M CC 160 Calculus for Physical Scientists I



MATH 160 Calculus for Physical Scientists I Name:

Spring, 2008

Calculator Laboratory Section:

Date due:

Calculator:

The Differentiation Theorems Graphically

Connecting differentiation formulas and tangent lines.

The investigations in this lab require a calculator that can produce traceable graphs and generate tables of values of elementary functions. While many makes and models of calculators have these capabilities, the authors used Texas Instrument calculators as they wrote this lab. The lab does not include comprehensive 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 . Search for manuals for other makes and models of calculators at the manufacturers’ web sites. The Texas Instrument web site will also point you to tutorials for various models of TI calculators.

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 thoroughly.

• Mathematical and computational accuracy.

• Clarity and readability. Tables and graphs must be accurate and presented in a clear, readable format. Explanations must be written in complete sentences with correct spelling, capitalization, and punctuation. Handwriting must be legible.

Space for writing your report is provided within the lab. However, if you wish to word process your lab report, you may ask your instructor to 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.

Overview

From previous work, we know the derivative of a function y = f(x) at a point x = c can be interpreted graphically as the slope of the line tangent to the graph of y = f(x) at the point ( c, f(c) ). We also have sound intuitive understanding of the tangent line – it is the line the matches the position and the direction/alignment of the graph of y = f(x) at the point of tangency. So, knowing the tangent line is equivalent to knowing the derivative. In fact, in more advanced settings the derivative is defined in terms of the tangent line or tangent plane. This connection between derivatives and tangent lines can be used to discover the formulas for the derivatives of sums, products, reciprocals, and compositions of differentiable functions. Here’s the idea.

Suppose the two functions y = f(x) and y = g(x) are both differentiable at x = c. From their derivatives f((c) and g((c) we can write equations for the lines tangent to the graphs of the two functions y = f(x) and y = g(x) at

x = c. We can add the equations for these two tangent lines to get an equation for the line tangent to the sum function s(x) = f(x) + g(x) at x = c. Then the derivative of the sum of y = f(x) and y = g(x) at the point x = c is just the slope of this tangent line. An equation for the line tangent to the product function p(x) = f(x) g(x) at the point x = c can be found by multiplying the equations for these two tangent lines and using a little algebraic ingenuity. The slope of this tangent line is, of course, the derivative of the product. The differentiation formulas for reciprocals and compositions can also be discovered in a similar way.

In this lab, you will carry out the idea just described to find the formulas for the derivatives of sums and products of differentiable function. The idea works with any differentiable functions, we will use specific functions and the point c = 0 to make it easier to see how the idea works.

Preliminaries

P.1 Graph the function f(x) = sin(x) + 0.5x – 1 in the decimal window. Be sure your calculator is in radian mode.

(On the TI-83® and TI-84® calculators , enter this function as Y1, then press [Zoom] and choose 4:ZDecimal. )

(a) Describe what you see in the graph that tells you the function f(x) is differentiable at x = 0.

Use the Draw Tangent command on your calculator to add the tangent line at (0 , f(0)) to the graph and to find an equation for the tangent line at (0, f(0)). (On the TI-83® and TI-84® calculators, press [2nd] [draw] [5]. Position the cursor at the point on the graph of y = f(x) where x = 0. Then press [enter]. The tangent line will be added to the graph and an equation for the tangent line displayed at the bottom of the screen.)

(b) The tangent line is the graph of a function. Denote this function by t1(x) (to indicate “tangent to the first function”). Enter the function t1(x) as Y2. (Round the coefficient of x to one decimal place.)

An expression/equation for this function is t1(x) = _________________________ .

(c) Sketch the graphs of the functions y = f(x)

and y = t1(x) on the coordinate grid given.

(Draw a very accurate graph. Use a ruler to mark

units on the axes. Use a sharp pencil and draw

the graph as a thin, but clearly visible, line.)

P.2 Turn off Y1 and Y2 by moving the cursor over the equal sign and pressing [enter] so the equal sign is not highlighted. Enter the function g(x) = 2 – 0.5x – x2 as Y3. Press [graph].

(a) Describe what you see in the graph that tells you the function g(x) is differentiable at x = 0.

Use the Draw Tangent command on your calculator to add the tangent line at (0 , g(0)) to the graph and to find an equation for the tangent line at (0, g(0)). (On the TI-83® and TI-84® calculators, press [2nd] [draw] [5]. Then move the cursor to the point on the graph of y = g(x) where x = 0 and press [enter]. The tangent line will be added to the graph and an equation for the tangent line displayed at the bottom of the screen.)

(b) The tangent line is the graph of a function. Denote this function by t2(x) (to indicate “tangent to the second function”). Enter the function t2(x) as Y4. (Round the coefficient of x to one decimal place.)

An expression/equation for this function is t2(x) = _________________________ .

(c) Sketch the graphs of the functions y = g(x)

and y = t2(x) on the coordinate grid given.

(Use a sharp pencil. Use a ruler to mark units

on the axes. Draw a very accurate graph.)

Investigation I. The Differentiation Formula for Sums

I.1 Keep the functions you entered earlier in Y1, Y2, Y3, and Y4, but turn them all off.

Graph the sum function s(x) = f(x) + g(x) in the decimal window. Since the expressions sin(x) + 0.5 x – 1 and 2 – 0.5 x – x2 for f(x) and g(x) are already entered as Y1 and Y3, you don’t need to enter them again to enter their sum. Simple enter s(x) in Y5 as Y1(X) + Y3 (X). Then press [graph]. (Or, if you’ve changed the window, press [Zoom] and choose 4:ZDecimal.)

(a) Describe what you see in the graph that tells you the sum function s(x) = f(x) + g(x) is

differentiable at x = 0.

Use your calculator as you did in P.1 to add the line tangent to the graph of s(x) at (0 , s(0)) and to find an equation for the tangent line at (0, s(0)).

(b) The tangent line is the graph of a function. Denote this function by t3(x) (to indicate “tangent to the third function”). Enter the function t3(x) as Y6. (Round the coefficient of x to one decimal place.) An expression/equation for this function is t3(x) = __________________________

(c) Sketch the graphs of the functions y = s(x)

and y = t3(x) on the coordinate grid given.

(Use a sharp pencil. Use a ruler to mark units

on the axes. Draw a very accurate graph.)

I.2 Enter the sum t1(x) + t2(x) of the lines tangent to the graphs of f(x) and g(x) at the point 0 as Y7. Since the expressions for t1(x) and t2(x) are already entered as Y2 and Y4, you don’t need to enter them again to enter their sum. Simple enter Y7 = Y2(X) + Y4(X).

Set the graphing mode to “animated” so as the graph of Y7 is generated the leading point on the graph is traced by a moving ball. (To change the graphing mode on a TI-83® or TI-84®, move the cursor to the left of Y7 and press [enter] repeatedly until a small oval appears.)

Press [graph]. What do you observe as the graph is generated?

Algebraically simplify the sum t1(x) + t2(x). t1(x) + t2(x) = _____________________________

Show the algebraic work below.

How is the slope of the tangent line t3(x) related to the slopes of the tangent lines t1(x) and t2(x) ?

I.3 Explain how the graphs you just created show that, at least for the specific functions f(x) and g(x) and the specific point x = 0, the derivative of the sum is the sum of the derivatives.

I.4 We developed the formula for the derivative of a sum using the specific functions f(x) = sin(x) + 0.5x – 1 and g(x) = 2 – 0.5x – x2 and the specific point c = 0. Explain why one would expect the reasoning in Investigation I and the conclusion reached in I.3 to be valid for all pairs of differentiable functions at all points where both functions are differentiable.

Investigation II. The Differentiation Formula for Products

II.1 Keep the functions you entered earlier in Y1, Y2, Y3, Y4, Y5, and Y7, but turn them off. (Move the cursor over the equal sign and press [enter] so the equal sign is not highlighted.) Delete Y6.

Graph the product function p(x) = f(x)(g(x). Since the expressions for f(x) and g(x) are already entered as Y1 and Y3, and their sum is already entered as Y5, to graph the product, simply edit Y5. On the Y= screen, move the cursor over the “+” sign and type “(” to get Y5 = Y1(X)* Y3(X).

To get a good viewing window, start with the decimal window, then subtract the same quantity from Ymin and Ymax to get Ymin = -5.5. (On a TI-83® and TI-84® calculator, subtract 2.4 to get Ymin = -5.5 and Ymax = .7. If you have some other calculator, will need to subtract a different number.)

(a) Describe what you see in the graph that tells you the product function p(x) = f(x)(g(x) is

differentiable at x = 0.

Use your calculator as you did in P.1 to add the tangent line at (0 , p(0)) to the graph and to find an equation for the tangent line at (0, p(0)).

(b) The tangent line is the graph of a function. Denote this function by t4(x) (to indicate “tangent to the fourth function”). Enter the function t4(x) as Y6. (Round the coefficient of x to one decimal place.)

An expression/equation for this function is t4(x) = _______________________ .

(c) Sketch the graphs of the functions y = p(x)

and y = t4(x) on the coordinate grid given.

(Use a sharp pencil. Use a ruler to mark units

on the axes. Draw a very accurate graph.)

II.2 Enter the product t1(x)(t2(x) of the lines tangent to the graphs of f(x) and g(x) at the point 0 as Y7. Since the expressions for t1(x) and t2(x) are already entered as Y2 and Y4 and their sum is entered as Y7, you don’t need to enter them again to enter their product. Simply edit Y7 by replacing “+” with “(” to get

Y7 = Y2(X) ( Y4(X). Set the graphing mode to “animated” so as the graph of Y7 is generated the leading point on the graph is traced by a moving ball. Press [graph].

(Change the window if you want to see more of the graph.)

What single word accurately describes the shape of the graph of t1(x)(t2(x)? _________________________

II.3 (a) Algebraically simplify the product t1(x)(t2(x) by expanding the product and collecting like terms.

Show details of the simplification.

t1(x) (t2(x) = _____________________________________

(b) Explain how to get the expression t4(x) for the line tangent to the product function p(x) = f(x)(g(x) at

x = 0 from the expression for the product t1(x)(t2(x) of the tangent lines t1(x) and t2(x) .

II.4 (a) Use the expression t1(x) = f(0) + f'(0) x and t2(x) = g(0) + g'(0) x for t1(x) and t2(x) to find an expression for the product t1(x)(t2(x) in terms of f(0), g(0), f'(0), and g'(0).

(b) Use the result from II.4(a) with what you observed in II.3 to write an expression for the slope of the line tangent to the product function p(x) = f(x)(g(x) at x = 0 in terms of f(0), g(0), f'(0), and g'(0).

II.5 Explain how the graphs and calculations you just completed show that, at least for the specific functions f(x) and g(x) and the specific point x = 0, the derivative of the product p(x) = f(x)(g(x) at x = 0 is given by the familiar formula.

II.6 We developed the formula for the derivative of a product using the specific functions f(x) = sin(x) + 0.5x – 1 and g(x) = 2 – 0.5x – x2 and the specific point c = 0. Explain why one would expect the reasoning in Investigation II and the conclusion reached in II.5 to be valid for all pairs of differentiable functions at all points where both functions are differentiable.

Investigation III. Combining Differentiation Formulas

III.1 (a) Explain how to see from its graph that the derivative of a constant function is zero.

(b) Use the result of part (a) and differentiation formula for products to develop the differentiation formula for a constant times a differentiable function. Show details of your development.

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