5. calculator allowed\na table of values for a continuous function ( f ) is shown above. if four equal…
5. calculator allowed\na table of values for a continuous function ( f ) is shown above. if four equal subintervals of (0,2) are used, which of the following is the trapezoidal approximation of ( int_{0}^{2} f(x) d x )?\n(a) 8 (b) 12 (c) 16 (d) 24 (e) 32
Answer
Explanation:
Step1: Recall the trapezoidal rule formula
The trapezoidal rule formula for (n) sub - intervals on ([a,b]) is (T=\frac{\Delta x}{2}\left(f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n - 1})+f(x_n)\right)), where (\Delta x=\frac{b - a}{n}). Here, (a = 0), (b = 2), and (n = 4). So, (\Delta x=\frac{2-0}{4}=0.5).
Step2: Identify the function values
We have (x_0 = 0), (f(x_0)=3); (x_1 = 0.5), (f(x_1)=3); (x_2 = 1.0), (f(x_2)=5); (x_3 = 1.5), (f(x_3)=8); (x_4 = 2.0), (f(x_4)=13).
Step3: Substitute into the trapezoidal rule formula
[ \begin{align*} T&=\frac{0.5}{2}(f(0)+2f(0.5)+2f(1.0)+2f(1.5)+f(2.0))\ &=0.25(3 + 2\times3+2\times5+2\times8+13)\ &=0.25(3+6 + 10+16+13)\ &=0.25\times48 \end{align*} ]
Answer:
(12), so the answer is B.