INTRODUCTION TO DERIVE - by M. Yahdi

Size: px

Start display at page:

Download "INTRODUCTION TO DERIVE - by M. Yahdi"

Alexis Nicholson
6 years ago
Views:

1 Math 111/112-Calculus I & II- Ursinus College INTRODUCTION TO DERIVE - by M. Yahdi This is a tutorial to introduce main commands of the Computer Algebra System DERIVE. You should do (outside of class) all parts of this tutorial lab and have it with you for all the future labs. Please check the COMMON MISTAKES (section 3) and read the section 21 on Saving your DERIVE work from the Algebra Window; As a Microsoft WORD-FILE. Most of the lab-work takes part outside of class, so it should come as no surprise that you cannot finish a lab in class. You should finish this lab outside of class and let me know if you have any questions. For future labs, you must plan at least two long meetings with your lab partners in order to finish a lab and to write/edit the required lab-report. Contents 1. Starting Derive 2 2. Entering a mathematical expression 2 3. Common Mistakes 3 4. Simplifying and Approximating 3 5. Factoring 3 6. Substituting 3 7. Solving Equations Algebraically 3 8. Solving Equations Numerically 4 9. Plotting a Graph from a formula Plotting a Graph from a table of values Declaring a function Make list and table of values from a function Limits using Derive Derivative Using Derive Antiderivative of Indefinite Integral using Derive Definite Integral using Derive Including text on the Algebra Window Including Annotations on the Plot Window Printing your Work Saving your DERIVE work from the Algebra Window Saving your DERIVE work from the Plot Window Other buttons Help 8 1

2 2 1. Starting Derive (1) It can be done two different ways: Look for the Derive for Windows icon : on the computer desktop and double click it. An Algebra window will appear. The other way is to go to START/Programs/DERIVE for Windows/ DERIVE for Windows. An Algebra window will appear. (2) The Algebra window is used to enter all expressions and to display the calculations. (3) For the GRAPHS, you will need to open a Plot window: click on the graph button and the plot window will appear. (4) To view the algebra and plot window together, choose Window/Tile Vertically from the menu. This will split the screen into 2 windows. 2. Entering a mathematical expression Examples: Follow the procedure below to type in all of the expressions in the Table below. Focus on the importance of the parentheses and use the list of symbols displayed on the entry form. (1) Select the Algebra Window (just click on). (2) Click the picture of a pencil button (called Author): the Author Expression window, also called Entry Form, will appear. Type the mathematical expression (from the table below), then press the Enter key or else click OK. DERIVE will place the corresponding output in the Algebra window. It will not perform the calculations yet! You enter (3+5)/2 You get / xˆ2 x 2 5ˆ2x 5 2 x 5ˆ(2x) 5 (2x) sin x sin x sin 3x sin (3)x sin(3x) sin (3x) asin(x) asin(x); which means arcsin(x) atan(x) atan(x); which means arctan(x) You enter You get asin x asin(x) (means arcsin x) (5 xˆ2 - x)/(4xˆ3-7) 5x 2 x 4x 3 7 (4+x)ˆ(1/2) (4+x) or sqrt(4 + x) (4 + x) 1/2 4 + x ê (from the list) or [Ctrl]& e ê (for the exponential e) ê ˆ 5 ê 5 pi π (or choose it from the list) ln x LN(x) (means logarithm) abs(5+3x) 5 + 3x

3 3 3. Common Mistakes The standard order of operations is Exponentiation before Multiplication and Division, and then Addition and Substraction. To avoid mistakes, use parentheses to be sure that the operations are performed in the desired order. For example (3 + 5)/4 is not the same as 3 + 5/4. Be sure to use round parentheses rather than square brackets [ ] or curly braces {}, which have other meanings in DERIVE. Be aware that the number e for the exponential function appears with a hat in DERIVE: ê. It can be accessed from the list on the entry form or by entering [Ctrl]+e. For DERIVE, e alone means just a variable like x. 4. Simplifying and Approximating Examples: Read the procedure below, then apply it to the two following examples (2 4 + ( π ) 32)/6 and sin 4 (1) If the expression is not already in the Algebra Window, then you have to enter it like above by Authoring the expression on entry form and clicking on OK. (2) Highlight the expression from the Algebra Window, and click the = button (called simplify ). The = button is used to perform the operations or calculations of entered expressions (3) Derive uses exact calculation. To see a decimal approximation, you click the button. (4) The number of decimal places used can be changed to any number: choose Simlify/Approximate from the menu and enter the number of decimals. (5) An alternative method is to click Simplify instead of OK on the entry form. Try it for the same examples. But by doing this you will see the answer without the original expression which is not convenient if you want to keep track of the all operations! I recommend using the first method for your reports to keep track of your work and to avoid typing mistakes. 5. Factoring Example: Apply the procedure below to the expression x 4 9x 2 4x (1) Choose Simplify/Factor from the menu, enter the expression and click Factor. (2) If you want to keep track of your work I recommend that you click on OK instead of Factor, then click on =. Try it for the same example above. 6. Substituting Example: To evaluate x 4 9x 2 4x + 12 for x = 2, x = 3 and x = 4, follow the steps below. (1) Author the expression. (2) Highlight it, click the SUB button and then fill in the substitution value, then click OK. (3) Finally, to perform the calculation click on = and on if needed. 7. Solving Equations Algebraically Examples:Apply the procedure below to each of the following examples x 2 x 2 = 0 and 5x = 3x

4 4 (1) Author the equation. (2) Highlight the equation and click the = button (called Solve). Then click = to perform the factorization. (3) An alternative method is to choose Solve/Algebraically from the menu. Try it! 8. Solving Equations Numerically Example: Follow the steps below to solve x 4 = e x (1) Derive cannot solve algebraically this equation (try it). (2) But Derive can give an approximation of the solution using numerical techniques. (3) To do this, choose Solve/Numerically from the menu. (4) Fill in the equation. (If the equation is already in the Algebra Window, just Highlight it, then choose Solve/Numerically.) (5) You will be asked to choose an appropriate interval that you believe contains the desired solution, which is at the intersections of the two curves; (see next paragraph, in the meantime, try the interval [0, 2]). You may need to repeat this to cover all possible intersections between the two curves. (6) Click OK, then = or if needed. 9. Plotting a Graph from a formula Examples: Graph each of the following function: f(x) = x 4, g(x) = e x andh(x) = sin x. (1) Author the expression x 4 (without the f(x)). (2) Then click the button. The plot window will be selected. Then click a second time the button (note that its position is different in the plot window menu!). (3) Do the same for the two other functions g(x) and h(x). (4) Take a look at the set of buttons on the plot window; you will see several buttons for zooming horizontally in, zooming horizontally out, zooming vertically out, etc..., use them to familiarize with. 10. Plotting a Graph from a table of values Example: x y (1) Click the [...] button (picture of a matrix). (2) Select the number of rows and columns. (3) Fill in the values in the form (without x and y) and click OK. (4) Highlight the data and click the button twice (note that its position changes after the first click). (5) You will see a set of points that you can choose to connect them with line segments (if it is not already done) so that a graph will appear. You do this by choosing Options/Points from the menu of the plot window, and then check Yes, then click on button again.

5 5 (6) Have a look at the Options from the menu, there are lots of other interesting features). 11. Declaring a function Example:Apply the procedure below to the functions: f(x) = x 4, g(x) = e x andh(x) = sin x. (1) This helps to enter a name of a function in DERIVE to avoid retyping the expression of the function, to simplify the calculations and to improve the presentation of your work. (2) Choose Declare/Function from the menu. A Declare Function Definition dialogue box will appear. (3) Fill in the function name on Name and Arguments : for example f(x). (4) Fill in its expression on Definition : for example x 4. (5) DERIVE will then show F(x) := x 4 for example. (6) REMARK that the :=sign is used to declare a function, and the =sign is used for an equation, please do not confuse them. You can also define a function by using :=in the Author dialogue box instead of the Declare Function Definition dialogue box; e.g., f(x):=x^4. (7) Whenever a function is Declared on DERIVE, substituting values of x can be done directly with the name of the function without using the formula of the function. For example, define F(x) := x 4, then Author f(2) and click =. Repeat it for f(5.5), etc... (8) Try it for the functions g(x) and h(x) above. 12. Make list and table of values from a function This is a good technique for studying patterns in data and limits in particular. Example1: Suppose we want to have f(x) = x 2 for many values of x where x=1,2,3,4,5.(note that x follows a specific pattern!) (1) As a LIST of outputs only: Choose Calculus/Vector from the menu and fill in the form starting with the function. If f(x) := x 2 was previously Declared, enter f(x), otherwise you have to enter its expression x 2. (2) As a TABLE of inputs and outputs: The same as above with only one modification; replace f(x) (or x 2 ) with [x, f(x)] (or with [x, x 2 ]). It is important to note the square brackets!. You will get a table with the first column containing the x-values and the second column containing the f(x)-values. (3) Example 2: Make a 2-columns table with values of x and f(x) = x 2 for x = 1, 0.8, 0.6, 0.4, 0.2, 0. (4) Example 3: Make a 2-columns table with the values of x = 0.1, 0.01, 0.001, in the first columns and the corresponding values g(x) = sin x x in the second columns. Because the step size between the values of x is not the same, the previous method is not adequate. One good method in this case is: Declare the function g(x) := sin x x. Click the [...] button (picture of a matrix), select the number of rows (4) and columns (2). Fill in the values of x in the first column and the corresponding values g(x) in the second column, i.e. g(0.1), g(0.01), g(0.001), g(0.0001). Click OK, then click = and click if needed.

6 6 Click the lim button and fill in the form. 13. Limits using Derive Example: lim x 0 g(x). Consider the functions f(x) = sin x x and g(x) = 1 x. Calculate lim f(x), x 0 lim g(x) and x 0 + Notation: 0 means at the left of 0, and 0 + means at the right of Derivative Using Derive On Derive, DECLARE the function f(x) := (6x x 2 ) arctan x. In this problem, you will find f (2) using several methods. click button, a Calculus Differentiate Window will open, type in the Function f(x) (without its formula), select the Variable x and the Order 1, click OK then =. The result is the derivative f (x) To find f (2), highlight the derivative and click on SUB button to substitute x by 2 then click = or The result is the value of f (2). 15. Antiderivative of Indefinite Integral using Derive Follow the steps below to find the Indefinite integral where f(x) = 3x (1) Click on the button with icon, called calculate integral. (2) Type the expression of the function (for our example, type 3x) (3) Select the variable (x in this case) (4) Choose Indefinite in the integral box down to the left. (5) Enter the value of the constant if it is given or just enter c. (6) Click on OK. (7) The integral 3 x dx will appear highlighted on the Algebra Window. (8) Click on the = button to get the answer. f(x) dx, i.e. the antiderivative of f(x), 16. Definite Integral using Derive Follow the steps below to find the Definite integral 4 1 f(x) dx, where f(x) = 3x (1) Click on the button with icon. (2) Type the expression of the function (for our example, enter 3x) (3) Select the variable (x in this case) (4) Choose Definite in the integral box down to the left. (5) Then choose the Upper Limit 4 (in our example), and the Lower Limit 1 (in our example)

7 7 (6) Click on OK. (7) The integral 4 1 3x dx will appear highlighted on the Algebra Window. (8) Then click on the button = for the answer, then for an approximation. 17. Including text on the Algebra Window To type a text on the Derive Algebra Window, you click on the picture of a pencil button and you should start typing your sentence with a quotation mark. Example: Type This is the first lab with and without the quotation marks to see the difference. 18. Including Annotations on the Plot Window Do the following to type a text or expression on the Plot window, for example the names or the expressions of the functions near their corresponding graphs, the problem number, or any comment on the graphs: Click the position on the plot window where you want the expression to appear. (a cross + will appear at the selected position.) Select Edit/Create Annotation... from the menu (or click on the picture of a pencil button). A form will appear. Type the expression and click OK To remove this expression, select Edit/Delete Annotation... appear then click on Yes. from the menu, a small form will 19. Printing your Work Just select print from the menu. But if you do not want or cannot print it yet, you should save your work as described below. 20. Saving your DERIVE work from the Algebra Window As a Microsoft WORD-FILE: This is alternative way that is more practical for typing and presenting your lab-report. First, in the Word file, select Format/Font and set the font to be DfW Printer, then simply Copy the work on the DERIVE Algebra window, and Paste it on the Word file. The trick of changing the font is necessary otherwise it will not look nice. Beside the fact that you are more familiar with Word files, this method allows you to better present your work since you can type in easily directly on the Word file (and even between the DERIVE Inputs) any explanations or comments, and insert any graphs from the Drive Plot Window (see below). This method is required for your lab report. As a DERIVE-FILE: Select File/Save or Save As from the menu. A form will appear. Select a location where you want it to be saved and give a name to the file. You can save it in a Disc, a shared file or it as an Attachment to you and the members of your group so you can access the file from any other computer to finish the work later. How to open a saved DERIVE-FILE? A DERIVE file has a name that ends with.mth. You CANNOT open it by just clicking on, you need first to open DERIVE, select File/Open... from the menu, then finally select the file and click on Open.

8 8 21. Saving your DERIVE work from the Plot Window As a DERIVE-FILE: You CANNOT save the curves on the Plot Window! You should then immediately print any work or save it as a Microsoft Word file. As a Microsoft WORD-FILE: Like in the previous section, this method is required for your lab report because it saves pictures of the curves and improves the presentation of your report. Just Copy the work on the Plot Window, and Paste it on a Microsoft Word file. You may need to copy just a part of the Plot Window, to do that you select Edit/Mark and Copy... from the menu then, using the mouse, select a part in the plot window that you want to copy. 22. Other buttons Several buttons are on the top of the Algebra window and the Plot window as well as other features in the menu bar. We will learn more all along the semester. I recommend that you try some of them such as: In the algebra window, Remove to remove expressions, Renumber to renumber all inputs in the algebra window, etc. In the plot window, see the boutons for zooming horizontally in, zooming horizontally out, zooming vertically out, an others boutons such as Zoom vertical in, Zoom both in, Zoom both out, Center on cross, Center on origin, etc. 23. Help Select Help/Index or Help/Contents from the menu of the Derive Window. Then type a keyword from the topis you are looking for, and/or select topic from the list.

Lab#1: INTRODUCTION TO DERIVE

Math 111-Calculus I- Fall 2004 - Dr. Yahdi Lab#1: INTRODUCTION TO DERIVE This is a tutorial to learn some commands of the Computer Algebra System DERIVE. Chapter 1 of the Online Calclab-book (see my webpage)