Activity 7. The Slope of the Tangent Line (Part 2) Objectives. Introduction. Problem
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1 Activity 7 Objectives Use the CellSheet App to find the approximate slope of a tangent line of a curve Compare the x-slope relationship of parabolic and cubic curves Introduction In Activity 6, you found the approximate slope of a line tangent to a parabolic curve by finding the slope of a secant line. You saw that the slope of the secant line changes the same amount that the value of x changes. In other words, there is a linear relationship between the slope of the secant line and the value of x. You also found that the smaller the x, the more accurate the approximation of the slope of the tangent line. In this activity, you will look at the slope of a tangent line of a cubic curve to determine if the same relationship holds with the cubic curve. You will find how the slope of a tangent line of a cubic curve changes as the value of x changes. Problem The Slope of the Tangent Line (Part 2) How does the slope of a tangent line of a cubic curve change as the value of x changes? Is there a linear relationship between the two values as there is with tangent lines of parabolic curves? What is the relationship between the two sets of values? In this activity, you will graph the function y = x 2 + 4x 5 and the function y = x 3 + 2x 2. You will also graph the slope values of the tangent lines for each of the graphs. Because you will approximate the slope of the tangent line using a secant line, you want the increment for x to be rather small. From Activity 6, you found that the smaller the x, the more accurate the approximation of the slope of the tangent line.
2 68 Exploring Mathematics with the CellSheet TM Application Exploration 1. Create a new spreadsheet in the CellSheet App, and name it SLOPETAN. 2. Enter "X in cell A1 and "Y in cell B1. You want to fill column A with values of x from L5 to 7 in increments of To do so quickly and easily, you will use the Sequence command. 3. Select Menu > Options > Sequence. Enter A2 as the 1st cell. Beside the seq( prompt, type,,l5,7,.25 and close the parentheses. Select Down. Press Í three times. This will fill the column with a sequence that starts at L5 and goes to 7 with increments of You will enter a formula to compute the y-values. This will be done using the Fill Range option. 4. Move to cell B2. Select Menu > Options > Fill Range. Enter B2:B50 as the range. Enter the formula =A2^2+4*A2 5 for y at the Formula: prompt. Press Í. The result is a complete table of values. Next, you will determine the slope of the secant line through the two points (L5, 0) and (L4.75, L1.438). 5. Enter the formula =(B3 B2)/(A3 A2) in cell C3. The value in cell C3, L5.75, is an approximation of the slope of the tangent line through the point (L5, 0). As this formula is copied down column C, the slope of the curve, as approximated by a secant line at each point, is calculated. What do you notice about the values in column C? Is there a pattern? What does that tell you about the relationship between the x and the slope?
3 Activity 7: The Slope of the Tangent Line (Part 2) Use the FILL RANGE option to copy the formula through cell C50. At the Formula prompt, type =(B3 B2)/(A3 A2). Columns A and B contain a table of values for the curve y = x 2 + 4x 5. The points (x, M T ) are in columns A and C. Because there is no value in cell C2, the graphs will be plotted with points in rows 3 through Select Menu > Charts > Line. You can graph up to 3 curves on the same coordinate axis. Enter the values as shown on the screen. 8. Press Í until the graph appears. The graph of the parabola and the graph of the slope of the curve appear. You will notice that the graph of the slope is a straight line, representing a linear function. This result agrees with what you found in the Activity 6. The slope of the secant changed in direct proportion to the change in x. Now you want to determine if the same x-slope relationship holds for cubic curves. You can use the same spreadsheet, but will change the function to a cubic one, y = x 3 + 2x Use the FILL RANGE option to enter the function =A2^3+2*A2^2 for y in cells B2 through B50. It may take some time for the graphing handheld to carry out all the changes to columns B and C. 10. Is there a constant difference between the values in column C as there was with the parabolic curve? Looking at the first four values listed, what are the differences?,,. Is the x-slope relationship a linear one with a cubic curve? Explain.
4 70 Exploring Mathematics with the CellSheet TM Application 11. Graph these two curves using the LINE CHART option, as before. The appropriate XRange, YRange1, and YRange2 are still present. 12. Press p to access the LINE WINDOW settings. Change the settings to those shown. Look at the graphs shown. The cubic graph shows the function of the cubic equation, y = x 3 + 2x 2. What is the shape of the second graph? What does the graph tell you about the relationship between the slope of the tangent line and the x? Is the relationship the same as with the quadratic function?
5 Student Worksheet Name Date Solving the Problem 1. Complete the table using data from this activity. Slope of the Line Tangent to the Curve (estimated by the slope of the secant line) x-coordinate of the Point Slope of the Cubic y = x 3 + 2x 2 Change in the Slope (from one point to the next) Slope of the Quadratic y = x 2 + 4x 5 x = 0 L x = x = x = x = Change in the Slope (from one point to the next) Analyzing the Data 2. The data show that the slope of the quadratic changes, that is, for each change of 1 in the x-direction, there is a change of in the y-direction. 3. The data show that the slope of a cubic does not change. For each change of one in the x-direction, the change in the y-direction is. For example, the change in the slope from x = 0 to x = 1 is, and the change in the slope from x = 1 to x = 2 is. 4. From the table above and the graph shown, you can determine that the change in the slope for the quadratic equation is while the change for the cubic equation is in nature. 5. Looking at the graph in Question 4, why is the vertex of the parabola at (0, 0)?
6 72 Exploring Mathematics with the CellSheet TM Application Extending the Activity 1. Recall that the degree of a polynomial in x is the value of the highest power of x present in the polynomial. The degree of a cubic is 3, the degree of a quadratic is 2, and the degree of a linear function is 1. In each of the two examples in this activity, how does the degree of the function compare with the degree of the curve made up of the points (x, M T )?
7 Teacher Notes Activity 7 The Slope of the Tangent Line (Part 2) Objectives Use the CellSheet App to find the approximate slope of a tangent line of a curve Compare the x-slope relationship of parabolic and cubic curves Materials Time TI-84 Plus/TI-83 Plus 60 minutes Preparation The appropriate preparation for this activity is Activity 6. However, another activity which focuses on the same concepts can be substituted. Students should already realize the relationship between the x and the slope of a line tangent to a parabolic curve. Elicit Questions Consider how the slope of each curve shown here changes as x changes. What characteristics do you expect? What patterns might be present? Management As the concepts in this exploration require prior learning, it may be useful to have students work in pairs to help those students whose prior knowledge of the essential concepts is not as instantiated. Answers to Exploration Questions 5. There is a constant difference between the values. The relationship is linear in nature , 5.75, 5.375; No. The difference between the values is not constant.
8 74 Exploring Mathematics with the CellSheet TM Application 12. Parabolic; The graph shows that the relationship is quadratic in nature. It is not the same as that of the quadratic function, which is linear in nature. Answers to the Student Worksheet Solving the Problem 1. x-coordinate of the Point Analyzing the Data 2. Linearly; 2 3. Linearly; increasing; 6.25; Linear; quadratic Slope of the Curve (estimated by the slope of the secant line) Slope of the Cubic y = x 3 + 2x 2 Change in the Slope (from one point to the next) Slope of the Quadratic y = x 2 + 4x 5 x = 0 L Change in the Slope (from one point to the next) x = x = x = The parabola is a graph of the slope of the cubic. Because the cubic flattens out (has a zero slope) at x = 0, the value of the quadratic at 0 must be 0. Extending the Activity 1. In each case, the degree of the curve made up of the points (x, M T ) is 1 less than the degree of the function.
Objective. m y 1 y = x 1 x 2
Objective Use the CellSheet App to approximate the slope of a line tangent to a curve Activity 6 Introduction The Slope of the Tangent Line (Part 1) You have learned that the equation y = mx + b is a linear
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MEI GeoGebra Tasks for A2 Core
Task 1: Functions The Modulus Function 1. Plot the graph of y = x : use y = x or y = abs(x) 2. Plot the graph of y = ax+b : use y = ax + b or y = abs(ax+b) If prompted click Create Sliders. What combination
Quadratic Functions CHAPTER. 1.1 Lots and Projectiles Introduction to Quadratic Functions p. 31
CHAPTER Quadratic Functions Arches are used to support the weight of walls and ceilings in buildings. Arches were first used in architecture by the Mesopotamians over 4000 years ago. Later, the Romans
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Summer Math Assignments for Students Entering Algebra II
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The Tangent Line as The Best Linear Approximation
The Tangent Line as The Best Linear Approximation Mark Howell Gonzaga High School Arlington, VA A tangent line to a curve, y f (x), at a point where x a has two important properties: it contains the point
PreCalculus Chapter 9 Practice Test Name:
This ellipse has foci 0,, and therefore has a vertical major axis. The standard form for an ellipse with a vertical major axis is: 1 Note: graphs of conic sections for problems 1 to 1 were made with the
1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check
Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's
MATHS METHODS QUADRATICS REVIEW. A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation
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MEI Desmos Tasks for AS Pure
Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x² 4x + 1 2. Add a line, e.g. y = x 3 3. Select the points of intersection of the line and the curve. What
STATISTICS MEAN Know the TOTAL # of points MEDIAN MIDDLE ($) Arrange the scores in order MODE most frequent. RANGE DIFFERENCE in high and low scores
HSPE Mathematics Hints for SUCCESS The BASICS Be positive, be reassuring. Tell the students that if they have done what you have asked in preparation, then they are prepared for the test. They will pass
Properties of Graphs of Quadratic Functions
H e i g h t (f t ) Lesson 2 Goal: Properties of Graphs of Quadratic Functions Identify the characteristics of graphs of quadratic functions: Vertex Intercepts Domain and Range Axis of Symmetry and use