2413labI9 - Laboratory I.9 Applications of the Derivative...

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Laboratory I.9 Applications of the Derivative Goals • The student will determine intervals where a function is increasing or decreasing using the first derivative. • The student will find local minima and maxima of functions. • The student will determine the intervals where a function is concave up or concave down. • The student will use the derivative of a function to solve an application problem. Before the Lab The material in this lab is essentially the same as that learned in Chapter 4, sections 3 and 7 in the text. This lab gives you a chance to practice the ideas without worrying about the algebra. In the Lab (90 pts) 1. As you work through this problem, fill in your answers in “Table 1” in the After the Lab part. (a) Author and Plot f ( x ) := (2 x^ 2 + x 1)/(2 x^ 2 + 5 x + 4). Use the Trace function of Derive to find the approximate coordinates of the local minimum and the local maximum that you see. Put the ( x , y ) coordinates of these two points in Table 1. (b) Recall from class that a function is increasing where its derivative is positive. We can get Derive to find where this is true for f ( x ). Use Calculus Differentiate f(x) to find the derivative, then Author (the derivative you just found) > 0 . To avoid having to type the derivative in, make sure the derivative is highlighted when you open the Author box, then use the F3 key to copy it in. Now Author solve(the inequality you just authored,x) to solve the resulting inequality. Repeat for f ’( x ) < 0 to see where the function is decreasing, and fill in the appropriate lines in Table 1. (c) To find the minimum and maximum, Author ( your derivative ) = 0, then Solve and Approximate the results. Use f ( x ) to obtain the y values, and fill in the appropriate places in Table 1. (d) Finally, the concavity of the function f ( x ) depends on the second derivative: where f ’’( x ) > 0, the function is concave up; where f ’’( x ) < 0, the function is concave down. Go through steps (b) and (c) again with the second derivative to find where the function is concave up, concave down, and the points where the concavity changes. 2. Repeat the above for f ( x ) := x 3 – 3 x 2 + 3 x – 5. Put the results in Table 2. 3. A customer wants to know the cost of a water tank that will hold 800 gallons. Your firm can make the tank from a rectangular piece of metal that you will roll into a cylinder and two circular ends that will be welded to the cylinder. The rectangular piece is made of a malleable alloy that cost \$2.36 per square

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MATH 2413 Lab 9 Page 2 of 6 foot, but the circular ends can be made of a cheaper metal costing \$1.44 per square foot. Welding costs \$1.20 per foot. In general the firm adds 23% of the total cost to get the selling price of any tank they
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This note was uploaded on 03/27/2012 for the course CALC 2413 taught by Professor Moody during the Spring '10 term at Texas A&M University, Corpus Christi.

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2413labI9 - Laboratory I.9 Applications of the Derivative...

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