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08-Bisection-Method-4UP

# 08-Bisection-Method-4UP - IVT Bisection method Examples...

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IVT Bisection method Examples Implementation details The Bisection Method Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) The Bisection Method MATH 2070U 1 / 16 IVT Bisection method Examples Implementation details The Bisection Method 1 The Intermediate value theorem 2 The Bisection method 3 Examples 4 Implementation details c D. Aruliah (UOIT) The Bisection Method MATH 2070U 2 / 16 IVT Bisection method Examples Implementation details The Intermediate value theorem Theorem (Intermediate value theorem) Given a continuous function f : [ a , b ] R such that f ( a ) f ( b ) < 0 , then there exists at least one ξ ( a , b ) such that f ( ξ ) = 0 . x y y = f ( x ) a b ( a, f ( a )) ( b, f ( b )) ( ξ 1 , 0) ( ξ 2 , 0) ( ξ 3 , 0) ( ξ 4 , 0) c D. Aruliah (UOIT) The Bisection Method MATH 2070U 4 / 16 IVT Bisection method Examples Implementation details Bracketing a zero Assume f is a continuous function Any interval [ a , b ] for which either f ( a ) > 0 and f ( b ) < 0; or f ( a ) < 0 and f ( b ) > 0 is said to bracket a zero of f Interval can be called a bracket Equivalent criterion: f ( a ) f ( b ) < 0 Assumption of continuity essential for interval to bracket a zero c D. Aruliah (UOIT) The Bisection Method MATH 2070U 5 / 16

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IVT Bisection method Examples Implementation details Example: estimating 2 Let f
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