MATH FOR COLLEGE



Problem Set#1

Multiple Choice Test

Chapter 03.06 False-Positiion Method

COMPLETE SOLUTION SET

1. The false-position method for finding roots of nonlinear equations belongs to a class of a (an) ____________ method.

(A) open

(B) bracketing

(C) random

(D) graphical

Solution

The correct answer is (B).

The false-position method brackets the root with the two initial guesses.

2. The newly predicted root for false-position and secant method can be respectively given as

[pic]

and

[pic] ,

While the appearance of the above 2 equations look essentially identical, and both methods require two initial guesses, the major difference between the above two formulas is

A) false-position method is not guaranteed to converge

B) secant method is guaranteed to converge

C) secant method requires the 2 initial guesses [pic] to satisfy [pic]

D) false-position method requires the 2 initial guesses [pic] to satisfy [pic]

Solution

The correct answer is (D).

In order for the two initial guesses to bracket the root, [pic] must be true. In other words, the function must change signs between the two guesses for there to be a root present.

3. Given are the following nonlinear equation

[pic]

two initial guesses, [pic] and [pic], and a pre-specified relative error tolerance of 0.1%. Using the false-position method, which of the following tables is correct = predicted root)?

A)

|Iteration |[pic] |[pic] |[pic] |

|1 |1 |4 |? |

|2 |? |? |2.939 |

B)

|Iteration |[pic] |[pic] |[pic] |

|1 |1 |4 |? |

|2 |? |? |2.500 |

C)

|Iteration |[pic] |[pic] |[pic] |

|1 |1 |4 |? |

|2 |? |? |1.500 |

D)

|Iteration |[pic] |[pic] |[pic] |

|1 |1 |4 |? |

|2 |? |? |2.784 |

Solution

The correct answer is (A).

First check given function is of the form [pic], then proceed as follows:

Iteration 1:

Initial guesses are [pic] and [pic]. Check that they satisfy the requirement [pic].

[pic], which is 0, the root must lie between [pic] and [pic]. Therefore, [pic]and [pic].

Iteration 2:

Initial guesses are [pic]and [pic]. Again, check requirement: [pic]

[pic], which is ................
................

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