record: 3 score: 3 select value of the twice differentiable function f and its derivative are shown in the…

record: 3 score: 3 select value of the twice differentiable function f and its derivative are shown in the table below. what is the value of the expression ∫₅⁷ f(x)dx? x 1 3 5 7 f(x) 3 9 15 21 f(x) 2 3 4 2 3 -2 -1 6

record: 3 score: 3 select value of the twice differentiable function f and its derivative are shown in the table below. what is the value of the expression ∫₅⁷ f(x)dx? x 1 3 5 7 f(x) 3 9 15 21 f(x) 2 3 4 2 3 -2 -1 6

Answer

Explanation:

Step1: Apply the fundamental theorem of calculus

By 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_{5}^{7}f''(x)dx), let (F'(x)=f''(x)), then (F(x)=f'(x)). So (\int_{5}^{7}f''(x)dx=f'(7)-f'(5)).

Step2: Find values from the table

From the table, (f'(7) = 2) and (f'(5)=4).

Step3: Calculate the result

Substitute the values into the formula: (f'(7)-f'(5)=2 - 4=-2).

Answer:

(-2)