Exercise 3. Suppose F: [a, b] →ℝ is continuous on [a, b] and differentiable on [a, b] ∖ S, where

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Exercise 3. Suppose $F: [a, b] \to \mathbb{R}$ is continuous on $[a, b]$ and differentiable on $[a, b] \setminus S$, where $S$ is a finite set. Suppose there exists an $f \in \mathcal{R}[a, b]$ such that $f(x) = F'(x)$ for $x \in [a, b] \setminus S$. Show that \begin{equation} \int_a^b f = F(b) - F(a). \end{equation}

          Exercise 3. Suppose $F: [a, b] \to \mathbb{R}$ is continuous on $[a, b]$ and differentiable on $[a, b] \setminus S$, where $S$ is a finite set. Suppose there exists an $f \in \mathcal{R}[a, b]$ such that $f(x) = F'(x)$ for $x \in [a, b] \setminus S$. Show that \begin{equation*} \int_a^b f = F(b) - F(a). \end{equation*}

Exercise 3. Suppose F: [a, b] →ℝ is continuous on [a, b] and differentiable on [a, b] ∖ S, where S is a finite set. Suppose there exists an f ∈ℛ[a, b] such that f(x) = F'(x) for x ∈ [a, b] ∖ S. Show that
^b f = F(b) - F(a).

Added by Todd C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Kayla Ashcroft

08/14/2021

Step 1

Step 1: Since F is differentiable on [a, b] \ S, it is also continuous on [a, b] \ S. Show more…

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Exercise 3. Suppose F: [a, b] → R is continuous on [a,b] and differentiable on [a, b] \ S, where S is a finite set. Suppose there exists an f ∈ R[a,b] such that f(x) = F'(x) for x ∈ [a,b] S. Show that $$ \int_a^b f = F(b) - F(a). $$