Explanation Given: The function f ( x ) = x 3 − 8 x − 1 The function has a root in the interval [2.75,3] Key concepts applied: The Bisection method is a method used to obtain a rough approximation to a solution of a given function. It is based on the intermediate value theorem for continuous functions. The bisection method is an iterative process, in which roots of a given equation are identified through bisection of a known interval that contains a root and then selecting a subinterval in which a root must lie for further processing. Note that the given function f ( x ) is continuous for all values of x. Calculation: The steps of bisection can be carried out as follows, Description f ( x ) = x 3 − 8 x − 1 Step 1: We know that f ( x ) = x 3 − 8 x − 1 has a zero in [2.75, 3], which is an interval of length 1 4 . Bisect this interval to create two new sub-intervals [2.75,2.875] and [2.875,3] of length 1 8 . Find in which of these intervals does f ( x ) have a root. Interval : [2.75, 2.875] f ( 2.75 ) = ( 2.75 ) 3 − 8 ( 2.75 ) − 1 = 20.797 − 22 − 1 = − 2.203 < 0 f ( 2.875 ) = 2.875 3 − 8 ( 2.875 ) − 1 = 23.764 − 23 − 1 = − 0.236 < 0 Since both values are negative, we cannot infer from IVT that there is a root for the function in the interval [2.75,2.875]. Interval : [2.875,3] f ( 2.75 ) = ( 2.75 ) 3 − 8 ( 2

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Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 2.8, Problem 22E

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An interval of length 116 containing the root of the given function found in the given interval by applying the Bisection method twice.

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Chapter 2.8, Problem 21E

Chapter 2.8, Problem 23E

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You are constructing a box out of cardboard with the dimensions 5 m by 6 m. You then cut equal-size squares from each corner so you may fold the edges. Let x be the side length of each square. Find that maximizes the volume of the box. Answer exactly. 8 x x H x ४ x ४ ४ m

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Chapter 2 Solutions

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An interval of length 1 16 containing the root of the given function found in the given interval by applying the Bisection method twice.

Solution Summary: The author explains the bisection method, based on the intermediate value theorem for continuous functions.

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An interval of length 116 containing the root of the given function found in the given interval by applying the Bisection method twice.

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Chapter 2 Solutions