No, it would not.XodoX said:Homework Statement
Give the upper sum for f(x)=4-x2 on the interval [-1,1] with respect to the partition P{ -1, -0.5, 0.5, 1]
Give the lower sum for f(x)=4-x2 on the interval [-1,1] with respect to the partition P{ -1, -0.5, 0.5, 1]
Homework Equations
You might try writing the definition of upper and lower sum with respect to a partition here.
The Attempt at a Solution
How do I do this again? Lower sum should be the area under the curve.
You'd just integrate then on the interval given (-1,1) ?
Also don't know anymore how to do upper sum.
The function f(x)=4-x2 is a quadratic function, meaning it has the form ax2+bx+c where a, b, and c are constants. In this case, a=-1, b=0, and c=4. It is a downward facing parabola that crosses the y-axis at (0,4) and has roots at x=-2 and x=2.
"Upper" and "lower" sums refer to the ways in which a function can be approximated by dividing the interval on which it is defined into smaller subintervals and using the values of the function at the endpoints of each subinterval to calculate the area under the curve. The upper sum uses the maximum value of the function on each subinterval, while the lower sum uses the minimum value.
To calculate the upper and lower sums for a given function, you first need to divide the interval on which the function is defined into smaller subintervals. Then, you calculate the width of each subinterval by subtracting the endpoint of the previous subinterval from the endpoint of the current subinterval. Next, you find the maximum and minimum values of the function on each subinterval. Finally, you multiply the width of each subinterval by the maximum and minimum values respectively, and add up all of the resulting products to find the upper and lower sums.
The upper and lower sums are approximations of the area under a curve. As the number of subintervals used to calculate the sums increases, the approximations become more accurate and approach the actual area under the curve. This is known as the Riemann sum, and by taking the limit as the number of subintervals approaches infinity, we can find the exact area under the curve.
Calculating upper and lower sums for a function allows us to approximate the area under a curve, which has many real-world applications. It can be used to determine the total distance traveled by an object with varying velocity, the total work done by a varying force, or the total amount of product produced by a varying production rate. It also allows us to better understand the behavior of a function and make predictions about its values at different points.