multiple choice 1 point the function f is continuous on the closed interval 0, 6 and has the values given in…

multiple choice 1 point the function f is continuous on the closed interval 0, 6 and has the values given in the table above the trapezoidal approximation for ∫₀⁶f(x) dx found with 3 subintervals of equal length is 52. what is the value of k? (a) 2 (b) 6 (c) 7 (d) 10 (e) 14

multiple choice 1 point the function f is continuous on the closed interval 0, 6 and has the values given in the table above the trapezoidal approximation for ∫₀⁶f(x) dx found with 3 subintervals of equal length is 52. what is the value of k? (a) 2 (b) 6 (c) 7 (d) 10 (e) 14

Answer

Explanation:

Step1: Determine the width of sub - intervals

The interval is $[0,6]$ and $n = 3$ sub - intervals. The width $\Delta x=\frac{b - a}{n}=\frac{6-0}{3}=2$.

Step2: Write the trapezoidal rule formula

The trapezoidal rule for $n$ sub - intervals is $T_n=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n - 1})+f(x_n)]$. Here, $n = 3$, $x_0 = 0$, $x_1=2$, $x_2 = 4$, $x_3=6$, $f(x_0)=4$, $f(x_1)=k$, $f(x_2)=8$, $f(x_3)=12$, and $\Delta x = 2$. So $T_3=\frac{2}{2}[4 + 2k+2\times8 + 12]$.

Step3: Set up the equation and solve for $k$

We know that $T_3 = 52$. So $4+2k + 16+12=52$. Combine like terms: $2k+32 = 52$. Subtract 32 from both sides: $2k=52 - 32=20$. Divide both sides by 2: $k = 10$.

Answer:

D. 10