Calculus X
Calculus Name: ______________________________
Date: ______________________________
Lab – Derivation of the Slope Function
Objective: Given a function, [pic], students will generate the graph and equation of the corresponding slope function, [pic].
I. Pre-Lab Questions
To understand this activity, you will need to refresh your memory on some previously defined terms. Answer the following questions before proceeding.
1. Define “Local Linearity.”
2. What is meant by an instantaneous rate of change?
3. On a graph, what gives us the instantaneous rate of change at a point?
4. When an instantaneous rate of change is negative, how would you describe the graph?
5. When an instantaneous rate of change is positive, how would you describe the graph?
II. Directions – Please follow directions carefully!
Throughout this activity, you will be working with the function [pic], defined at the top of the worksheet titled, “Charts, Data and Computations.” Note: Everyone’s [pic] is not the same!
1. On your graphing calculator, graph [pic] in the window –6 < x < 6 and –20 < y < 20.
2. Carefully transfer your graph to the graph paper (provided). Use the trace function or table to generate points so your graph can be transferred to paper as accurately as possible. (Your graph should look “taller” than on your calculator, since you will be using a square scale on your graph paper.)
3. Because of local linearity, we can say that a curve has a slope at any given point. We will find the slope of the curve at the x-values given in Chart #1, then later we will graph these values as a separate function called the slope function.
4. To complete the table, first recall that when we discuss the slope of a curve at a point, we really mean the slope of the line tangent to the curve at that point. If we draw the tangent line at a point, and figure out the equation of that tangent line (written in slope-intercept form), then we also know the slope of the curve at that point.
5. Fortunately, we do not have to draw and calculate equations of tangent lines by hand; our calculators will do this for us. To do this, first be sure that you are working in the graphing screen. You should be looking at the graph of your function.
6. Press the DRAW button. (It’s the 2nd function on the PRGM key.) Since you wish to draw a tangent line, select option 5: Tangent ( from this menu. Once you do this, the calculator will return you to your graph. It is now waiting for a prompt from you. You need to enter the x-value at which you want to draw the tangent line. Type the first x-value on Chart #1 and press ENTER. The calculator will construct the tangent line, and give the equation of that line in slope-intercept form. Copy the equation of the tangent line onto your table, and enter the slope in the slope column.
7. Now repeat these steps for each of the x-values on the table, and complete the chart.
8. Notice that the slope changes as the value of x changes. Therefore, the slope at a point on the curve depends on the x-value of the point, so slope is a function of x. (Slope = [pic]) Transfer the values in the slope column to the T-chart column labeled,[pic].
9. Since [pic] is its own function, if has its own graph, and your T-chart now has the coordinates of several points on the graph of [pic]. Using a different color pen or pencil, accurately plot these points on the same coordinate axis you graphed [pic] at the beginning. Connect these points with a smooth curve. Your coordinate axis should now have two functions, [pic] and [pic], graphed simultaneously, but in different colors. Be sure each function is clearly labeled on your graph.
10. Look at the points on [pic], along with its graph. Use this information to derive the equation of the function [pic]. Show how you derived the equation. Do this work in the space provided on the Charts, Data and Computations sheet.
11. Lastly, on the back of the Charts, Data and Computations sheet, use the limit definition of the derivative to find [pic]. Compare your result with the equation of the slope function.
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