11 the table below gives values of a function f and its derivative at selected values of x. if f is…

11 the table below gives values of a function f and its derivative at selected values of x. if f is continuous on the interval -4,-1, what is the value of ∫-4,-1f(x)dx? x -4 -3 -2 -1 f(x) 0.75 -1.5 -2.25 -1.5 f(x) -3 -1.5 0 1.5 a) 0 c) -2.25 e) 4.5 b) -4.5 d) 2.25

11 the table below gives values of a function f and its derivative at selected values of x. if f is continuous on the interval -4,-1, what is the value of ∫-4,-1f(x)dx? x -4 -3 -2 -1 f(x) 0.75 -1.5 -2.25 -1.5 f(x) -3 -1.5 0 1.5 a) 0 c) -2.25 e) 4.5 b) -4.5 d) 2.25

Answer

Explanation:

Step1: Apply fundamental theorem of calculus

By the fundamental theorem of calculus, if (F(x)) is an antiderivative of (f^{\prime}(x)), then (\int_{a}^{b}f^{\prime}(x)dx=F(b)-F(a)), and since (F(x)) is an antiderivative of (f^{\prime}(x)), (F(x) = f(x)+C). So (\int_{-4}^{-1}f^{\prime}(x)dx=f(-1)-f(-4)).

Step2: Substitute values from the table

We know from the table that (f(-1)= - 1.5) and (f(-4)=0.75). Then (f(-1)-f(-4)=-1.5 - 0.75=-2.25).

Answer:

C) -2.25