# p1812tut05 - x = 1 5 and the function ϕ x in part ii to...

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School of Mathematics and Statistics The University of Sydney PHAR1812 Calculus Tutorial 5 1. Use the graph method to determine the number of roots of each of the following equations. For each root, name an interval (not too big) which contains the roots: ( i ) sin x = 2 x - 1 ( ii ) x 2 = 1 . 3 + sin x ( iii ) 2 x 3 + 3 x 2 - 36 x + 5 = 0 2. Use the bisection method to ﬁnd each of the roots of the equation x 2 = 1 . 3 + sin x correct to 1 decimal place. 3. ( i ) Sketch y = ln x and y = 2 - x on the same diagram and explain why the equation x - 2 + ln x = 0 has exactly one solution α . ( ii ) Before doing any numerical calculation, explain why one would expect the iteration method using ϕ ( x ) = 2 - ln x to converge to α beginning with any initial value between 1 and 2. ( iii ) Apply the iteration method using initial value
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Unformatted text preview: x = 1 . 5 and the function ϕ ( x ) in part ( ii ) to ﬁnd α correct to 5 decimal places. Check your answer using a change of sign test. 4. Sketch the graph of f ( x ) = x 4 + x-3 . Now use Newton’s Method to ﬁnd each of the roots of the equation x 4 + x-3 = 0 correct to 4 decimal places. 5. Sketch the graph of f ( x ) = x 5 + x-3 . Now use Newton’s Method to ﬁnd the single root of the equation x 5 + x-3 = 0 correct to 4 decimal places. 6. Consider the equation x 3-x = 0 . What happens if we use Newton’s method to ﬁnd roots starting with any of x = 1 √ 3 ,-1 √ 3 , 1 √ 5 ,-1 √ 5 . ?...
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## This note was uploaded on 03/07/2010 for the course GENERAL MA General Ma taught by Professor Not sure during the Spring '10 term at École Normale Supérieure.

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