Set1 - Math 21B - Homework Set 1 Section 5.1: 2. f (x) = x3...

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Math 21B - Homework Set 1 Section 5.1: 2. f ( x ) = x 3 between x = 0 and x = 1. a. Estimate using lower sum with two rectangles of equal width: If we want two rectangles of equal width, we will let Δ x = 1 2 . The function f ( x ) is increasing on [0 , 1], so the height of each rectangle is given by the value of f at its left endpoint. f (0) = 0 f ± 1 2 ² = 1 8 Thus we get: A 0 · 1 2 + 1 8 · 1 2 = 1 16 b. Estimate using lower sum with four rectangles of equal width: We will let Δ x = 1 4 and the heighths of the rectangles are given by the value of f at their respective left endpoints. f (0) = 0 f ± 1 4 ² = 1 64 f ± 1 2 ² = 1 8 f ± 3 4 ² = 27 64 Thus we get: A 0 · 1 4 + 1 64 · 1 4 + 1 8 · 1 4 + 27 64 · 1 4 = 36 256 = 9 64 1
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c. Estimate using upper sum with two rectangles of equal width: We will let Δ x = 1 2 and the heighths of the rectangles are given by the value of f at their respective right endpoints. f ± 1 2 ² = 1 8 f (1) = 1 Thus we get: A 1 8 · 1 2 + 1 · 1 2 = 9 16 d. Estimate using upper sum with four rectangles of equal width: We will let Δ x = 1 4 and the heighths of the rectangles are given by the value of f at their respective right endpoints. f ± 1 4 ² = 1 64 f ± 1 2 ² = 1 8 f ± 3 4 ² = 27 64 f (1) = 1 Thus we get: A 1 64 · 1 4 + 1 8 · 1 4 + 27 64 · 1 4 + 1 · 1 4 = 100 256 = 25 64 3. f ( x ) = 1 x between x = 1 and x = 5. a. Estimate using lower sum with two rectangles of equal width: If we want two rectangles of equal width, we will let Δ x = 2. The function f ( x ) is decreasing on [1 , 5], so the height of each rectangle is given by the value of f at its right endpoint. 2
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f (3) = 1 3 f (5) = 1 5 Thus we get: A 1 3 · 2 + 1 5 · 2 = 16 15 b. Estimate using lower sum with four rectangles of equal width: We will let Δ x = 1 and the heighths of the rectangles are given by the value of f at their respective right endpoints. f
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Set1 - Math 21B - Homework Set 1 Section 5.1: 2. f (x) = x3...

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