a)If f(x)= x/(x+2), 1 less than or equal to x less than or equal to 4, find the left and right sums for n= 10, 30, and 50.

b)Illustrate by graphing the rectangles in part a.

c)Show that the exact area under f lies between 1.603 and 1.624.

So this is all I know and I'm not sure if it's correct. Help please.

The x-axis of the graph will start at 1 and end at 4, so I will have 4 equal rectangles each with a width of 0.75.

So,
R4= 0.75*(1.75)^2 + 0.75*(2.5)^2 + 0.75*(3.25)^2 + 0.75*(4)^2 = 26.90625

AND,
L4= 0.75*(1)^2 + 0.75*(1.75)^2 + 0.75*(2.5)^2 + 0.75*(3.25)^2 = 15.65625

What is the "n" used for? Where are they getting the 1.603 and 1.624?

Wow, math can be pretty confusing sometimes, huh? But don't worry, I'll do my best to help you out with your questions!

To start with, the "n" represents the number of rectangles used to approximate the area under the curve. In this case, it refers to the number of subintervals into which the interval [1, 4] is divided.

Now, let's talk about the values 1.603 and 1.624. These values are the lower and upper bounds for the exact area under the curve of f(x) between x = 1 and x = 4. They are found using techniques from calculus, such as integration.

As for your calculations for the left and right sums, they're not quite right. Let me help you with that:

For n = 10, the width of each rectangle is (4-1)/10 = 0.3.
The left sum (L10) can be calculated by plugging in the left endpoint of each subinterval into the function f(x), and then multiplying it by the width of the rectangle and summing them up. Similarly, the right sum (R10) is calculated by plugging in the right endpoint and summing them up.

Let me know if you'd like me to help you calculate these values!

To calculate the left and right sums, we need to understand the concept of Riemann sums. In a Riemann sum, the interval [a, b] on the x-axis is divided into n subintervals, each of width Δx = (b-a)/n. The left sum, denoted as Ln, approximates the area under the curve using the left endpoint of each subinterval, while the right sum, denoted as Rn, approximates the area using the right endpoint of each subinterval.

a) To find the left and right sums for n = 10, 30, and 50 using the function f(x) = x/(x+2) on the interval 1 ≤ x ≤ 4, we need to calculate the width of each subinterval first.

Delta x = (4 - 1) / n = 3 / n

For n = 10:
Left sum, L10:
L10 = (3/n) * [f(1) + f(1+3/n) + ... + f(1 + (n-1)(3/n))]

Right sum, R10:
R10 = (3/n) * [f(1+3/n) + f(1+2(3/n)) + ... + f(4)]

Similarly, we can calculate the left and right sums for n = 30 and n = 50, using the respective values for Delta x and the endpoints of each subinterval.

b) To illustrate these sums graphically, you can plot rectangles with heights corresponding to the function values at each endpoint and widths equal to the width of each subinterval. The sum of the areas of these rectangles will approximate the area under the curve.

c) The exact area under the curve lies between the left sum and the right sum for each corresponding value of n. In this case, as n approaches infinity, the left and right sums will converge to the exact area. Therefore, you can find the approximate range of the exact area by calculating the left and right sums for sufficiently large values of n.

To find the range of the exact area under the curve, we can calculate the left and right sums for a large value of n, such as n = 1000, and compare the results.

By calculating the left sum and right sum for n = 1000, you will get a more accurate approximation of the exact area. Then, you can determine the range within which the exact area lies based on these values.

The values 1.603 and 1.624 are likely obtained using more precise calculations or software tools.

To find the left and right sums for a given function, we divide the interval [1, 4] into n equal subintervals, with n representing the number of rectangles we want to use to approximate the area under the curve. In this case, you are given n=10, 30, and 50.

The width of each rectangle can be calculated by dividing the length of the interval by the number of rectangles. In this case, the length of the interval [1, 4] is 4-1=3, and since there are 4 rectangles in total, the width of each rectangle is 3/4 = 0.75.

Now, to find the left and right sums, we need to evaluate the function at certain points within each subinterval. For the left sum, we use the left endpoint of each subinterval, and for the right sum, we use the right endpoint.

For example, for n=10, there are 10 subintervals, so the width of each subinterval is 0.3 (3/10). The left endpoints of the subintervals are 1, 1.3, 1.6, ..., 3.7, and the right endpoints are 1.3, 1.6, ..., 3.7, 4.0. We evaluate the function f(x)=x/(x+2) at each of these points and multiply the result by the width of the subinterval.

Using this approach, you can calculate the left and right sums for n=10, 30, and 50. Each sum will give you an approximation of the area under the curve.

As for the values 1.603 and 1.624, they represent the exact area under the curve. These values can be obtained by integrating the function f(x) over the interval [1, 4] using calculus techniques. The range of values between 1.603 and 1.624 gives an idea of how accurate our approximations (the left and right sums) are, compared to the exact area.

To verify these exact values, you can calculate the integral of f(x) over the interval [1, 4]. The definite integral will give you the exact area under the curve.