 4: 
and 
Graph 5: 
and 
Graph 6: 
and 
Graphs 5 and 6 should lead the students to the conclusion you want.
The Second Derivative Test
If 
and 
, then 
has a local minimum at 
.
If 
and 
, then 
has a local maximum at 
.
If 
and 
, then the test fails and more analysis must be done or 
is undetermined.
Use the second derivative test to determine whether x values at which the first derivative is 0 correspond to local maxima or local minima. Note that if 
and 
or 
is undefined, then the second derivative test has failed. Use the first derivative test to see if 
is a local maximum, minimum, or neither.
Discuss examples similar to the two that follow.
Example 1: Determine the intervals of concavity and the inflection points for the function 
. Confirm with a graphing calculator.
Solution: 
. A sign analysis follows.
From the sign analysis: 
is concave up for 
because 
; 
is concave down for 
because 
. 
changes its sign at 
and 
. Thus 
and 
, namely 
and 
, are the coordinates of the inflection points. The graph appears below.
The window used was 
and 
.
Example 2: Find the local extrema for the function 
. First, find the critical numbers and then use the second derivative test. If the second derivative test fails at a critical number, use the first derivative test.
Solution: 
. Since 
, the critical numbers are 
, 
, 
. Evaluate 
at each critical number.
Now 
. Complete the following chart.
Since 
was neither positive nor negative, the second derivative test failed. To determine if there is a local maximum or minimum at 
, apply the first derivative test. A sign analysis of 
follows.
As 
does not switch signs at 0, then 
is neither a local maximum or minimum. It might best be described as a flat spot. Had we done a sign analysis on 
, we would notice that there is an inflection point at 
. Note that upon completing a sign analysis of 
, it is pointless to use the second derivative test to determine local extrema since the first derivative sign analysis will provide that information.
Also note that when doing the first derivative test, it is important that all critical points be included to give an accurate picture and prevent the students from analyzing the sign in a different interval.
Examples/Activities
You can use the graphing calculator to plot the graphs of 
, and/or 
, and/or 
simultaneously without first having to determine 
and 
. Suppose you wanted to illustrate the sign nature of 
in regions in which 
is concave up and concave down for the function 
. Enter the functions into the Y= menu as shown below if using a TI-83 calculator.
You will find nDeriv by pressing MATH 8. You will find Y 1 by pressing VARS ? 1 1.
To draw only 
and 
, deselect Y 2 by moving the cursor over the = sign and pressing ENTER. Notice that the = sign is no longer shaded in.
To have the graphs appear simultaneously, press MODE and choose Simul instead of Sequential as shown below.
Now press s and the graph of 
and 
appears.
Students can determine which graph is which by using the TRACE button and looking at the upper left corner of the screen where it indicates the function that the cursor is on. By pressing the up and down arrows, the students can move from one function to the other.
Note that 
is negative (below the x -axis) where 
is concave down and 
is positive (above the x -axis) where 
is concave up.
This strategy is particularly useful for functions whose derivatives require considerable work. In the case of 
, putting 
for Y 2 and 
for Y 3 is quicker.
Shown below is the graph of 
and 
. Which is which?
Shown below is the graph of 
, 
, and 
. Which is which?
Find 
. On the basis of a sign analysis of 
, determine the intervals in which the function is concave up or concave down. Confirm with a graphing calculator.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
Examples/Activities
1. Complete the following chart by referring to the graph of 
below. Determine the sign of 
, 
, and 
at each of the indicated points. Respond with 
, 
, or 0.
2. If 
exists, what conclusion, if any, can be made about the point 
if it is known that:
(a) 
and 
(b) 
and 
(c) 
and 
(d)
3. Find the critical numbers for each function. Then use the second derivative test to determine the relative extrema. If the second derivative test fails, use the first derivative test.
(a) 
(b) 
(c) 
(d) 
(e) 
Suggestions/Extensions
Have students try the technique at left for the function 
.
Have pairs of
students create sketches of function and their first and second derivatives for their partners to identify as 
, 
, and . As a class, discuss the strategies that students used in order to determine which curves were 
, 
and 
(CCT, COM).
A 24-hour flu victim's temperature after x hours is modelled by the function 
The graph of the function for 
and 
is shown below.
Some would argue that at the first inflection point, the victim is considered to be recovering because at this time, the rate of increase in the victim's temperature is starting to decrease. Do you agree? How many hours into the 24-hour flu is the first inflection point? Give your answer to two decimal places. You will need the quadratic formula. If you do not agree, when do you think the victim is beginning to recover?
If money is spent on advertising, usually there is an initial increase in sales. As additional money is spent on advertising, sales continue to increase but at a declining rate. See the figure below.
Point P is called the point of diminishing returns . In the interval 
, the graph is concave upward and each additional dollar spent on advertising results in more units sold than the previous dollar. In the interval 
, the function is concave down and each additional dollar spent on advertising results in fewer units sold than the previous dollar. Thus, the point of diminishing returns is the inflection point of the graph above. Finding the point of diminishing returns is important to someone in business because money invested in advertising beyond the point of diminishing returns is usually not considered wisely spent.
Suppose that if x dollars is spent on advertising, the number of bicycles sold is given by the function 
. Find the point of diminishing returns. How many bicycles will be sold at the point of diminishing returns? Confirm with a graphing calculator.
Objective
E.5 Back to Top
To determine the asymptotes (vertical, horizontal) of a function.
Instructional Notes
Ask the students to draw the graphs of 
, 
, and 
. This should be enough for students to recall what was mentioned in Math B30 as well as earlier in this course (A.3, C.2) that if 
causes the denominator of a function to be zero, but not the numerator, then the function has a vertical asymptote line with equation 
.
Definition of Vertical Asymptotes
If 
or 
, then 
is a vertical asymptote of 
.
Consolidate students' understanding by having them draw the graph of 
, noting the vertical asymptote line at 
.
Students have also experienced horizontal asymptotes in Math B30 as well as earlier in this course (C.2). Students should recall that if for large positive or negative values of x , the function approaches a constant value of L , then 
is a horizontal asymptote to the function. This should be formalized as follows.
Definition of a Horizontal Asymptote Line
The line 
is a horizontal asymptote of the function 
if either 
or 
.
It is important to guide students to a complete understanding of these ideas through some examples such as the ones that follow.
Example 1 : Find the equations of the vertical and horizontal asymptotes of the function 
.
Solution: To find the vertical asymptotes, find the values of x that make the denominator 0 but not the numerator: 
. Thus, the equations of the vertical asymptotes are 
and 
.
To find the horizontal asymptotes, find 
and 
:
. The 
is found in the same way and also yields 2. Note that in some cases, the two limits can be found simultaneously by evaluating 
. Thus, 
is the horizontal asymptote line. Support with a graphing calculator.
Example 2 : Find the horizontal and vertical asymptotes of the function 
.
Solution: The equation of the vertical asymptote is 
because 
causes the denominator to become 0 while making the numerator nonzero.
To find the horizontal asymptotes, examine 
and 
.
Now 
. As 
, it is positive and thus we can replace 
by x . Thus, 
. Thus 
is a horizontal asymptote line as 
.
To find 
, students proceed in a similar manner except that 
is replaced by 
. This results in 
. Thus, 
is a horizontal asymptote line as 
. Support with a graphing calculator.
Rather than show the work for 
each time, it is helpful for students to recognize that any rational function, for which the degree of the numerator is less than the degree of the denominator, will approach 0 as 
.
Examples/Activities
Students should be able to match a function with its graph by examining asymptotes.
1. Match each function with its graph by examining asymptotes.
(a) 
(b) 
(c) 
(d) 
(e) 
Determine the equations of the vertical and horizontal asymptotes of each function.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. Suppose that the number of people in a developing city neighbourhood x months from today is given by the function 
.
(a) How many people are in the neighbourhood right now? (Hint:
)
(b) By examining, prove that the population of the neighbourhood is always increasing.
(c) What is the equation of the horizontal asymptote of this function?
(d) Interpret your answer from part (c).
10. A tank currently contains 100 L of pure water. A salt-water solution of concentration 20g/L is added to the tank.
(a) If x L of salt water is added to the tank, then show that the concentration of the salt water is
.
(b) If the functionis graphed, find the equation of the horizontal asymptote line.
(c) Interpret your answer in part (b).
11. Rectangle ABCD has a length of x m and a width of 10 m.
(a) You walk around the perimeter of the rectangle while your brother walks along the diagonal from A to C and back. Find an expression for the ratio of the distance you walk to the distance your brother walks.
(b) What is the value of the ratio if
?
(c) Find
.
(d) Explain why the answer of 1 in part (b) “make sense”.
Suggestions/Extensions
Many students of mathematics think that the graph of a function does not cross an asymptote to the function. Consider the function 
. What is 
? To answer that, consider what possible values the numerator could have and consider what is happening to the value of the denominator. What then is the equation of the horizontal asymptote to the function? Draw the graph of the function using a graphing calculator. What do you notice?
As the students are introduced to the definitions of the asymptotes, have students sketch graphs and discuss how they visualize both the function and graphs of the functions with that type of asymptote.
Although slant asymptotes are not specified in the learning objective, students should be able to identify slant asymptotes on graphs.
Students can also explore how to determine the equation of a slant asymptote by applying the definition: if 
, and if 
, then 
is an asymptote of 
. If 
is of the form 
, then 
is said to have a slant asymptote line .
Recall that if 
results in 
, then 
has a hole at 
provided 
. Find all asymptotes and holes for the function 
.
(Hint: There are two holes, one vertical asymptote line, and one slant asymptote line.)
You may want to lead students to observe that for rational functions: if the degree of the numerator exceeds the degree of the denominator by j , then the asymptote is k . See table.
Find the slant asymptote to the function 
.
Solution: Note that the degree of the numerator is greater than the degree of the denominator. Carry out the long division.
As a result of the long division, we see that 
.
Notice that 
.
Thus as 
, 
( 
approximately equal to). Thus 
is the equation of the slant asymptote line. Support with a graphing calculator.
Objective
E.6 Back to Top
To sketch the graph of a function by analyzing the first and second derivatives, the asymptotes, and the intercepts of the function.
Instructional Notes
In this section students will use the knowledge and skills learned in the previous section of this unit to sketch the graphs of functions. Questions of this type require considerable time – up to 30 minutes each. With student input and assistance, work through a few typical examples like the ones shown below.
It is extremely difficult and time consuming to make up questions so that the x- intercepts, intervals of increase and decrease, intervals of concavity, and coordinates of relative extrema are cooperative values. Questions that address this objective can create difficulties for students. Should a student obtain the wrong first derivative, and thus the wrong second derivative, the conclusions reached will be incorrect. If the student then draws the graph of the function based on this analysis, it will not resemble the correct graph. It is difficult for teachers to assess such responses.
The examples that follow model what to expect from the students.
Example 1: For the function 
, find each of the following. Support all conclusions reached.
(a) a sign analysis of 
.
(b) the open intervals on which 
is increasing and/or decreasing.
(c) the critical numbers.
(d) the relative extrema.
(e) a sign analysis of 
.
(f) the intervals on which 
is concave up and concave down.
(g) the coordinates of any inflection points.
(h) the x and y- intercepts.
(i) the equations of any asymptotes.
(j) a careful sketch of the function that supports all of the above features.
Solution:
(a) 
. For the polynomial 
, notice that 
. Thus 
is a factor of 
. Dividing synthetically and factoring completely, we find that 
. Thus, 
. A sign analysis follows.
(b) Thus 
increases for 
because 
and decreases for 
because 
. True, at 
, 
. However, single points at which 
, instead of being negative, do not exclude them from being in the decreasing interval. See E.3.
Example 1 Solution (continued):
(c) The critical numbers are 
and 
because 
and 
are 0.
(d) 
, that is 
, is a local minimum because 
went 
at 
. 
, that is 
, is neither a local maximum nor a local minimum because 
does not switch signs at 
.
(e) 
. A sign analysis follows.
(f) 
is concave up for 
because 
. 
is concave down for 
because 
.
(g) There are inflection points at 
and 
, that is 
and 
, because there is a sign change in 
at 
and 
.
(h) To find the y -intercept, find 
. We found this to be 0 in (g). Thus, the y -intercept is 0. To find the x -intercept, set 
and find x . If 
, then 
. Thus 
or 
. Since none of 
, 
, 
, 
, 
give 0, 
cannot be factored over the rationals. Thus the best answer is that 
is an x -intercept and there will be no more than three others, as the degree of 
is 3.
(i) Polynomial functions never have asymptotes. Both 
and 
yield 
.
(j) To sketch the graph of the function, begin by drawing all asymptotes. Then mark any intercepts. Next plot all points you have already found. At the local minima, make a small È and at the local maxima, make a small Ç . Next, work with the increasing/decreasing intervals and the concavity intervals to complete the graph. Insist that students choose a suitable scale so that all features of the graph are shown.
We know the function increases to the right of 
, but we cannot be sure just where the graph will cross the x -axis. Evaluating 
would be helpful.
Example 2: Repeat example 1 for the function 
.
Solution:
(a) 
. The details are omitted.
(b) Note that 
and 
are not in the domain of the function.
is increasing on the intervals 
because 
. 
is decreasing on the intervals 
because 
.
(c) The critical numbers are 
, 
because 
. ( 
are not critical numbers because they are not in the domain.)
(d) 
is a local maximum because 
went 
. 
is a local minimum because 
went 
. 
is not a local extremum because there is no sign switch in 
at 
.
(e) 
. 
(f) 
is concave up for 
because 
.
is concave down for 
because 
.
(g) 
is an inflection point because 
switched signs at 
.
(h) Since 
, the y -intercept is 0. To find the x -intercept, let 
and solve for x . 
implying 
so 
is the x -intercept.
Example 2 Solution (continued):
(i) 
has vertical asymptotes with equations 
and 
as these values cause the denominator to be 0 but not the numerator.
If students have considered slant asymptotes previously students can do so for this graph. Since the degree of the numerator is one greater than the degree of the denominator, there will be a slant asymptote. Perform long division.
Thus 
. It can be shown that 
. Thus, as 
, 
. Thus 
is a slant asymptote line.
Note: Slant asymptote is optional.
Examples/Activities
An activity for an individual hand-in assignment, gr
oup project, or exploration follows.
Consider the cubic function with 
. (a) How are a , b , and c related if 
has two local extrema. ( 
)
(b) Assuming 
has two local extrema, find their x -coordinates. ( 
)
(c) How are a , b , and c related if 
has no local extrema. ( 
)
(d) Why is it impossible for 
to have only one local extrema?
(e) Why will 
always have an inflection point?
(f) What is the x -coordinate of the inflection point? ( 
)
(g) If 
has two local extrema, prove that the x -coordinate of the inflection point always lies midway between the x- coordinates of the local extrema.
(h) Under what conditions is the inflection point on the y -axis? ( 
)
(i) Find an expression for the slope of the tangent line at the inflection point. ( 
)
(j) Under what condition(s) will the slope of the tangent line at the inflection point be positive?
(k) Under what condition(s) will the y -intercept of 
be negative?
(l) Under what condition(s) will there be at least one positive x -intercept?
(m) Under what condition(s) will there be two local extrema both of which have positive x -coordinates?
(n) Under what condition(s) will there be two local extrema both of which have negative x -coordinates?
(o) Under what condition(s) will there be two local extrema whose x -coordinates are opposite in sign?
These questions use a lot of paper and take up a good deal of time. Students may want to use one sheet of loose-leaf paper (both sides) per question. This assignment could be phased in earlier in the unit. When students have completed E.3, they can do parts (a) through (d) of every question. When objective E.4 has been achieved, students could do parts (e) and (f) of every question.
For each function, find the following. Support all conclusions reached.
(a) a sign analysis of 
.
(b) the open intervals on which 
is increasing and/or decreasing.
(c) the critical numbers.
(d) the relative extrema.
(e) a sign analysis of 
.
(f) the intervals on which 
is concave up and concave down.
(g) the coordinates of any inflection points.
(h) the x and y- intercepts.
(i) the equations of any asymptotes.
(j) a careful sketch of the function that supports all of the above features.
Most of the questions should lead students to “what do I do now?” moments so that they can really come to grips with the ideas of this unit. Here are a few suggestions of the different situations that could arise.
1. 
(no complications)
2. 
(will need the quadratic formula to find all three x -intercepts)
3. 
(one of the critical numbers does not yield a local extrema)
4. 
(need quadratic formula to determine that there are no x -intercepts, a slant asymptote, a vertical asymptote)
5. 
(two horizontal asymptote lines)
6. 
(rational exponents, two inflection points, one that occurs where 
, the other where 
does not exist; y -coordinates are messy – use a calculator for them; 0 is a critical number ( 
does not exist) but there is no local extrema there).
7. 
(a horizontal asymptote line and two vertical asymptote lines)
8. 
(rational exponent, three x -intercepts, two of which are radicals, 
does not exist at the inflection point 
, 
does not exist at 
, but there is no relative extrema at 
, no asymptotes)
Suggestions/Extensions
Students could work in pairs or small, groups and reflect on each others work on the same or different questions (PSVS).
See Appendix A for additional objective and activities.