Preparation open tasks for the exam in Mathematical Analysis. BISECTION METHOD 1. Using the bisection method...
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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 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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Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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