a) Estimate the area under the graph of f(x)=7+4x^2 from x=-1 to x=2 using three rectangles and right endpoints.
R3= ???
Then improve your estimate by using six rectangles.
R6= ???
Sketch the curve and the approximating rectangles for R3 and R6?
b) Repeat part (a) using left endpoints.
L3= ???
L6= ???
Sketch the curve and the approximating rectangles for L3 and L6?
c) Repeat part (a) using midpoints.
M3=????
M6=???
Sketch the curve and approximating rectangles for M3 and M6?
d) from your sketched in parts (a)-(c) which appears to be the best estimate??
R6 or M6 or L6 ????
Please help me with this I only have one try left on WebAssign.
I can't sketch it. The instructions are really clear. Sketch the curve, and limits, draw the rectangles. Put the endpoints on the curve, add the areas.
To estimate the area under the graph of the function f(x) = 7 + 4x^2, we can use rectangles and right, left, or midpoint endpoints. Let's go through each part step by step:
a) Estimate the area using three rectangles and right endpoints (R3):
First, we need to determine the width of each rectangle. Since we are dividing the interval from x = -1 to x = 2 into three equal parts, the width of each rectangle is (2 - (-1)) / 3 = 3/3 = 1.
Next, we calculate the height of each rectangle by evaluating the function at the right endpoint of each rectangle.
- For the first rectangle, the right endpoint is -1, so the height is f(-1) = 7 + 4(-1)^2 = 7 + 4 = 11.
- For the second rectangle, the right endpoint is 0, so the height is f(0) = 7 + 4(0)^2 = 7 + 0 = 7.
- For the third rectangle, the right endpoint is 1, so the height is f(1) = 7 + 4(1)^2 = 7 + 4 = 11.
Now, we multiply each rectangle's width by its height and sum up the results:
R3 = (1 * 11) + (1 * 7) + (1 * 11) = 11 + 7 + 11 = 29.
b) Improve the estimate using six rectangles and right endpoints (R6):
Following the same steps as in part (a), we divide the interval into six equal parts, so the width of each rectangle is 1/2.
The heights of the rectangles using right endpoints are:
- For the first rectangle, the right endpoint is -1, so the height is f(-1) = 11.
- For the second rectangle, the right endpoint is -1/2, so the height is f(-1/2) = 7 + 4(-1/2)^2 = 7 + 4(1/4) = 8.
- For the third rectangle, the right endpoint is 0, so the height is f(0) = 7.
- For the fourth rectangle, the right endpoint is 1/2, so the height is f(1/2) = 8.
- For the fifth rectangle, the right endpoint is 1, so the height is f(1) = 11.
- For the sixth rectangle, the right endpoint is 3/2, so the height is f(3/2) = 7 + 4(3/2)^2 = 7 + 4(9/4) = 16.
Now, we multiply each rectangle's width by its height and sum up the results:
R6 = (1/2 * 11) + (1/2 * 8) + (1/2 * 7) + (1/2 * 8) + (1/2 * 11) + (1/2 * 16) = 5.5 + 4 + 3.5 + 4 + 5.5 + 8 = 31.5.
You can sketch the graph of the function and the rectangles to visualize it.
c) Repeat using midpoints (M3 and M6):
For M3, the midpoints of the three rectangles are -1/2, 0, and 1/2. Calculate the heights of these rectangles similarly to part (b) and find the area.
For M6, find the midpoints of six rectangles, evaluate the heights, and calculate the area.
d) Compare the estimates:
To determine which estimate is the best, compare the areas obtained with R6, M6, and L6. The estimate with the highest area is likely to be the best as it covers more area under the curve.
Once you have the areas for R6, M6, and L6, compare them to determine which one is the largest. That estimate would be the best one.
Make sure to sketch the curve and the approximating rectangles for each part as it will help you visually understand the process and evaluate the accuracy of the estimates.
Good luck with your WebAssign submission!