Learning Scientific Notebook

Transcription

1 Learning Scientific Notebook Reading and Writing in Scientific Notebook Viewing On-Screen: Zoom Factor, Invisibles Math and Text Producing the Math Symbols from Toolbars and from the Keyboard Using the Drag and Drop Features Creating a Document, Item Tags, Section Tags, Text Tags Using Fragments Creating Piecewise Defined Functions Creating and Using Math Names Using Automatic Recognition Placing Mathematical Documents in Websites Customizing your Installation Working with Long Documents Inserting Markers and Hyperlinks Typing Math Names Examples of Writing 1

2 Solving an Equation x 2 2x 1 3x 1 2x 1 x 2 3 Adding Fractions x 2 2x 1 x 3 3x 1 2x 1 x 2 6x 2 2x 1 x 3 x 2 Solving an Equation log 7 6x 1 log 7 x 2 1 Prove the Identity cos tan 1 sin csc 1 Prove the Identity 1 cos 1 cos 1 cos sin Working with an Expression Suppose that we have previously obtained the identities cos cos cos sin sin sin sin cos sin cos and that we want to expand tan. We proceed as follows: Proving the Product Rule for Differentiation 2

3 Using L Hôpital s rule Integration by Parts lim x 0 sinx tanx x 3 0 /2 cos 7 xcosxdx Integration by Parts 1 2 x 3 logxdx Evaluating a Limit We shall prove that as n. 2n 2 3n 4 3n 2 5n Computing with Scientific Notebook Expanding an Expression and then Factorizing 5x 7y 100 Solving an Easy Equation Solving a Hard Equation 3x 2 7x x 3 6x 1 0 Factorizing an Integer factor Greatest Common Divisor of a Pair of Integers gcd 2346,9724 Greatest Common Divisor of a Pair of Polynomials gcd x 2 9,x 2 x 6 Least Common Multiple of a Pair of Integers lcm 2346,9724 3

4 Least Common Multiple of a Pair of Polynomials gcd x 2 9,x 2 x 6 Inverting A Matrix and Finding Eigenvalues and Eigenvectors Finding a Derivative d dx x2 3x 2 5 sin 3 1 x 4 Finding an Integral 0 3 x 2 9 x 2 3/2 dx assume real Solving for Two Unknowns Find where the line meets the curve 3x 2y 1 0 2x 2 3xy y 2 x 3y 2 0 4

5 Standard Deviation 2,3,3,5,6,7,1, 11,9, 4, 1,7,2,3,1,0 The Rational Roots Theorem We shall demonstrate that none of the roots of the equation Can be rational numbers. First we define for every number x. 8x 3 2x 2 6x 15 0 f x 8x 3 2x 2 6x 15 Annuities An investment of $P is made monthly into an account that pays interest at the rate of r percent compounded monthly. The problem is to find the rate of interest that will cause the amount of money in the account after 31 payments to be double the amount invested. Working with Scientific Units An Area Problem with Mixed Units One length of side of a rectangle is miles and the length of the other side is meters. Find the area of the rectangle in acres. 5

6 Units of Energy How many calories in 47 foot pounds? xcal 47ft lbf Motion of a Car with a Horrible Mix of Units A car with a mass of killograms has an engine with a power of 6349 BTU per minute. How long does it take for the car to reach a speed of 86.3 miles per hour, starting from rest? kg dv dt mi /h/s V Btu/min Vmi /h Testing a Series for Convergence Test each the following series for convergence 2n! 4 n n! 2 2n! 4 n n! 2 2n! 4 n n! Using Scientific Notebook to Apply Newton s Method. To illustrate the method we shall look for some solutions of the equation e x cos2x x 2 0 that lie in the interval 4,4. We supply the definition f x e x cos2x x 2 to Scientific Notebook and then we supply the definition g x x f x f x. Start with 1, with 1, with 0 and with??? Draw the graph. 6

7 Curvature Exercise Before We Begin assume real Now We Begin Work out s approximately. s Define F t t 2 cost,t 2 sint,t s t 0 t F u du T t 1 s t F t Ask Scientific Notebook to evaluate T t using the simplify button. T t Redefine T t as the latter expression. Define k t 1 s t T t Evaluate and simplify. k t Define Evaluate N t and simplify it. N t N t 1 T t T t Some Graphing Techniques with Scientific Notebook A Rectangular 2D Plot f x x 3 x 1 x 3 e x2 Define this function, draw its graph and drag in f x and f x. 7

8 An Implicit Plot x 3 y 3 3xy 0 Another Implicit Plot x 5 y 5 3x 2 y 0 Drawing a Circle x 3 2 y In this circle, x runs from 3 to 9 and y runs from 10 to 2. Another Rectangular 2D Plot Restrict the vertical variable to 6,6. f tan 2sin Finding a Volume Find the volume of the region that lies above the cone and below the paraboloid z x 2 y 2 z 6 x 2 y 2 Sketch the graphs with domain 2.3, ,2.3. 8

9 Möbius Band x 1 tsin cos2 y 1 tsin sin2 z tcos where.4 t.4 and 0. By pointing at the expression Drawing an Animated Musical Chord First draw 1 tsin cos2, 1 tsin sin2,tcos sinx sin 4 3 x sinx sin 4 3 t 300 x Make x run from 0 to 300. Plot thin with 600 sample points. For the animation, make t run from 1 to 1 A Knotted Tube In this subsection we draw a thickened form of the parametric graph Click on the expression x 10cost 2cos 5t 15sin 2t y 15cos 2t 10sint 2sin 5t z 10cos 3t 10cost 2cos 5t 15sin 2t, 15cos 2t 10sint 2sin 5t,10cos 3t 9

10 andclickonplot 3D and then Tube. If you allow t to vary from 0 to 6.3 and choose the radius of the tube to be 2. MacLaurin Polynomials of A Function Try a polynomial of degree 30. f x xsin x 2 3x 1 Fitting a Curve to Data This question takes you through the process of analysing data that you might have obtained in a science laboratory. We assume that you took 12 readings at one second intervals, starting at time 0 and ending at time 11. The readings you obtained are as follows: 2.3, 3.1, 3.9, 5.3, 7.5, 10.1, 12.8, 15.2, 18.7, 24.4, 29.7, 35.4 Part 1 ClickontheInsert menu at the top of your screen andclickonmatrix. When the matrix dialogue box opens, set the number of rows at 12 and the number of columns at 2. Fill the first column with the numbers 012up to 11 and the second column with the data numbers shown above. 10

11 Part 2 Place your cursor inside or the immediate right of the matrix you obtained in Part 1 and then click on the Plot 2D graphing button. Make your graph light blue with medium thickness. Now highlight your matrix, place your cursor in the highlighted region, hold down the left mouse button, and drag into your sketch. You won t immediately see any difference. Click on the graph to give it handles and then click on the button on its right bottom corner to bring up the dialogue box of the graph. Go the the Items Plotted page and to the second item and set its Plot Style to Point and make the Point Markers appear as Boxes. Part 3 Place a new copy of your matrix here. You don t have to go to the trouble of retyping it. All you need to do is highlight the one you have above, copy it to the clipboard, move the cursor to here and paste. Place your cursor in or to the right of the matrix, open the Compute menu, point the mouse at Statistics and slide it to the right and on the option Fit Curve to Data.You will see the 11

12 following dialogue box: Set the curve as a polynomial of degree 5 with the dependent variable in the last column, as shown in the above figure. Click on OK. When the polynomial appears, highlight it. (Do not highlight the y ) and then click on the Plot 2D graphing button. Make the graph light blue with medium thickness. Set the domain interval as to run from 0 to 11. Then, as you did in Part 2, drag a copy of the matrix into the graph and set the second plot as a point plot with boxes for the point markers. 12

13 We begin by defining Midpoint Riemann Sums f x 3 1 x 2 a 0 b 1 x a,b,j,n a j b a n a a 0 b a n a 1 b Ý a n a 2 b Ý a n a b n b a n n M a,b,n j 1 b a n f x a,b,j 1,n x a,b,j,n 2 g n M a,b,n Introducing the Number e Numerical Motivation of the Number e We are looking for a value of a 0 for which Insert this expression as a formula. Then define some values of a and h. lim a h 1 h 0 h 1 h ah 1 a h 1. Graphical Motivation of the Number e Clear all definitions. Define a 2 Draw the graph y ax 1 x with domain 1,1. Define Try more values. a 13

14 Using the Copy as Picture Feature Creating Drawings Making Simple Drawings with SmartDraw C B a b 2 c 2 2bc cos c A b Using the Flood-Fill Tool in FotoFinish Pasting Graphs into a SmartDraw Drawing Graph of an Inequality Sketch the region x,y 4 x 8 and 3 4 x 3 y 9 8x x2 14