Estimate area, 4 left endpoints and rectangles. Find \(\displaystyle L_{4}\) Is this an overestimate or underestimate?
\(\displaystyle f(x) = 5\sqrt{x}\) on interval \(\displaystyle [0,4]\) given \(\displaystyle [a, b]\)
\(\displaystyle \sum\limits_{i=4}^n \Delta x [f(a + i \Delta x - \Delta x)]\)
\(\displaystyle \Delta x = \dfrac{b - a}{n}\)
\(\displaystyle \Delta x = \dfrac{\dfrac{4} - 0}{4} = 1\)
\(\displaystyle n = 4\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[f(0 + (1)(1) - (1)] + [f(0 + (2)(1) - (1)] + [f(0 + (3)(1) - (1)] + [f(0 + (4)(1) - (1)]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[f((1)(1) - (1)] + [f((2)(1) - (1)] + [f((3)(1) - (1)] + [f((4)(1) - (1)]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[f((1) - (1)] + [f((2) - (1)] + [f((3) - (1)] + [f((4) - (1)]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[5\sqrt{(0)}] + [5\sqrt{(1)}] + [5\sqrt{(2)}] + [5\sqrt{(3)}]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[5(0)] + [5(1)] + [5\sqrt{3}] + [5\sqrt{2}]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[0] + [5] + [5\sqrt{3}] + [5\sqrt{2}]]\)
\(\displaystyle \sum\limits_{i=4}^n [[0] + [5] + [5\sqrt{3}] + [5\sqrt{2}]]\)
On the right track?
\(\displaystyle f(x) = 5\sqrt{x}\) on interval \(\displaystyle [0,4]\) given \(\displaystyle [a, b]\)
\(\displaystyle \sum\limits_{i=4}^n \Delta x [f(a + i \Delta x - \Delta x)]\)
\(\displaystyle \Delta x = \dfrac{b - a}{n}\)
\(\displaystyle \Delta x = \dfrac{\dfrac{4} - 0}{4} = 1\)
\(\displaystyle n = 4\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[f(0 + (1)(1) - (1)] + [f(0 + (2)(1) - (1)] + [f(0 + (3)(1) - (1)] + [f(0 + (4)(1) - (1)]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[f((1)(1) - (1)] + [f((2)(1) - (1)] + [f((3)(1) - (1)] + [f((4)(1) - (1)]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[f((1) - (1)] + [f((2) - (1)] + [f((3) - (1)] + [f((4) - (1)]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[5\sqrt{(0)}] + [5\sqrt{(1)}] + [5\sqrt{(2)}] + [5\sqrt{(3)}]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[5(0)] + [5(1)] + [5\sqrt{3}] + [5\sqrt{2}]]\)
\(\displaystyle \sum\limits_{i=4}^n (1) [[0] + [5] + [5\sqrt{3}] + [5\sqrt{2}]]\)
\(\displaystyle \sum\limits_{i=4}^n [[0] + [5] + [5\sqrt{3}] + [5\sqrt{2}]]\)
On the right track?
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