Math  /  Calculus

QuestionLet ff be a continuous function on the interval [0,8][0,8]. If we use the Trapezoidal Rule with n=4n=4 to approximate the integral 08f(x)dx\int_{0}^{8} f(x) d x , which of the following is the required approximation?
Select one: (f(0)+f(2)+f(4)+f(6)+f(8))2(f(0)+2f(2)+2f(4)+2f(6)+f(8))12f(0)+2f(2)+2f(4)+2f(6)+f(8)f(0)+2f(1)+2f(2)+2f(3)+2f(4)+؛\begin{array}{l} (f(0)+f(2)+f(4)+f(6)+f(8)) 2 \\ (f(0)+2 f(2)+2 f(4)+2 f(6)+f(8)) \frac{1}{2} \\ f(0)+2 f(2)+2 f(4)+2 f(6)+f(8) \\ f(0)+2 f(1)+2 f(2)+2 f(3)+2 f(4)+؛ \end{array}

Studdy Solution

STEP 1

1. The function f(x) f(x) is continuous on the interval [0,8][0,8].
2. We are using the Trapezoidal Rule with n=4 n = 4 subintervals to approximate the integral.
3. The Trapezoidal Rule formula for n n subintervals is given by: $ \frac{b-a}{2n} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right] \]

STEP 2

1. Determine the width of each subinterval.
2. Identify the points x0,x1,,x4 x_0, x_1, \ldots, x_4 .
3. Apply the Trapezoidal Rule formula.
4. Match the result with the provided options.

STEP 3

Determine the width of each subinterval. The interval [0,8][0, 8] is divided into n=4 n = 4 subintervals. The width h h of each subinterval is calculated as:
h=ban=804=2h = \frac{b-a}{n} = \frac{8-0}{4} = 2

STEP 4

Identify the points x0,x1,,x4 x_0, x_1, \ldots, x_4 . With h=2 h = 2 , the points are:
x0=0,x1=2,x2=4,x3=6,x4=8x_0 = 0, \quad x_1 = 2, \quad x_2 = 4, \quad x_3 = 6, \quad x_4 = 8

STEP 5

Apply the Trapezoidal Rule formula:
ba2n[f(x0)+2f(x1)+2f(x2)+2f(x3)+f(x4)]\frac{b-a}{2n} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right]
Substitute the values:
802×4[f(0)+2f(2)+2f(4)+2f(6)+f(8)]\frac{8-0}{2 \times 4} \left[ f(0) + 2f(2) + 2f(4) + 2f(6) + f(8) \right]
Simplify the coefficient:
88[f(0)+2f(2)+2f(4)+2f(6)+f(8)]=1[f(0)+2f(2)+2f(4)+2f(6)+f(8)]\frac{8}{8} \left[ f(0) + 2f(2) + 2f(4) + 2f(6) + f(8) \right] = 1 \left[ f(0) + 2f(2) + 2f(4) + 2f(6) + f(8) \right]

STEP 6

Match the result with the provided options. The correct expression is:
f(0)+2f(2)+2f(4)+2f(6)+f(8)f(0) + 2f(2) + 2f(4) + 2f(6) + f(8)
This matches the third option.
The required approximation is f(0)+2f(2)+2f(4)+2f(6)+f(8) \boxed{f(0) + 2f(2) + 2f(4) + 2f(6) + f(8)} .

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