Q. 6 - Calculus

Question

Fill in the blanks to complete each of the following theorem statements: 

6. The Second Fundamental Theorem of Calculus: Suppose $f$ is _____ on $[a, b]$ and, for all $x \in [a, b]$, we define

$F\left(x\right)={\int }_a^{x}f\left(t\right)dt$Then,

 $F$ is _____  on $[a, b]$ and ____ on $(a, b)$, and $F$ is an _____ of $f$ , or in other words, ____ = _____ .

Step-by-Step Solution

Answer

The Second Fundamental Theorem of Calculus: Suppose $f$ is continous on $[a, b]$ and, for all $x \in [a, b]$, we define

$F\left(x\right)={\int }_a^{x}f\left(t\right)dt$

Then $F$ is continous on $[a, b]$ and differentiable  on $(a, b)$, and $F$ is  an antiderivative  of  $f$, or in other words, $F\text{'}\left(x\right)=f\left(x\right)$ .

Step 1. Given data

We have to complete the statement,

The Second Fundamental Theorem of Calculus: Suppose $f$ is _____ on $[a, b]$ and, for all $x \in [a, b]$, we define

$F\left(x\right)={\int }_a^{x}f\left(t\right)dt$

Then $F$ is _____  on $[a, b]$ and differentiable on $(a, b)$, and $F$ is an ____ of $f$ , or in other words,  ____ = _____ .

Step 2. Fill in the blanks

The Second Fundamental Theorem of Calculus: Suppose $f$ is continous  on $[a, b]$ and, for all $x \in [a, b]$, we define

$F\left(x\right)={\int }_a^{x}f\left(t\right)dt$

Then $F$ is differntiable on $[a, b]$ and antiderivative on $(a, b)$, and $F$ is an antiderivative of $f$, or in other words,  $F\text{'}\left(x\right)=f\left(x\right)$