Project AMP Antonio R. Quesada – Director, Project AMP
Introduction to Tangent Lines using TI-nspire CAS
Section 1: A Teacher Page
1. A summary of the activity
2. Assumptions of prior student knowledge
3. Extensions
4. State standards addressed
5. AP standards addressed
Solutions
Introduction to Tangent Lines using TI-nspire CAS
Summary of Activity:
Students will use the TI-Nspire to draw the tangent lines to a point on a particular graph, grab the point and move it around on the graph, and observe the changes in the tangent line. Students will be asked to observe the relationship of the tangent to the curve, the signs of the slopes of the tangent lines, slopes of tangents at extrema, and the position of the tangents at points of inflection and concavity.
Assumptions of Previous Knowledge
Students can solve and find derivatives algebraically and know that the definition of derivative is the slope of the tangent line. This activity is designed to be an introduction to curve sketching, finding extrema, identifying where a function increases and decreases, determining concavity, and finding points of inflection that bridges the two approaches.
Students must also be able to tell on what intervals a function is increasing or decreasing and if a line is above or below the curve. They must also understand what minimums and maximums are on a graph.
Students must also be familiar with the basic operations of the TI-Nspire, specifically graphing a function, drawing a tangent to the curve, measuring the slope and dragging the tangent around. Directions are given for these procedures for the first problem only.
Extension to the Activity
Two extensions are included. The first one asks students to explore the relationship between position, velocity, and acceleration. The second focuses on linear approximations and whether the estimates are underestimates or overestimates.
Ohio State Standards
Grade:12 Strand: Patterns, Functions, and Algebra
#7: Make mathematical arguments using the concept of limit
#10 Use the concept of limits to find instantaneous rate of change for a point on a graph as a slope of tangent at a point.
A.P. Objectives
This activity emphasizes the Advanced Placement objectives listed below. These objectives are found in the AP Course Description handbook distributed to AP teachers.
- The slope of a curve at a point. Examples are emphasized, including points at which
there are vertical tangents and points at which there are no tangents.
- Tangent line to a curve at a point and local linear approximation
- Relationship between the concavity of ƒ and the sign of ƒ
- Points of inflection as places where concavity changes
- Corresponding characteristics of graphs of ƒ and ƒ
- Relationship between the increasing and decreasing behavior of ƒ and the sign of f
Solutions
Introduction to Tangent Lines using TI-nspire CAS
Student Activity Sheet
Objectives: Students will graph a function, draw the tangent line to a point, move the tangent along the function, and make observations regarding the relationship of the tangent to the curve, the signs of the slopes of the tangent lines, slopes of tangents at extrema, and the position of the tangents at points of inflection and concavity.
To the student: You have been finding the derivative of a variety of functions using Calculus rules such as the power rule, product and quotient rules, and the chain rule. You have also found the slope of a tangent line at a point and used it to write the equation of the tangent line. This activity will help you explore the behavior of tangents and the concept of the derivative.
Part One: Increasing and Decreasing Functions and the Signs of Slopes
1) Using the TI-Nspire, graph. Set the window to [-10,10] by [-10,10].
Choose the home icon, select 2:Graphs
Enter f1(x)= x2
Choose menu button, select 4:Window→1:Window Settings. Set the x-min to -10, hit TAB. Set the x-max to 10 and hit TAB. Set the y-min to -10 and hit TAB. Set the y-max to 10, and hit enter.
2)Now draw a point on the curve y = x2
Choose the menu icon, select 6:Points and Lines→2:Point on
Move the cursor to the curve until “point on” and a number appears.
Hit enter and the point should appear and coordinates should darken. Hit the esc key.
You can move the coordinates and drag them to an empty area of the graph, if you wish: Move the cursor to the coordinates until “text” appears. Hold the hand button down until the finger and thumb pinch together and drag it to an empty portion on the graph. Hit esc key to release.
3)Now draw a tangent to the curve y = x2 at the point you drew in step 2.
Choose the menu icon, select 6:Points and Lines→7:Tangent
Move the cursor to the curve until “point on” appears.
Hit enter and the tangent line should appear. Hit the esc key.
4)Next calculate the slope of the tangent line you drew in step 3.
Choose the menu icon, select 7:Measurement→3:Slope
Move the cursor to the tangent line until “line” appears and hit enter twice. The number should darken. Hit the esc key.
(You can move this number to an empty area of the graph by following the directions at the end of step 2.) You can add text to the screen using the menu icon. Select 1: Actions→5 Text.
Your Screen should now look like the screen at the right
5)Drag the point around the graph and the tangents should move as well. Move the point around and observe what happens to the tangents and the slope of the tangent line. You will use this process to fill out the tables in this activity.
Move the cursor to the point until the word “point” appears.
Hold down the hand button until the finger and thumb pinch together and the point is flashing.
Use the navigation pad to move the point along the graph. If you wish, hit the esc key to release.
6)Fill out the chart below. Move the point as close to the coordinates listed in the table, AS CLOSE AS YOU CAN. You may or may not be able to get the actual coordinate point, but get as close as you can. Fill in positive or negative for the sign of the slope.
Coordinate Point / Is y = x2 /4 increasing or decreasing? / Sign of the slope of the tangent linex = -3 / decreasing / negative
x = -2 / decreasing / negative
x = -1 / decreasing / negative
x = -0.5 / decreasing / negative
x = 0.5 / increasing / positive
x = 1 / increasing / positive
x = 2 / increasing / positive
x = 3 / increasing / positive
7)Now, on a new graph page, draw the graph of . Keep the same window setting. Repeat the process for drawing a tangent at a point for the function . Enter the results in the table below. Remember to drag the point as close to the x-value as you can.
Choose the home icon, select 2:Graphs. This gives you a new graph as well as a new page.
Enter
Choose menu button, select 4:Window→1:Window Settings. Set the x-min to -10, hit TAB. Set the x-max to 10 and hit TAB. Set the y-min to -10 and hit TAB. Set the y-max to 10, and hit enter. Follow the rest of steps 2-5.
Coordinate Point / Is y = -x2/4 increasing or decreasing? / Sign of the slope of the tangent linex = -3 / increasing / positive
x = -2 / increasing / positive
x = -1 / increasing / positive
x = -0.5 / increasing / positive
x = 0.5 / decreasing / negative
x = 1 / decreasing / negative
x = 2 / decreasing / negative
x = 3 / decreasing / negative
NOTE: For each of the new tables that follow, begin the graph on a new graph page.
8)Repeat the process for drawing a tangent at a point for the function y = x3, keeping the same window settings. Enter the results in the table below.
Coordinate Point / Is y = x3 increasing or decreasing? / Sign of the slope of the tangent linex = -2 / increasing / positive
x = -1.5 / increasing / positive
x = -1 / increasing / positive
x = -0.5 / increasing / positive
x = 0.5 / increasing / positive
x = 1 / increasing / positive
x = 1.5 / increasing / positive
x = 2 / increasing / positive
9)Repeat the process for drawing a tangent at a point for the function y = ex, keeping the same window settings. Enter the results in the table below. Remember to drag the point as close to the x-value as you can.
Coordinate Point / Is y = ex increasing or decreasing? / Sign of the slope of the tangent linex = -4 / increasing / positive
x = -2 / increasing / positive
x = -1 / increasing / positive
x = -0.5 / increasing / positive
x = 0.5 / increasing / positive
x = 1 / increasing / positive
x = 1.5 / increasing / positive
x = 2 / increasing / positive
10)Repeat the process for drawing a tangent at a point for the function, changing the window settings to [-1, 35] by [-5, 5]. Enter the results in the table below. Remember to drag the point as close to the x-value as you can.
Coordinate Point / Is y = ln(x) increasing or decreasing? / Sign of the slope of the tangent linex = 0.5 / increasing / positive
x = 1 / increasing / positive
x = 3 / increasing / positive
x = 10 / increasing / positive
x = 15 / increasing / positive
x = 20 / increasing / positive
x = 25 / increasing / positive
x = 30 / increasing / positive
11)Repeat the process for drawing a tangent at a point for the function. Use the window settings [-3, 6] by [-2,2]. Enter the results in the table below. Remember to drag the point as close to the x-value as you can.
Coordinate Point / Is y = sin(x) increasing or decreasing? / Sign of the slope of the tangent linex = -2.7 / decreasing / negative
x = -2.5 / decreasing / negative
x = -1 / increasing / positive
x = -0.5 / increasing / positive
x = 0.5 / increasing / positive
x = 1 / increasing / positive
x = 2.5 / decreasing / negative
x = 3 / decreasing / negative
x = 4 / decreasing / negative
x = 5 / increasing / positive
12)
(a) Looking at all of the charts, what conclusion can be made regarding the sign of the slope of the tangent when the function is increasing?
The slope is positive.
(b) When the function is decreasing?
The slope is negative.
13)Where do you think that the sign of the tangent switches from positive to negative?
When the slope is neither positive or negative. When the slope equals zero.
14) Without graphing, can you tell if a function is increasing or decreasing by looking a given point on the function? Consider y = 2x2 + 4x – 3. Take the derivative by hand at the points x = -2 and at x = 3. Is it increasing or decreasing at each of those points? Show all of your work.
The slope is negative, consequently the curve is decreasing at
The slope is positive, consequently the curve is increasing at
Part Two: Slopes of Tangents and Local Extrema
NOTE: Local extrema are the mimimums or maximums of a function on a given interval.
1)Graph , on a new graph screen using a window setting of [-10, 10] by [-10, 10]. Draw a point on the curve and draw a tangent to that point. Drag the tangent to the points where the graphs reach a minimum value. Fill in the chart below:
Refer to steps 1-5 from Part One.
Graph the point and drag it to the minimum value. You know you are at the minimum because a lower case m in a box appears to the lower right of the “hand.”
Remember that if you cannot read some of the information because it is all bunched together, you can drag the coordinates and slopes to another area of the screen. Refer to step 2 in part one.
Coordinates of Minimum / Value of the slope at the minimum / The line is increasing, decreasing, horizontal, or vertical…(0, 0) / 0 / horizontal
2)Graph on a new graph screen using, the same window settings. Draw a point on the curve and draw a tangent to that point. Drag the tangent to the points where the graphs reach a maximum value. Fill in the chart below:
Graph the point and drag it to the maximum value. You know you are at the maximum because an upper case M in a box appears to the lower right of the “hand.”
Coordinates of Maximum / Value of the slope at the maximum / The line is increasing, decreasing, horizontal, or vertical…(0, 0) / 0 / horizontal
3) Graph on a new graph screen using the window [-3, 6] by [-2, 2]. Draw a point on the curve and draw a tangent to that point. Drag the tangents to the points where the graphs reach a maximum or minimum value. Fill in the chart below:
Min or Max / Coordinates of Min or Max / Value of the slope at the min or max / The line is increasing, decreasing, horizontal, or verticalMinimum / (3π/2, -1) / 0 / horizontal
Maximum / (π/2, 1) / 0 / horizontal
Minimum / (-π/2, -1) / 0 / horizontal
4) Now make a conclusion about your observations from #1-3. Fill in the blanks:
At an extrema, the slope of the tangent is __ 0 __ and the tangent line is __ horizontal ______.
5) Now we want to examine the behavior of a function before and after an extrema point. Consider ,. From part two, #1, the coordinates of the minimum were (0, 0). Fill in the table below, using information from part one and part two, as well as your graph:
interval / f(x) increasing or decreasing? / Is the slope of tangent positive or negative?(-∞, 0) / Decreasing / negative
(0, ∞) / Increasing / positive
Conclusion: The point (0, 0) is a minimum or maximum?
The point is a minimum.
6)Consider ,. From part two, #2, the coordinates of the maximum were (0, 0). Fill in the table below, using information from part one and part two, as well as your graph:
interval / f(x) increasing or decreasing? / Is the slope of tangent positive or negative?(-∞, 0) / Increasing / positive
(0, ∞) / Decreasing / negative
Conclusion: The point (0, 0) is a minimum or maximum?
The point is a maximum.
7)The function on the given interval had two minimums and one maximum. From part 2 #3, what were the coordinates of these three points?
Mimimums:Maximum:
Fill in the table below, using information from part one and part two, as well as your graph:
intervals / f(x) increasing or decreasing? / Is the slope of tangent positive or negative?(-3, -1.5708) / decreasing / negative
(-1.5708, 1.5708) / increasing / positive
(1.5708, 4.7123) / decreasing / negative
(4.7123, 6) / increasing / positive
8) Make a conclusion about determining if a point is a maximum or minimum based on your observations:
A point is a maximum if the slope at the point is ___ 0 ______and the slope of the tangents
switch from __ positive __ before the point to ___ negative ___after the point.
A point is a minimum if the slope at the point is ____0____ and the slope of the tangents switch
from __ negative ____before the point to ___ positive ___after the point.
9)Consider the other functions from part 1: y = x3, y = ex, and . Look at the graphs and your tables from part one. Do these functions have any extrema? If so, are they minimums or maximums?
They have no extrema.
Explain your answer based on your conclusions from part two, #8.
The curves are always increasing. The slopes of the tangent lines never switch signs.
10) How can find extrema without looking at the graph? Consider y = 2x2 + 4x – 3. Take the derivative by hand. Based on your answers from #8, find where an extrema occurs.
An extrema can occur when the slope of the tangent line equals zero.
Determine if the extrema is a maximum or minimum. Show all work and explain your answer.
The derivative (slope of the tangent line = 0 at x = -1).
The slope is negative, consequently the curve is decreasing at
The slope is positive, consequently the curve is increasing at
Thus, the function has a minimum at x = -1, since the slopes change from negative to positive.
Part Three: Tangents Above or Below the Curve and Concavity
Tangents can lie above or below the curve. All tangent lines that lie below the curve have y-values that are less than the corresponding y-values on the function. In other words, if f(x) are the y-values on the curve and T(x) are the y-values on the tangent curve, then T(x) ≤ f(x) for all values of x.
All tangents lines that lie above the curve have y-values that are more than the corresponding y-values on the functions. In other words, T(x) ≥ f(x) for all values of x.
1) Let’s revisit the function ,. As you move the point along the graph, are the tangents above the graph or below the graph?
They are below the graph.
What is the general shape of the function, specifically looking at whether it opens upward or downward?
The graph opens upward.
A graph that opens upward is referred to as being concave up. A graph that opens downward is concave down. Is concave up or concave down?
is concave up.
2) Now look at . As you move the point along the graph, are the tangents above the graph or below the graph?
They are above the graph.
Is concave up or down?
is concave down
3)Now look at y = ex. As you move the point along the graph, are the tangents above the graph or below the graph?
They are below the graph.
Is y = ex concave up or down?
y = ex is concave up
4)Now look at . As you move the point along the graph, are the tangents above the graph or below the graph?
They are below the graph.
Is concave up or down?
is concave up
5) Now make a conclusion:
With respect to tangent lines, a function is concave up when:
the tangent lines lie below the graph.
With respect to tangent lines, a function if concave down when:
the tangent lines lie above the graph.
Part Four: Points of Inflection and Concavity
Some functions won’t have a single concavity. Let’s look at some examples of those functions.
As you move the point along the graph, the tangents lines may switch positions. The point at which the concavity switches is called the point of inflection. At this point, some of the points of the tangent line are above the curve and some of the points are below.
1)Consider y = x3. Drag the tangents along the curve to find at which point on y = x3 this occur?
This occurs at x = 0.
On what interval is y = x3 concave up? On what interval is y = x3 concave down?