Preparation open tasks for the exam in Mathematical Analysis BISECTION METHOD 1 Using the bisection method within three steps and initial interval 3, 4 , find a decimal approximation of 100 and estimate the error of the approximation Sample solution We use the interval 3, 4 as our initial interval Since we are considering the fourth root, to determine whether the approximation is an overestimate or an underestimate, we will examine the fourth power of the approximation Step 1 Interval Midpoint Power of 4 3, 4 3 5 (3 5) 10 3, 3 5 3 25 (3 25) 10 3, 3 25 3 125 (2 125) 10 2 3 The value is in the interval (3 125, 3 25) If we assume that 100 3 1875 (midpoint), then the error of this approximation will not exceed 0 03125 (half of the length of the interval) 2 Using the bisection method within three steps and initial interval 1, 3 , find a decimal approximation of 35 and estimate the error of the approximation 3 Using the bisection method within three steps and initial interval 2, 3 , find a decimal approximation of 40 and estimate the error of the approximation 4 Using the bisection method within three steps and initial interval 4, 6 , find a decimal approximation of 31 and estimate the error of the approximation STUDY OF A SINGLE VARIABLE FUNCTION Find (by investigating sign of derivatives) where each of the following functions are increasing and where they are decreasing Then find the extrema, and say whether each extremum is a maximum or a minimum 1 f(x) x 3x 6 Answer increasing on ( 0, 0 and 2, c), decreasing on 0,2 f(2) 2,(0) 6 2 f(x) x Answer increasing on ( 00, 1) and 1, 0), decreasing on 1, 0) and (0, 1) fin(1) 2, max( 1) 2 3 f(x) x Answer increasing on ( 0, 0 and c), decreasing on 0, () 2 4 f(x) x Answer increasing on ( , 0 and 2, ), decreasing on (0, 2) min(2) 2 INTEGRAL SUMS 1 Given a function f 1,0 0, 1 , f(x) x sketch (accurately ) the graph of this function Divide the interval 1,0 into 4 equal parts and, choosing the right bound of each interval to calculate the heights

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Question: Preparation open tasks for the exam in Mathematical Analysis. BISECTION METHOD 1. Using the bisection method within three steps and initial interval [3, 4],

Preparation open tasks for the exam in Mathematical Analysis. BISECTION METHOD 1.

Preparation open tasks for the exam in Mathematical Analysis. BISECTION METHOD 1. Using the bisection method within three steps and initial interval [3, 4], find a decimal approximation of 100 and estimate the error of the approximation. Sample solution: We use the interval [3, 4] as our initial interval. Since we are considering the fourth root, to determine whether the approximation is an overestimate or an underestimate, we will examine the fourth power of the approximation. Step 1 Interval Midpoint Power of 4 [3, 4] 3.5 (3.5) 10 [3, 3.5] 3.25 (3.25) > 10 [3, 3.25] 3.125 (2.125)

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