Q. 65 The proof of the latter part of ... [FREE SOLUTION]

Chapter 4: Q. 65 (page 401)

The proof of the latter part of the Second Fundamental Theorem of Calculus in the reading covered only the case $h 鈫� 0^{+}$ . Rewrite this proof in your own words, and then write a proof of what happens as $h 鈫� 0^{-}$ .

Short Answer

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The Second Fundamental Theorem states that if $f$ is continuous on $[a, b]$ then role="math" localid="1648741945639" $F (x) = {鈭�}_{0}^{x} f (t) d t$ for all $x 鈭� [a, b]$ , then

$F$ is continuous on $[a, b]$ and differentiable on $(a, b)$ and $F$ is an antiderivative of $f$ , i.e., $F' (x) = f (x)$ .

Step by step solution

Step 1. Given information

We have to proof the latter part of the Second Fundamental Theorem of Calculus the case when $h 鈫� 0^{+}$ .

Step 2. Proof of the given question

It is required to prove that $F$ is an antiderivative of $f$ , i.e. $F' (x) = f (x)$ .

Considering the right derivative, $h$ is positive.

So,

$F (x) = {鈭�}_{0}^{x} f (t) d t$

Finding its derivative with respect to $x$ ,

$F' (x) = \lim_{h 鈫� 0^{+}} \frac{F (x + h) - F (x)}{h} = \lim_{h 鈫� 0^{+}} \frac{{鈭�}_{a}^{x + h} f (t) d t - {鈭�}_{a}^{x} f (t) d t}{h} = \lim_{h 鈫� 0^{+}} \frac{{鈭�}_{a}^{x + h} f (t) d t + {鈭�}_{x}^{a} f (t) d t}{h} = \lim_{h 鈫� 0^{+}} \frac{{鈭�}_{x}^{x + h} f (t) d t}{h}$

The Mean Value Theorem is,

$f (c) = \frac{{鈭�}_{a}^{b} f (x) d x}{b - a}$

Applying it to $\lim_{h 鈫� 0^{+}} \frac{{鈭�}_{x}^{x + h} f (t) d t}{h}$

$\lim_{h 鈫� 0^{+}} \frac{{鈭�}_{x}^{x + h} f (t) d t}{h} = \lim_{h 鈫� 0^{+}} f (c)$

This means that there is some $c$ on $[x, x + h]$ .

In the limit as $h$ goes to $0^{+}$ , $c$ gets squeezed down to $x$ .

Since $f (x)$ is continuous,

$\lim_{h 鈫� 鈭�} f (c) = \lim_{c 鈫� x} f (c) = f (x)$

Therefore, $F' (x) = \lim_{h 鈫� 0^{+}} \frac{F (x + h) - F (x)}{h} = f (x)$ is proved.

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The proof of the latter part of the Second Fundamental Theorem of Calculus in the reading covered only the case $h 鈫� 0^{+}$ . Rewrite this proof in your own words, and then write a proof of what happens as $h 鈫� 0^{-}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof of the given question

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91影视

The proof of the latter part of the Second Fundamental Theorem of Calculus in the reading covered only the case h鈫�0+. Rewrite this proof in your own words, and then write a proof of what happens as h鈫�0-.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof of the given question

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Geometry

Discrete Mathematics

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

The proof of the latter part of the Second Fundamental Theorem of Calculus in the reading covered only the case $h 鈫� 0^{+}$ . Rewrite this proof in your own words, and then write a proof of what happens as $h 鈫� 0^{-}$ .