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NCSU MA 242 - Assignments - MA 242

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iMA 242 Assignment 1 Due: February 9, 1996Name:SSN:Instructor:Section:Welcome to Maple assignment 1 for MA 242.1. Consider the functiong( x, y ) = sin( x ) sin( y ) e( −x2)(a) Produce a contour plot for this function in the square region−π ≤ x ≤ π, −π ≤ y ≤ π.Show the command that you use and at one of the following prompts paste in a copy of the contourplot. 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 youuse and paste in a copy of the plot which uses the boxed axes, the hidden line style option, and whichis rotated so that Theta = 135, and Phi = 75 degrees. You can change all the options in your pictureusing the buttons at the top of the plot window and can adjust the angle by clicking on the picture itselfwith the left mouse button. Press the middle mouse button to replot the picture.>>2. Consider the functionh( x, y ) = ln(¯¯x2+ y2−6¯¯).(a) Produce a three dimensional plot for this function over the region inside the circlex2+ y2= 25with the following features: style is patch, axes is framed, there are 4 tickmarks along the x and ydirection and 4 in the z direction, Theta =40, and Phi = 30. Paste the plot into this worksheet at one ofthe following prompts.>>>ii(b) One can obtain the level curves forz=1forthis 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 naturallogrithm. (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 nso 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 copla-nar or non-coplanar:SetA := (212, 3875, 613), (674, 312, −413), (611, 520, 718), (1, 2, 885775392/42791)SetB :=(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 thetetrahedron(s) which are formed.Hint: You may use vectors and Maple V . For example, the vector determined from the line segmentfrom A to B can be obtained as follows:> with(linalg):Warning: new definition for normWarning: 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.>>>vMA 242 Assignment 2 Due: March 5, 1996Name:SSN:Instructor:Section:In this assignment, you will be doing problems related to the material covered in Lesson 2. Please give your an-swers 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 normWarning: new definition for trace1. Letf( x, y ) = cos( 3 x ) y3.Find the gradient of f(x, y).>>>>2. Letf ( x, y ) = sin( 2 x ) sin( 2 y ) +110sin( 16 x ) sin( 16 y )define a hill on the region0 ≤ x ≤ π/2, 0 ≤ y ≤ π/2.The function f (x, y) provides the elevation of the hill at the point (x, y). Think of the (x,y) as coordinateson a flat map. Suppose there is a all-terrain vehicle (ATV) on top of the hill at the point (π/4,π/4,1). Thehill 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 viewfrom the hill from above so that the hill is seen as though on a flat map. By driving directly, we mean thatthe path on the map, that is the xy-plane, should be a straight line from (π/4,π/4) to (0, 0). So the ATVrides down the hill along this path.First, define the function f(x, y) below. Be sure it is a function!>vi00.511.5x00.511.5y00.51Figure 90: A HillNow find a vector corresponding to the direction of the ATV along the xy-plane, i.e. this should be a twodimensional vector.>Now find the directional derivative of the ATV as it follows the path specified above. (The answer shouldbe 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 canbe found in terms of either x or y. Make a substitution into your expression for the directional derivative toexpress it in terms of only x.>>>Now plot this new expression for the directional derivative. Paste a copy into your worksheet. NOTE: Therange of x should be from 0 to π/4.>>>Assume that this particular ATV has been booby trapped. It has a bomb that will explode if the line runningfrom the front to the back of the ATV tilts more than 50.625 degrees (that is about the same as 9π/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(−9π/32). Then examine the graph you madeabove. You will have to specify an interval when you use fsolve. Also remember which way the ATV isdriving.)>>>viiThe ATV explodes at the point (, , ).3. Find a unit vector which is normal to the graph ofz = x3− y2+esin( x )at the point (π, 1,π3).>>>4. Find the equation of the tangent plane to the paraboloidz =124x2+116y2at the point (3, 2, 5/8).>>>5. Locate and classify all critical points of the functionf ( x, y ) = x3+ y3+3 xy2−15 x −15 y>>>6. A rectangular box


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