91Ó°ÊÓ

A function is said to be decreasing on an interval if, as you move from left to right across the interval, the function values decline. For example, if you have a function like \(f(x) = -x\), it is a decreasing function because it has a negative slope, indicating that as \(x\) increases, \(f(x)\) gets smaller.
This means for any two values \(x_1\) and \(x_2\) within the interval where \(x_1 < x_2\), you have \(f(x_1) > f(x_2)\).
Using a decreasing function affects how we compute sums in calculus, particularly in Riemann sums, where we assess the area under a curve.

The left-hand sum is a method to approximate the integral of a function, specifically for calculating the area under a curve. For a decreasing function, the left-hand sum always picks the value of the function at the left endpoint of each subinterval.
The formula for this is: \[\sum_{i=0}^{n-1} f(x_i) \Delta x\] Where \(x_i\) represents each left endpoint, and \(\Delta x\) is the width of the subintervals.
In a decreasing function scenario, each \(f(x_i)\) is larger than the next endpoint \(f(x_{i+1})\), which results in a higher sum because we continually choose the larger value from the interval.

The right-hand sum is another method to approximate the integral of a function. It focuses on using the value at the right endpoint of each subinterval to calculate the sum. The formula for the right-hand sum is: \[\sum_{i=1}^{n} f(x_i) \Delta x\] Where \(x_i\) is the right endpoint of each subinterval.
In a decreasing function like \(f(x) = -x\), each \(f(x_i)\) is smaller because the function is decreasing. This results in a lower sum since we're generally picking the smaller value of each interval, meaning the sum of these areas is less compared to the left-hand sum.

A velocity function, noted typically as \(f(x)\), represents how velocity varies with time or another independent variable. When we look at the exercise, our velocity function is \(f(x) = -x\).
This particular velocity function is decreasing, meaning as time progresses, velocity decreases. When applying Riemann sums to velocity functions, the resulting sums can be interpreted as distance. - **Left-Hand Sum**: Takes initial point velocity, over-estimating compared to the actual decreasing function.- **Right-Hand Sum**: Takes ending point velocity, providing a more accurate but lower estimation for a decreasing function. Thus, understanding how the velocity function behaves is important when evaluating how accurately these sums reflect actual movement over a given interval.