select value of the twice differentiable function f and its derivative are shown in the table below. what is…
select value of the twice differentiable function f and its derivative are shown in the table below. what is the value of the expression ∫₂⁵f″(x)dx?\n|x|0|2|3|5|\n|f(x)|2|8|11|15|\n|f′(x)|2|2|2|0|\n-2/3 7 7/3 -2
Answer
Answer:
$-2$
Explanation:
Step1: Recall the fundamental theorem of calculus
If $F'(x)=f(x)$, then $\int_{a}^{b}f(x)dx = F(b)-F(a)$. For the integral $\int_{2}^{5}f''(x)dx$, let $F'(x)=f''(x)$, so $F(x)=f'(x)$.
Step2: Apply the fundamental theorem
By the fundamental - theorem of calculus, $\int_{2}^{5}f''(x)dx=f'(5)-f'(2)$.
Step3: Use the values from the table
From the table, $f'(5) = 0$ and $f'(2)=2$. Then $f'(5)-f'(2)=0 - 2=-2$.