Chapter 4: Q. 6 (page 403)

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
$F (x) = \int_{a}^{x} f (t) d t$ Then,
$F$ is ___ on $[a, b]$ and on $(a, b)$ , and $F$ is an _ of $f$ , or in other words, = ___ .

Short Answer

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

$F (x) = \int_{a}^{x} f (t) d t$

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

Step by step solution

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

$F (x) = \int_{a}^{x} f (t) d t$

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

$F (x) = \int_{a}^{x} f (t) d t$

Then $F$ is differntiable on $[a, b]$ and antiderivative on $(a, b)$ , and $F$ is an antiderivative of $f$ , or in other words, localid="1649352465544" $F' (x) = f (x)$

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Fill in the blanks to complete each of the following theorem statements:6. The Second Fundamental Theorem of Calculus: Suppose fis _____ on [a,b]and, for all x∈[a,b], we defineF(x)=∫axf(t)dtThen,Fis _____ on [a,b]and ____ on (a,b), and Fis an _____ of f, or in other words, ____ = _____ .

Short Answer

Step by step solution

Step 1. Given data

Step 2. Fill in the blanks

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Discrete Mathematics

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Probability and Statistics

Mechanics Maths

Calculus

Geometry

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