7/21/05 DUE DATE: MATH 171 – CALCULUS IPROJECT 2VERSION A1. Below is the graph of the functionf x( ).Place the following quantities in order, from smallest to largest: a. slope of tangent at point Ab. slope of tangent at point Cc. slope of secant line CDd. slope of secant line ABe.( ) ( )f t f st s--Be sure to give justification for your answers. Hint: As part of your justification, either print a copy of the graph or trace it by hand and then by hand sketch each ofthe lines described in a. to e. onto the graph. 2. Examine the function ( )2 524 1( )6xf xx x+=- - for points of discontinuity and non-differentiability.a. Obtain a graph of the function on 8 8 and 3 4x y     . Either print out this graph or paste it into your lab report. Label the graph with its equation.b. Examine the graph of this function. At what points is the function discontinuous? Give reasons for your answers.c. At what points is the function non-differentiable? Give reasons for your answers.Note: You may need to examine “suspicious” points more closely by using a close-up graph or some slope tables.1ABCDstxy7/16/053. Find the point where the graphs of 3( ) 2f x x x= - and 2( ) 0.5 1.5g x x= - are tangent to each other; that is, have a common tangent line. a. Use your knowledge of the derivative to solve this problem. Write out a complete solution with explanations. (Note: Solutions by guessing or just by inspection of the graph are not acceptable).b. Illustrate your solution with a graph containing f(x), g(x), and the tangent line you found.4. You went to the freezer and scooped out a large bowl of ice cream at 8:00 PM. You forgot about the ice cream because you got a phone call, and the bowl of ice cream saton the counter for a long time. The temperature of the ice cream is I (in degrees Fahrenheit) and is a function of time, t, in minutes since you scooped it. The equation of its temperature is:0.1575 45tI e-= -a. Graph ( )I t for 0 40t� �. Include the graph in your report.b. What does the I-intercept tell you about your ice cream?c. What will happen to the temperature of the ice cream after a long period of time? What feature of the graph does this fact correspond to? Verify your statement withan appropriate limit. d. Thinking about the temperature of the ice cream over any period of time, should( )I t�be positive or negative? Why?e. Find( )I t�. Show all your work. What are the units of( )I t�?f. Find (5)I� and interpret your answer in terms of the temperature of the ice cream.g. Should (20)I� be larger or smaller than (5)I�? Why?5. A rotated hyperbola is defined implicitly by the equation: 2 22 9 4 8 36x xy y x+ + - =.a. Write the equation of the line tangent to this curve at the point (0, 3). Show your work, with explanations.b. Graph the equation. Follow the directions in the 3.6 Computer Example 1 in the supplement. Use the window 3 3 and 6 6x y- � � - � �. Use 200 rows for the plot. Then use Post-graph, overlay any type of graph, to add the graph of the tangent line. Include this graph in your report.c. How many other tangent lines are also parallel to the line you found in part