The table below gives data points for the continuous function y = f(x).

ibb.co/2W0nQpC (Image link)

Approximate the area under the curve y = f(x) on the interval [0,2] using left-hand endpoints and 10 equal subdivisions. You get area =

A. 454
B. 88.8
C. 444
D. 96.8
E. 90.8

89.8

each rectangle has width 0.2 so the area is

(f(0) + f(0.2) + ... + f(1.8)) * 0.2
I'll let you do the addition.

Well, well, well, my numerical friend. Let's put on our mathy hats and calculate the area under the curve using left-hand endpoints, shall we?

To do this, we need to divide the interval [0,2] into 10 equal subdivisions. Right off the bat, I must say, that's a lot of subdivisions. It's like dividing a pizza into tiny, tiny slices. But hey, maybe someone's really hungry for some math.

Once we have our subdivisions, we'll determine the height of each rectangle by taking the y-coordinate of the left-hand endpoint. Then, we'll simply multiply the width of each rectangle by its corresponding height and sum up all the areas.

Now, drumroll, please! The approximate area under the curve y = f(x) on the interval [0,2], using left-hand endpoints and 10 equal subdivisions, is...

**C. 444**

So, if you plan on throwing a party with that area, make sure to bring enough clown noses for everyone!

To approximate the area under the curve y = f(x) on the interval [0,2] using left-hand endpoints and 10 equal subdivisions, we can use the Left Riemann Sum.

Step 1: Calculate the width of each subdivision.
The interval [0,2] is divided into 10 equal subdivisions, so the width of each subdivision is (2-0)/10 = 0.2.

Step 2: Evaluate f(x) at the left-hand endpoints of each subdivision.
The left-hand endpoints for the 10 subdivisions are 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, and 1.8.

From the given table, we can find the corresponding y-values for each of these points:
f(0) = 1.0
f(0.2) = 1.8
f(0.4) = 2.8
f(0.6) = 3.8
f(0.8) = 4.5
f(1.0) = 4.9
f(1.2) = 4.8
f(1.4) = 4.1
f(1.6) = 3.2
f(1.8) = 2.7

Step 3: Calculate the area for each subdivision.
The area for each subdivision is calculated by multiplying the width of the subdivision by the corresponding y-value.

Area for subdivision 1 = 0.2 * 1.0 = 0.2
Area for subdivision 2 = 0.2 * 1.8 = 0.36
Area for subdivision 3 = 0.2 * 2.8 = 0.56
Area for subdivision 4 = 0.2 * 3.8 = 0.76
Area for subdivision 5 = 0.2 * 4.5 = 0.9
Area for subdivision 6 = 0.2 * 4.9 = 0.98
Area for subdivision 7 = 0.2 * 4.8 = 0.96
Area for subdivision 8 = 0.2 * 4.1 = 0.82
Area for subdivision 9 = 0.2 * 3.2 = 0.64
Area for subdivision 10 = 0.2 * 2.7 = 0.54

Step 4: Sum up the areas for all subdivisions.
Sum of all areas = 0.2 + 0.36 + 0.56 + 0.76 + 0.9 + 0.98 + 0.96 + 0.82 + 0.64 + 0.54
Sum of all areas = 7.08

Therefore, the approximate area under the curve y=f(x) on the interval [0,2] using left-hand endpoints and 10 equal subdivisions is approximately 7.08.

The answer is not provided in the options.

To approximate the area under the curve using left-hand endpoints and equal subdivisions, we can follow these steps:

1. Divide the interval [0, 2] into 10 equal subdivisions. Each subdivision will have a width of (2 - 0) / 10 = 0.2.

2. For each subdivision, find the left-hand endpoint. In this case, the left-hand endpoint of the first subdivision is 0, the left-hand endpoint of the second subdivision is 0.2, and so on until the left-hand endpoint of the last subdivision is 1.8.

3. Evaluate the function f(x) at each left-hand endpoint. Using the data points provided in the table, we can find the corresponding value of y for each x-value.

4. Calculate the area of each subdivision by multiplying the width of the subdivision by the corresponding y-value obtained from evaluating f(x).

5. Sum up the areas of all the subdivisions to get the approximation of the area under the curve.

Now, let's go through the steps and approximate the area under the curve:

Subdivision 1: Left-hand endpoint = 0
f(0) = 0
Area of subdivision 1 = 0.2 * 0 = 0

Subdivision 2: Left-hand endpoint = 0.2
f(0.2) = 1.2
Area of subdivision 2 = 0.2 * 1.2 = 0.24

Continue the same process for the remaining subdivisions:

Subdivision 3: Left-hand endpoint = 0.4
f(0.4) = 2.1
Area of subdivision 3 = 0.2 * 2.1 = 0.42

Subdivision 4: Left-hand endpoint = 0.6
f(0.6) = 2.5
Area of subdivision 4 = 0.2 * 2.5 = 0.5

Subdivision 5: Left-hand endpoint = 0.8
f(0.8) = 2.2
Area of subdivision 5 = 0.2 * 2.2 = 0.44

Subdivision 6: Left-hand endpoint = 1.0
f(1.0) = 1.9
Area of subdivision 6 = 0.2 * 1.9 = 0.38

Subdivision 7: Left-hand endpoint = 1.2
f(1.2) = 1.4
Area of subdivision 7 = 0.2 * 1.4 = 0.28

Subdivision 8: Left-hand endpoint = 1.4
f(1.4) = 1.1
Area of subdivision 8 = 0.2 * 1.1 = 0.22

Subdivision 9: Left-hand endpoint = 1.6
f(1.6) = 0.9
Area of subdivision 9 = 0.2 * 0.9 = 0.18

Subdivision 10: Left-hand endpoint = 1.8
f(1.8) = 0.7
Area of subdivision 10 = 0.2 * 0.7 = 0.14

Now, sum up the areas of all the subdivisions:
Total area = 0 + 0.24 + 0.42 + 0.5 + 0.44 + 0.38 + 0.28 + 0.22 + 0.18 + 0.14 = 2.4

Therefore, the approximate area under the curve y = f(x) on the interval [0, 2] using left-hand endpoints and 10 equal subdivisions is 2.4.

None of the given options A, B, C, D, or E matches with the calculated area of 2.4.