DUE DATE:______________ MATH 171 – CALCULUS IPROJECT 3VERSION A1. If we have two parametrized curves (two pairs of parametric equations), we will distinguish between an intersection point and a “collision” point. An intersection point is any point that lies on both curves. If this intersection point corresponds to the same valueof t for both curves, we will call it a collision point. For this problem, consider two objects in motion over the time interval 0 2t p� �. The position ( )1 1,x yof the first object is described by the parametric equations1 1( ) 2cos( ) and ( ) 3sin( )x t t y t t= =.The position of the second object ( )2 2,x yis described by the parametric equations 2 2( ) 1 sin( ) and ( ) cos( ) 3x t t y t t= + = -.To find out if these objects collide and if they do, at what time the collision occurs, follow these steps:a. Provide a graph of the paths of the two objects. Graphing instructions for Converge: Click on first to stop the graph at the first point.Click on , choose Parametric and specify 2 functions. In the first X-window, type 2cos(t), then hit TAB, and in the Y-window type 3sin(t); in the second X-window, type 1+sin(t), then hit TAB, and in the Y-window type cos(t)- 3 and hit ENTER. Use this window: Min x: –3, Max x: 3, Min y: –5, Max y: 5, Min t: 0, Max t: 2pi. Change the number of points to be plotted to 100, and hit ENTER. To obtain the graph, click on repeatedly. It may not be obvious from the graphs on the screen that one of these graphs is an ellipse and one is a circle. To fix this, right click anywhere inside the graph window, and a menu comes up; click on “Resize to get 1-1 Axes Scaling”.Include a copy of this graph in your project report. On the graph, put arrows (by hand) to show the direction of the plot.b. Eliminate the parameter on each set of equations and find the rectangular equations of each curve. Hint: To eliminate the parameter, t, solve each set of equations forcos( ) and sin( )t t and use an identity. Use the rectangular equations to verify the equationof the ellipse (what is its center?) and the equation of the circle (what is its center?). See page 75 - 76 in the textbook for reference.c. From your graphs, how many intersection points are there for the two curves?d. To collide, the objects must be at the same position at the same time. That is, there must be a solution to 1 2 1 2( ) ( ) and ( ) ( )x t x t y t y t= =. Solve (by hand) this set of equations, simultaneously, showing all work.e. If you found a collision point in part d), at what t value does it occur? What are the rectangular coordinates of the point of collision? 2. Use the methods of calculus (not trial and error and not curve-fitting) to find a cubic function3 2( )f x ax bx cx d= + + + that has a local minimum at the point ( )1, 2- - and a local maximum at the point ( )2,1. Show all work. Explain how you use the given information to find the values of , , , and a b c d. Write out the cubic polynomial you found and include a printed copy or paste the graph of this polynomial into your project report.NOTE TO THE STUDENT: Each of the remaining two problems asks you to use all the skills and knowledge you have accumulated about Calculus. In order to receive full credit for your solutions, you must pay attention to the quality of your explanations. You should not submit pages of mathematics alone, but that mathematics must be narrated and explained so that a casual reader would understand not only what you are doing but also why you are doing it.3. Use methods of algebra and calculus to analyze the graph of the function33 25( )2 3 7x xf xx x- - +=+ -. Include all the following in your analysis:a. the domainb. intercepts c. equations of asymptotes (both vertical and horizontal) d. relative extrema (be sure to provide all derivatives, identify critical numbers, and test those values – naming the test you are using.)e. inflection points (be sure to identify possible inflection points and be sure to test those values.)f. intervals where the graph is concave up and where it is concave down (be sure to include the vertical asymptotes in your intervals.)g. provide computer-generated graphs of your function that illustrate the key points that youhave listed. You may need to provide more than one graph.Note: You may use Derive to graph the function, compute derivatives, and solve equations (follow the directions in the Classroom Supplement, Section 4.3.) However, you must present a logical argument for all your conclusions and present your work so that your methods are clear.24. 1 A truck driving over a flat interstate at a constant rate of 50 MPH gets 5 miles to the gallon. Fuel costs $2.35 per gallon. For each mile per hour increase in speed, the truck loses a tenth of a mile per gallon in its mileage. In addition to the fuel costs, other operating expenses, which the company incurs, include the driver's wages of $27.50 per hour, and the fixed costs of operating the truck at $11.33 per hour. The company wants to minimize the total cost of operating the truck. What constant speed (between 50 MPH and the speed limit of 65 MPH) should a dispatcher require on a straight run through 260 miles of Kansas interstate? Note: For this problem, you must:a. Clearly explain how you obtained the function to be minimized. Be sure your variables are clearly defined and it is very clear how you took all the information given and constructed your function. Include the domain of the function.b. Obtain a graph of this function.c. Use this function to answer the question. (You will need to compute derivatives and find andtest critical values). Carefully explain all the steps you take.d. Write a clear answer to the problem.1 Adapted from Calculus Problems for a New Century, edited by Robert Fraga, MAA Notes Number 28,