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

[mcastillo356]

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

 

[PeroK]

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.

 

[mcastillo356]

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.

 

[PeroK]

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.

 

[mcastillo356]

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.

 

[Mark44]

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.

 

[Dragon27]

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.

 

[mcastillo356]

Thanks, @Dragon27 , PF!