left endpoint
[left endpoint|left endpoint] The left endpoint sum is an underestimate when the function is increasing.
Definition
left endpoint The left endpoint sum is an underestimate when the function is increasing. This approximation method uses rectangles whose heights are determined by the function value at the left endpoint of each subinterval. The rule specifies constructing rectangles with width Δx and height f(x_{i-1}) on each subinterval [x_{i-1}, x_i].
Mechanism
The mechanism involves forming six rectangles by drawing vertical lines perpendicular to x;_ , with each line corresponding to the left endpoint of a subinterval. These lines are positioned at the left endpoint of each subinterval, creating vertical lines that define the rectangles. The process uses n=8 and selects left endpoint as the left endpoint of each interval.
Causes
left endpoint The left endpoint sum provides an underestimate of the actual value due to the increasing nature of the function. This occurs because the function's increasing trend leads to an underestimation at the left endpoint. The increasing function causes the left endpoint sum to miss the true value by not accounting for the rising trend.
Effects
left endpoint The left endpoint sum provides an underestimate of the actual value due to the increasing nature of the function. This occurs because the function's increasing behavior leads to an underestimation when using the left endpoint method. The relationship between the function's trend and the accuracy of the approximation is directly influenced by its monotonicity.
Constraints
left endpoint The analysis proceeds by evaluating two specific approaches: the left-endpoint approximation and the right-endpoint approximation. These methods impose constraints on how the function values are sampled across the interval. The left-endpoint method restricts the evaluation to the initial point of each subinterval, while the right-endpoint method applies the restriction to the terminal point. Both approaches are subject to the same fundamental limitations regarding approximation accuracy and interval partitioning.