Where did my textbook say clearly what it states at this point?

In summary: Sea J un intervalo abierto, y sea I un intervalo que contiene a todos los puntos de J, y posiblemente uno de sus extremos, o ambos. Sea f una función continua en I y diferencia-ble en J.(a) Si f'(x) > 0 para todo x perteneciente a J, entonces f es creciente en I.(b) Si f'(x) < 0 para todo x perteneciente a J, entonces f es decreciente en I.(c) Si f
  • #1
mcastillo356
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Hi, PF

I left reading for a while, and now I must revisit a quote from Spanish 6th edition of "Calculus", by Robert A. Adams. The quote is " 4.2 Extreme values problems (...) As we've seen, the sign of ##f'## shows if ##f## is increasing or decreasing".
Obvious, I shouldn't care and continue, but I'm kind of curious: does the book say it clearly previously, or must I read between lines what is written until now?

Best hopes, greetings
 
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  • #2
The derivative is the rate of increase or decrease of the function value. A positive derivative implies an increasing function and a negative derivative implies a decreasing function. If the derivative is zero, the function is neither increasing nor decreasing.

The author must have covered that somewhere when he introduced the derivative.
 
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  • #3
DEFINITION 3 The slope of a curve
The slope of a curve ##C## at a point ##P## is the slope of the tangent line to ##C## at ##P## if such tangent line exists. In particular, the slope of the graph of ##y=f(x)## at the point ##x_0## is

$$\displaystyle\lim_{h \to{0}}{\dfrac{f(x_0+h)-f(x_0)}{h}}$$

Well, here it is, I guess. Nevermind... It was just curiosity. The underlying true is clearly said at #2. Thanks.

On the road again, I hope.
 
  • #4
mcastillo356 said:
DEFINITION 3 The slope of a curve
The slope of a curve ##C## at a point ##P## is the slope of the tangent line to ##C## at ##P## if such tangent line exists. In particular, the slope of the graph of ##y=f(x)## at the point ##x_0## is

$$\displaystyle\lim_{h \to{0}}{\dfrac{f(x_0+h)-f(x_0)}{h}}$$

Well, here it is, I guess. Nevermind... It was just curiosity. The underlying true is clearly said at #2. Thanks.

On the road again, I hope.
That's the formal mathematical definition of the derivative. That didn't come out of nowhere. That definition is intended to define formally something very important that already had a well understood intuitive meaning: the rate of change of a function at a point.

When you learn that definition, you should check that it makes sense and does indeed have the properties that you expect of the derivative. It should make sense as a formal way to make the derivative rigorous.

For example, "the sign of ##f'(x_0)## determines whether the function is increasing or decreasing at ##x_0##" is something we should already know from our elementary understanding of functions - it doesn't need knowledge of the formal definition to tell us that.
 
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  • #5
PeroK said:
For example, "the sign of ##f'(x_0)## determines whether the function is increasing or decreasing at ##x_0##" is something we should already know from our elementary understanding of functions - it doesn't need knowledge of the formal definition to tell us that.
Definitely. Wrong, not accurate, my quote.
 
  • #6
mcastillo356 said:
As we've seen, the sign of f′ shows if f is increasing or decreasing".
Obvious, I shouldn't care and continue, but I'm kind of curious: does the book say it clearly previously, or must I read between lines what is written until now?
As @PeroK pointed out, you don't really need the definition of the derivative to determine whether a function is increasing or decreasing, assuming you are able to draw a graph of the function or are able to compare function inputs vs. their outputs.

Once you have the derivative as a tool to use, you can determine the intervals on which a function is increasing/decreasing just by noting whether the derivative is positive or negative, respectively.

So the quoted text seems pretty clear to me.
 
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  • #7
Capitulo 2, Teorema 12
Sea J un intervalo abierto, y sea I un intervalo que contiene a todos los puntos de J, y
posiblemente uno de sus extremos, o ambos. Sea f una función continua en I y diferencia-
ble en J.
(a) Si f'(x) > 0 para todo x perteneciente a J, entonces f es creciente en I.
(b) Si f'(x) < 0 para todo x perteneciente a J, entonces f es decreciente en I.
(c) Si f'(x) ≥ 0 para todo x perteneciente a J, entonces f es no decreciente en I.
(d) Si f'(x) ≤ 0 para todo x perteneciente a J, entonces f es no creciente en I.
 
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  • #8

1. Where can I find the information I need in my textbook?

The best place to start looking for information in your textbook is the table of contents. This will give you an overview of the topics covered in each chapter and help you narrow down your search. You can also use the index at the back of the book to search for specific keywords or terms.

2. How do I know if the information in my textbook is accurate?

Textbooks go through a rigorous review process before they are published, so the information presented is typically accurate and up-to-date. However, it's always a good idea to cross-check information with other sources to ensure its accuracy.

3. Can I rely on my textbook as my only source of information?

While textbooks are a great resource, they should not be your only source of information. It's important to consult other sources, such as scholarly articles, research papers, and reputable websites, to get a well-rounded understanding of a topic.

4. How do I know if I am interpreting the information correctly?

If you are unsure about your interpretation of the information in your textbook, it's always a good idea to discuss it with your peers or your instructor. They can provide additional insights and help clarify any confusion.

5. What should I do if I can't find the information I need in my textbook?

If you are unable to find the information you need in your textbook, try reaching out to your instructor or a librarian for assistance. They may be able to point you towards additional resources or provide guidance on where to find the information you need.

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