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vaieri [72.5K]
3 years ago
14

The constant of proportionality between the circumference and _______ of a circle is equal to π which means that for any given c

ircle, the length around the circle _______ by its diameter is a little over _______ , or more exactly π
Blank 1: Diameter or radius

Blank 2: Divided or multiplied

Blank 3: 4, 3, or 5
PLEASE ANSWER ASAP
Mathematics
1 answer:
katrin2010 [14]3 years ago
7 0
Blank 1= diameter
blank 2 =multiplied
blank 3=3.
Hope this helped!
Sorry  couldn't answer faster i didn't see this question.
Mark me as brainliest Please!!!
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Consider functions of the form f(x)=a^x for various values of a. In particular, choose a sequence of values of a that converges
sleet_krkn [62]

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A. As "a"⇒e, the function f(x)=aˣ tends to be its derivative.

Step-by-step explanation:

A. To show the stretched relation between the fact that "a"⇒e and the derivatives of the function, let´s differentiate f(x) without a value for "a" (leaving it as a constant):

f(x)=a^{x}\\ f'(x)=a^xln(a)

The process will help us to understand what is happening, at first we rewrite the function:

f(x)=a^x\\ f(x)=e^{ln(a^x)}\\ f(x)=e^{xln(a)}\\

And then, we use the chain rule to differentiate:

f'(x)=e^{xln(a)}ln(a)\\ f'(x)=a^xln(a)

Notice the only difference between f(x) and its derivative is the new factor ln(a). But we know  that ln(e)=1, this tell us that as "a"⇒e, ln(a)⇒1 (because ln(x) is a continuous function in (0,∞) ) and as a consequence f'(x)⇒f(x).

In the graph that is attached it´s shown that the functions follows this inequality (the segmented lines are the derivatives):

if a<e<b, then aˣln(a) < aˣ < eˣ < bˣ < bˣln(b)  (and below we explain why this happen)

Considering that ln(a) is a growing function and ln(e)=1, we have:

if a<e<b, then ln(a)< 1 <ln(b)

if a<e, then aˣln(a)<aˣ

if e<b, then bˣ<bˣln(b)

And because eˣ is defined to be the same as its derivative, the cases above results in the following

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but as "a"⇒e, the difference between f(x) and f'(x) begin to decrease until it gets zero (when a=e)

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