i MA 242 Assignment 1 Due February 9 1996 Name SSN Instructor Section Welcome to Maple assignment 1 for MA 242 1 Consider the function g x y sin x sin y e x 2 a Produce a contour plot for this function in the square region x y Show the command that you use and at one of the following prompts paste in a copy of the contour plot Remember to enter the with plots command to bring up the library containing contourplot with plots b Produce a three dimensional plot for this function over the same region Show the command that you use and paste in a copy of the plot which uses the boxed axes the hidden line style option and which is rotated so that Theta 135 and Phi 75 degrees You can change all the options in your picture using the buttons at the top of the plot window and can adjust the angle by clicking on the picture itself with the left mouse button Press the middle mouse button to replot the picture 2 Consider the function h x y ln x2 y2 6 a Produce a three dimensional plot for this function over the region inside the circle x2 y2 25 with the following features style is patch axes is framed there are 4 tickmarks along the x and y direction and 4 in the z direction Theta 40 and Phi 30 Paste the plot into this worksheet at one of the following prompts ii b One can obtain the level curves for z 1 for this surface with the command implicitplot ln abs x 2 y 2 6 1 x 9 9 y 9 9 grid 75 75 Explain why there are two such curves in this plot Either type or write by hand Explanation Explain why the following command gives two curves which are identical to the curves above NOTE In Maple the symbol E stands for the transcendental number e the base for the natural logrithm Either type or write by hand implicitplot x 2 y 2 6 E x 2 y 2 6 E x 9 9 y 9 9 grid 75 75 Explanation Paste the plot of these level curves into the worksheet at the following prompt 3 Consider the plane through the three points with vertices A 212 3875 613 B 674 312 413 and C 611 520 718 a Give a Maple V segment that results in the equation of the plane This means determine c m and n so that the points A B and C lie on the plane z c mx ny eqn1 613 c 212 m 3875 n eqn1 613 c 212m 3875n eqn2 eqn3 iii b Classify using Maple s ability to use exact arithmetic which of the following sets of points are coplanar or non coplanar Set A 212 3875 613 674 312 413 611 520 718 1 2 885775392 42791 Set B 212 3875 613 674 312 413 611 520 718 1 2 885775392042791 42791000000 Include all Maple segments to support your argument and type or write your arguments c If one or more of the sets of points SetA or SetB turn out to be non coplanar find the volume of the tetrahedron s which are formed Hint You may use vectors and Maple V For example the vector determined from the line segment from A to B can be obtained as follows with linalg Warning new definition for norm Warning new definition for trace AB vector 674 212 312 3875 413 613 AB 462 3563 1026 4 Let S be the triangle with vertices A 215 378 615 B 319 715 455 and C 211 568 1213 a Find the length of the shortest side of S b Find the number of degrees in angle BAC at vertex A 5 Consider the plane 317x 415y 725z 211 a Find a point that is on the x axis and on this plane b Find a unit vector which is perpendicular to this plane iv c Find a unit vector parallel to this plane v MA 242 Assignment 2 Due March 5 1996 Name SSN Instructor Section In this assignment you will be doing problems related to the material covered in Lesson 2 Please give your answers in simplest form Be sure to initiate the following command if you intend to use the linear algebra package with linalg Warning new definition for norm Warning new definition for trace 1 Let f x y cos 3 x y3 Find the gradient of f x y 2 Let f x y sin 2 x sin 2 y 1 sin 16 x sin 16 y 10 define a hill on the region 0 x 2 0 y 2 The function f x y provides the elevation of the hill at the point x y Think of the x y as coordinates on a flat map Suppose there is a all terrain vehicle ATV on top of the hill at the point 4 4 1 The hill is shown in Figure 90 below The ATV is driving slowly from the top of the hill to the point 0 0 0 along the surface Consider the view from the hill from above so that the hill is seen as though on a flat map By driving directly we mean that the path on the map that is the xy plane should be a straight line from 4 4 to 0 0 So the ATV rides down the hill along this path First define the function f x y below Be sure it is a function vi 1 0 5 0 0 0 0 5 0 5 y 1 1 1 5 x 1 5 Figure 90 A Hill Now find a vector corresponding to the direction of the ATV along the xy plane i e this should be a two dimensional vector Now find the directional derivative of the ATV as it follows the path specified above The answer should be in terms of x and y Since the ATV travels along the path x y in the xy plane the expression for the directional derivative can be found in terms of either x or y Make a substitution into your expression for the directional derivative to express it in terms of only x Now plot this new expression for the directional derivative Paste a copy into your worksheet NOTE The range of x should be from 0 to 4 Assume that this particular ATV has been booby trapped It has a bomb that will explode if the line running from the front to the back of the ATV tilts more than 50 625 degrees that is about the same as 9 32 radians below the horizontal Use Maple V to find the approximate x y z coordinates where the ATV explodes Hints Use Maple V to find the approximate value of tan 9 32 Then examine the graph you made above You will have to specify an interval when you use fsolve Also remember which way the ATV is driving vii The ATV explodes at the point 3 Find a unit vector which …