Math / Calculus

QuestionThe graph below shows $y=x^{2}$ . The right-hand sum for eight equal divisions is given by which expression? $0^{2}(0.5)+0.5^{2}(0.5)+1^{2}(0.5)+1.5^{2}(0.5)+2^{2}(0.5)+2.5^{2}(0.5)+3^{2}(0.5)+3.5^{2}(0.5)$ $0.5^{2}(0.5)+1^{2}(0.5)+1.5^{2}(0.5)+2^{2}(0.5)+2.5^{2}(0.5)+3^{2}(0.5)+3.5^{2}(0.5)+4^{2}(0.5)$ $0.5(0.5)+1(0.5)+1.5(0.5)+2(0.5)+2.5(0.5)+3(0.5)+3.5(0.5)+4(0.5)$ $0.5^{2}+1^{2}+1.5^{2}+2^{2}+2.5^{2}+3^{2}+3.5^{2}+4^{2}$

Studdy Solution

STEP 1

1. We are given the function $y = x^2$ .
2. We need to calculate the right-hand sum for eight equal divisions.
3. The interval for the divisions is from $0$ to $4$ .

STEP 2

1. Determine the width of each subinterval.
2. Identify the right endpoints of each subinterval.
3. Calculate the sum using the right endpoints and the function $y = x^2$ .

STEP 3

The interval is from $0$ to $4$ , and we have eight equal divisions.
To find the width of each subinterval, divide the total interval length by the number of divisions: $\text{Width} = \frac{4 - 0}{8} = 0.5$

STEP 4

The right endpoints of each subinterval are calculated by starting from the first endpoint and adding the width of the subinterval successively.
The right endpoints are: $0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4$ .

STEP 5

To find the right-hand sum, evaluate the function $y = x^2$ at each right endpoint and multiply by the width of the subinterval.
The right-hand sum is: $0.5^2(0.5) + 1^2(0.5) + 1.5^2(0.5) + 2^2(0.5) + 2.5^2(0.5) + 3^2(0.5) + 3.5^2(0.5) + 4^2(0.5)$
The correct expression for the right-hand sum is:
$\boxed{0.5^2(0.5) + 1^2(0.5) + 1.5^2(0.5) + 2^2(0.5) + 2.5^2(0.5) + 3^2(0.5) + 3.5^2(0.5) + 4^2(0.5)}$

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