To determine To sketch: The graph of the function y = x ( 6 − x ) 3 from the information obtained from the given function as well as the derivative and check the graph by using a calculator. Explanation Definition used: Critical value: A critical value of a function f is any number x in the domain of f such that f ′ ( x ) = 0 or f ′ ( x ) = 0 does not exist. Points of Inflection: A point ( c , f ( c ) ) where the graph of a function has a tangent line and where the concavity changes is a point of inflection Result used: The first derivative test for local maxima and minima : Suppose that c is a critical value of a continuous function f . 1. If the derivative changes from positive to negative at c , then there is a local maximum at c . 2. If the derivative changes from negative to positive at c , then there is a local minimum at c . 3. If the sign of the derivative does not change, there is neither a maximum nor a minimum point. The Second Derivative Test for Concavity: Let y = f ( x ) be twice differentiable on an interval I . 1. If f ″ > 0 on I , the graph of f over I is concave up. 2. If f ″ < 0 on I , the graph of f over I is concave down. Procedure used: 1. Find the critical values by equating the derivative of the unction to zero. 2. Use the critical values to subdivide the x -axis into test intervals, and then select a test value within each test interval to determine the sign of the derivative. 3. The critical values give the x -coordinates of any possible local minimum or maximum points. 4. Use the first derivative test for maxima and minima. 5. Now use the second derivative test to determine the concavity. 6. Sketch the graph based on all the findings. 7. Verify the graph using a calculator. Calculation: The given function is y = x ( 6 − x ) 3 . Substitute x = 0 in y = x ( 6 − x ) 3 to obtain the y -intercept. f ( x ) = 0 ( 6 − 0 ) 3 = 0 Hence, the y -intercept is ( 0 , 0 ) . Substitute y = 0 in y = x ( 6 − x ) 3 to obtain the x -intercept. y = x ( 6 − x ) 3 0 = x ( 6 − x ) 3 x = 0 or 6 − x = 0 x = 0 or x = 6 Hence, the x -intercepts are ( 0 , 0 ) and ( 6 , 0 ) Differentiate the function with respect to x . y ′ = d d x ( x ( 6 − x ) 3 ) = ( 6 − x ) 3 + x [ 3 ( 6 − x ) 2 ( − 1 ) ] = ( 6 − x ) 3 − 3 x ( 6 − x ) 2 = ( 6 − x ) 2 ( 6 − 4 x ) Obtain the critical value by setting y ′ = 0 . ( 6 − x ) 2 ( 6 − 4 x ) = 0 6 − x = 0 or 6 − 4 x = 0 x = 6 or x = 6 4 x = 6 or x = 3 2 Substitute, x = 6 in y = x ( 6 − x ) 3 . y = x ( 6 − x ) 3 = 6 ( 6 − 6 ) 3 = 0 Substitute, x = 3 2 in y = x ( 6 − x ) 3 . y = 3 2 ( 6 − 3 2 ) 3 = 3 2 ( 9 2 ) 3 = 3 2 ( 729 8 ) = 2187 16 Therefore, the critical values are ( 6 , 0 ) and ( 3 2 , 2187 16 ) . Use the critical value x = 6 , 3 2 to subdivide the x -axis into subintervals. Select a test value within each test interval to determine the sign of derivative. Test the value x = 0 for interval x < 3 2 . y ′ = ( 6 − 0 ) 2 ( 6 − 4 ( 0 ) ) = 36 ( 6 ) = 216 > 0 Test the value x = 4 for interval 3 2 < x < 6 . y ′ = ( 6 − 4 ) 2 ( 6 − 4 ( 4 ) ) = ( 2 ) 2 ( − 10 ) = − 40 < 0 Test the value x = 8 for the interval x > 6

11th Edition

ISBN: 9780134508290

Author: Evans

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 24, Problem 28RE

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To sketch: The graph of the function y=x(6−x)3 from the information obtained from the given function as well as the derivative and check the graph by using a calculator.

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Chapter 24, Problem 27RE

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Remix 4. Direction Fields/Phase Portraits. Use the given direction fields to plot solution curves to each of the given initial value problems. (a) x = x+2y 1111 y = -3x+y with x(0) = 1, y(0) = -1 (b) Consider the initial value problem corresponding to the given phase portrait. x = y y' = 3x + 2y Draw two "straight line solutions" passing through (0,0) (c) Make guesses for the equations of the straight line solutions: y = ax.

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Chapter 24 Solutions

EBK BASIC TECHNICAL MATHEMATICS

Ch. 24.1 - For the parabola y = 4 − x2, at the point (3, −5)...Ch. 24.1 - Prob. 2PE Ch. 24.1 - Prob. 1E Ch. 24.1 - Prob. 2E Ch. 24.1 - Prob. 3E Ch. 24.1 - Prob. 4E Ch. 24.1 - Prob. 5E Ch. 24.1 - Prob. 6E Ch. 24.1 - Prob. 7E Ch. 24.1 - Prob. 8E

Ch. 24.1 - Prob. 9E Ch. 24.1 - Prob. 10E Ch. 24.1 - Prob. 11E Ch. 24.1 - Prob. 12E Ch. 24.1 - In Exercises 11–14, find the equations of the...Ch. 24.1 - Prob. 14E Ch. 24.1 - Prob. 15E Ch. 24.1 - Prob. 16E Ch. 24.1 - Prob. 17E Ch. 24.1 - Prob. 18E Ch. 24.1 - Prob. 19E Ch. 24.1 - Prob. 20E Ch. 24.1 - Prob. 21E Ch. 24.1 - Where does the normal line to the parabola y = x —...Ch. 24.1 - Prob. 23E Ch. 24.1 - Prob. 24E Ch. 24.1 - A certain suspension cable with supports on the...Ch. 24.1 - Prob. 26E Ch. 24.1 - Prob. 27E Ch. 24.1 - Prob. 28E Ch. 24.1 - Prob. 29E Ch. 24.1 - Prob. 30E Ch. 24.2 - In Example 1, let x1 = 0.3, and find x2. 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Prob. 20E Ch. 24.4 - Prob. 21E Ch. 24.4 - Prob. 22E Ch. 24.4 - Prob. 23E Ch. 24.4 - Prob. 24E Ch. 24.4 - Prob. 25E Ch. 24.4 - Prob. 26E Ch. 24.4 - Prob. 27E Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - Prob. 31E Ch. 24.4 - Prob. 32E Ch. 24.4 - Prob. 33E Ch. 24.4 - Prob. 34E Ch. 24.4 - Prob. 35E Ch. 24.4 - Prob. 36E Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - Prob. 38E Ch. 24.4 - Prob. 39E Ch. 24.4 - Prob. 40E Ch. 24.4 - Prob. 41E Ch. 24.4 - Prob. 42E Ch. 24.5 - Prob. 1PE Ch. 24.5 - Prob. 2PE Ch. 24.5 - Prob. 1E Ch. 24.5 - Prob. 2E Ch. 24.5 - Prob. 3E Ch. 24.5 - Prob. 4E Ch. 24.5 - Prob. 5E Ch. 24.5 - Prob. 6E Ch. 24.5 - Prob. 7E Ch. 24.5 - Prob. 8E Ch. 24.5 - Prob. 9E Ch. 24.5 - Prob. 10E Ch. 24.5 - Prob. 11E Ch. 24.5 - Prob. 12E Ch. 24.5 - Prob. 13E Ch. 24.5 - Prob. 14E Ch. 24.5 - Prob. 15E Ch. 24.5 - Prob. 16E Ch. 24.5 - 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A microprocessor chip is being designed with a...Ch. 24.7 - Prob. 18E Ch. 24.7 - What are the dimensions of the largest rectangular...Ch. 24.7 - A rectangular storage area is to be constructed...Ch. 24.7 - Prob. 21E Ch. 24.7 - Prob. 22E Ch. 24.7 - Prob. 23E Ch. 24.7 - Prob. 24E Ch. 24.7 - Prob. 25E Ch. 24.7 - Prob. 26E Ch. 24.7 - Prob. 27E Ch. 24.7 - Prob. 28E Ch. 24.7 - Prob. 29E Ch. 24.7 - Prob. 30E Ch. 24.7 - Prob. 31E Ch. 24.7 - Prob. 32E Ch. 24.7 - Prob. 33E Ch. 24.7 - What is the minimum slope of the curve y = x5 −...Ch. 24.7 - Prob. 35E Ch. 24.7 - Prob. 36E Ch. 24.7 - Prob. 37E Ch. 24.7 - Prob. 38E Ch. 24.7 - Prob. 39E Ch. 24.7 - Prob. 40E Ch. 24.7 - Prob. 41E Ch. 24.7 - Computer simulation shows that the drag F (in N)...Ch. 24.7 - Prob. 43E Ch. 24.7 - The potential energy E of an electric charge q due...Ch. 24.7 - An open box is to be made from a square piece of...Ch. 24.7 - Prob. 46E Ch. 24.7 - Prob. 47E Ch. 24.7 - Prob. 48E Ch. 24.7 - An oil pipeline is to be built from a refinery to...Ch. 24.7 - 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