M CC 160 Calculus for Physical Scientists I
MATH 160 Calculus for Physical Scientists I Name:
Calculator Investigation
Spring, 2008 Section:
Date due:
Calculator:
Understanding the Derivative
What does it mean for a function to have a derivative at a point?
The investigations in this lab require a calculator that can produce traceable graphs. While many makes and models of calculators have these capabilities, the author used Texas Instrument calculators as he wrote this lab. The lab does not include instructions for using a calculator. Use the manual for your calculator to learn how to perform the tasks in this lab efficiently and accurately. Manuals for Texas Instrument calculators can be read from the Texas Instrument web site. Go to . You can find instructions for many different calculators at . You might also search for manuals for other calculators on the manufacturers’ web sites.
The calculator skills you develop doing this lab will serve you well throughout this and other courses. If you encounter difficulties, take your calculator and manual to your instructor and discuss the problem with him/her. Classmates may be able to help out, too
The following factors will be considered in scoring your lab report:
• Completeness. Each investigation must be completed entirely, recorded fully, and explained or interpreted clearly.
• Mathematical and computational accuracy.
• Clarity and readability. Explanations must be written in complete sentences with correct spelling, capitalization, and punctuation and with reasonable margins and spacing. Handwriting must be legible. Tables and graphs must be presented in a clear, readable format.
Space for writing your report is provided within the lab. However, if you wish to word process your lab report, your instructor will e-mail you a copy of this lab as an attached MS Word document. Submit your final lab report as a printed document.
PLEASE KEEP A COPY OF YOUR COMPLETED LAB REPORT.
You may need to refer to the work you did on this lab before it is graded and returned.
Investigation I. What is a Tangent Line?
I.1. Graph the function f(x) = [pic]x3 – 2x + 1 in the decimal window (on Texas Instruments calculators, press zoom ZDecimal). If your calculator doesn’t have this zoom feature, choose a square window (so the units are the same on the two axes) that includes most of the left “hump” of the graph and all the graph between x = 0 and x = 4.
Most graphing calculators will add a tangent line to the graph of a function. Add the tangent line at (3, f(3)) to the graph of f(x) = [pic]x3 – 2x + 1 on your calculator screen. On a TI-83® or TI-84®, display the graph and then press [2nd] draw. Scroll to 5:Tangent and press [enter]. Move the cursor along the graph to the point (3, f(3)) (or press [3] )and then press [enter] again. The calculator will draw the line tangent to the graph of the function at the point (3, f(3)) and display an equation for the tangent line at the bottom of the screen.
If you use some other make or model of calculator, refer to the manual for your calculator to learn how to get the calculator to draw a tangent line. If your calculator graphs the tangent line but doesn’t display an equation for the tangent line, it probably displays the slope of the tangent line. (On the TI-86®, use the dy/dx command from the [graph], [math] menu get the slope of the tangent line.)
Write the equation for the tangent line as displayed on your calculator screen below. If your calculator displays the slope, but not the equation for the tangent line, write the slope of the tangent line and “equation not displayed” as your answer.
Equation for tangent line (or slope, see above) as displayed by your calculator:
y = _______________________________________________________
Write the coefficients to as many decimal places as displayed on the calculator screen.
I.2. The slope of the tangent line at (3, f(3)) can be found by differentiating f(x) = [pic]x3 – 2x + 1 using differentiation formulas. Then the equation for the tangent line at ( 3, f(3) ) can easily be found.
(i) f′(x) = _____________________________________
(ii) Slope of tangent line at (3, f(3)): m = f′(3) = _____________________________________
(iii) Equation for tangent line in point-slope form: ______________________________________
Add the graph of the equation for the tangent line you found to the graph you created in I.1. If the equation you found for the tangent line is correct, its graph will coincide with the line you created using the “Draw Tangent” command. However, if these lines coincide you won’t be able to see the second line being graphed. You won’t be able to tell whether you have the correct equation for the tangent line or an equation so wrong that its graph lies entirely outside the window. To avoid this problem, set the graph style to Path. In this style a circle cursor traces the leading edge of the graph as it is drawn on the screen. On the TI-83® and TI-84® calculators set the style on the Y= screen by moving the cursor to the left of the expression and pressing [enter] repeatedly. On the TI-86® calculator, set the style from the graph menu. For other makes and models, refer to the manual. If the graph of the tangent line you computed doesn’t coincide with the tangent line drawn in #I.1, find and correct your mistake.
I.3. The tangent line matches the position and the direction/alignment of the graph of y = f(x) at the point (3, f(3)). You can see this graphically by zooming in on the point of tangency. Set the zoom factors in both the x and y directions to 2. (Refer the calculator manual to learn how to set the zoom factors.) Now, zoom in on the point (3, f(3)) repeatedly (at least five times). Describe how the graphs of the original function
f(x) = [pic]x3 – 2x + 1 and the tangent line compare in the immediate vicinity of the point (3, f(3)).
(trace the graphs to see that they aren’t identical. “Describe” means tell us in writing what you see.)
I.4. Mathematicians sometimes say that a function which is differentiable at a point is “locally linear” at that point. Why would they say this? Why “locally”? Why “linear” when the graph of the function f(x) = [pic]x3 – 2x + 1 obviously bends so no part of it is a straight line? (Write about it!)
Investigation II. Secant Lines, Tangent Lines, and Derivatives
Suppose we want to know the slope of the line tangent to the graph of y = f(x) at a point (a, f(a)). The slopes of secant lines drawn between (a, f(a)) and nearby points on the graph are very close to the slope of the tangent line. In fact, the slopes of the secant lines can be made to approximate slope of the tangent line as closely as one likes (though not necessarily be exactly the same as the slope of the tangent line) by drawing the secant lines between (a, f(a)) and points on the graph that are close enough to (a, f(a)). Usually these secant lines are hard to draw. In this investigation, we will use the calculator to show how close these secant lines come to the tangent line at a particular point on the graph of f(x) = [pic]x3 – 2x + 1.
II.1. Find the exact value (as a fraction or as a decimal with no rounding) of the slope of secant line through the points (3, f(3)) and (3.5, f(3.5)) on the graph of f(x) = [pic]x3 – 2x + 1. Write an equation for this secant line in point-slope form using the point (3, f(3)).
Graph the function and the secant line you just found in the decimal window. (Use zoom ZDecimal again.) zoom in on the point (3, f(3)) repeatedly until you can see clearly that this secant line matches the position but does not quite match the direction/alignment of the graph of f(x) = [pic]x3 – 2x + 1 at the point (3, f(3)).
(You are not expected to include this graph in your lab report.)
II.2. Find the exact values of the slopes of the secant lines through the point (3, f(3)) and each of the points
(3.5, f(3.5)), (3.25, f(3.25)), (3.125, f(3.125)), and (3.1, f(3.1)). You will find it more convenient to have the slopes as fractions (rather than exact decimals). Most Texas Instruments calculators have a “convert to fraction” command that makes it easy to do this. (On the TI-83® and TI-84®, press math and choose ►Frac.)
Record the exact coordinates of the points used to find the secant lines, the exact slopes of the secant lines, and the exact equations for the secant line in point-slope form using the point (3, f(3)) = (3, [pic]) .
You found the information for the first column of the table in problem #II.1 above.
[pic]
As in #II.1, graph each secant line and then zoom in on the point (3, f(3)) repeatedly until you can see clearly that this secant line matches the position but does not quite match the direction/alignment of the graph of
f(x) = [pic]x3 – 2x + 1 at (3, f(3)). Graph these secant lines one at a time so the screen doesn’t get too cluttered.
II.3. Repeat #II.2 with secant lines through (3, f(3)) and each of the points (2.5, f(2.5)), (2.75, f(2.75)),
(2.875, f(2.875)) and (2.9, f(2.9)).
[pic]
As in #II.2, graph each secant line and then zoom in on the point (3, f(3)) repeatedly until you can see clearly that this secant line matches the position but does not quite match the direction/alignment of the graph of
f(x) = [pic]x3 – 2x + 1 at (3, f(3)). Graph these secant lines one at a time so the screen doesn’t get too cluttered.
II.4. The tangent line at (3, f(3)) and the secant line between (3, f(3)) and (3.1, f(3.1) are almost indistinguishable on the calculator screen. However, they clearly aren’t the same line. Explain how to see from a graph that no secant line (not just the ones investigated above) between (3, f(3)) and a nearby point (3 + h, f(3 + h)) on the graph of f(x) = [pic]x3 – 2x + 1 coincides exactly with the tangent line at (3, f(3)).
Investigation II (continued)
II.5. In this situation, none of the secant lines coincide perfectly with the tangent line. However, the slope of the tangent line is certainly related to the slopes of the secant lines.
(a) Write a mathematical equation that expresses how the slope of the tangent line is related to the slopes of the secant lines.
(b) Explain in common, non-technical language what it means to say “the slope of the tangent line is the limit of the slopes of secant lines”. (In this case, the slopes of the secant lines do get closer and closer to the slope of the tangent line. But that’s not what limit means and not the best description of how the slopes of secant and tangent lines are related in general.)
II.6. The derivative of the function y = f(x) at x = 3 is defined by the equation f′(3) = [pic].
(a) Sketch the graph of f(x) = [pic]x3 – 2x + 1.
(b) On your drawing, illustrate and label the
points (3, f(3)) and (3 + h, f(3 + h))
and each of the quantities h,
f(3+h) – f(3), [pic] and f′(3) . _
(c) Explain the meaning of the symbol [pic] 1 _
and explain why it is needed to get f′(3).
-2 -1 0 1 2 3 4
_
_
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