ExponentialFcns | MathModel.uz

3.1-bo'lim: Asosiy eksponensial funktsiyalar

Birinchidan, $b > 0$ va $b \ne 1$ uchun eksponensial funktsiya shakldagi har qanday funktsiya ekanligini eslaylik.

\[f\left( x \right) = {b^x}\]

Quyidagi holatlarning oldini olish uchun biz $b \ne 1$ ni talab qilamiz:

\[f\left( x \right) = {1^x} = 1\]

Shunday qilib, agar biz $b = 1$ ga ruxsat bersak, biz faqat doimiy funktsiyani olamiz, 1.

Quyidagi holatlarning oldini olish uchun bizga $b > 0$ kerak:

\[f\left( x \right) = {\left( { - 4} \right)^x}\hspace{0.25in}\hspace{0.25in} \Rightarrow \,\hspace{0.25in}\,\,\,\,f\left( {\frac{1}{2}} \right) = {\left( { - 4} \right)^{\frac{1}{2}}} = \sqrt { - 4} \]

$b > 0$ ni talab qilish orqali biz manfiy sonlarning kvadrat ildizlari ehtimoli haqida tashvishlanishimiz shart emas.

$f\left( x \right) = {4^x}$, $g\left( x \right) = {\left( {\displaystyle \frac{1}{4}} \right)^x}$ va $h\left( x \right) = {4^{ - x}}$ ni $x = - 2, - 1,0,1,2$ da baholang.

Yechim

Bu erda gap, asosan, bunday funktsiyalarni baholashingiz mumkinligiga ishonch hosil qilishdir. Shunday qilib, javoblar bilan tezkor jadval.

	$\displaystyle x = - 2$	$\displaystyle x = - 1$	$\displaystyle x = 0$	$\displaystyle x = 1$	$\displaystyle x = 2$
$\displaystyle f\left( x \right)$	$\displaystyle f\left( { - 2} \right) = \frac{1}{{16}}$	$\displaystyle f\left( { - 1} \right) = \frac{1}{4}$	$\displaystyle f\left( 0 \right) = 1$	$\displaystyle f\left( 1 \right) = 4$	$\displaystyle f\left( 2 \right) = 16$
$\displaystyle g\left( x \right)$	$\displaystyle g\left( { - 2} \right) = 16$	$\displaystyle g\left( { - 1} \right) = 4$	$\displaystyle g\left( 0 \right) = 1$	$\displaystyle g\left( 1 \right) = \frac{1}{4}$	$\displaystyle g\left( 2 \right) = \frac{1}{{16}}$
$\displaystyle h\left( x \right)$	$\displaystyle h\left( { - 2} \right) = 16$	$\displaystyle h\left( { - 1} \right) = 4$	$\displaystyle h\left( 0 \right) = 1$	$\displaystyle h\left( 1 \right) = \frac{1}{4}$	$\displaystyle h\left( 2 \right) = \frac{1}{{16}}$

E'tibor bering, oxirgi ikki qator aynan bir xil javob beradi. Agar siz bu haqda o'ylasangiz, bu mantiqiy bo'lishi kerak, chunki,

\[g\left( x \right) = {\left( {\frac{1}{4}} \right)^x} = \frac{{{1^x}}}{{{4^x}}} = \frac{1}{{{4^x}}} = {4^{ - x}} = h\left( x \right)\]

$f\left( x \right) = {4^x}$, $g\left( x \right) = {\left( {\displaystyle \frac{1}{4}} \right)^x}$ va $h\left( x \right) = {4^{ - x}}$ grafigini bir xil o'q tizimida chizing.

Yechim

E'tibor bering, biz faqat $f\left( x \right)$ va $g\left( x \right)$ grafiklarini yaratishimiz kerak, chunki biz oldingi masalada $g\left( x \right) = h\left( x \right)$ ekanligini ko'rsatgan edik. Shuni ham yodda tutingki, bu erda juda ko'p ish yo'q. Biz 1-masalada qiymatlar to'plamini topdik, shuning uchun biz qilishimiz kerak bo'lgan narsa nuqtalarni chizish va keyin grafikni chizishdir. Mana eskiz,

$The graphs of $f\left( x \right) = {4^x}$ and $g\left( x \right) = h\left( x \right) = {\left( {\frac{1}{4}} \right)^x}$ on the same axis system for $ - 2 \le x \le 2$. Both graphs are always positive and both pass through $\left( {0,1} \right)$. The graph of $f\left( x \right) = {4^x}$ is nearly flat along the x-axis on the left and increases very steeply on the right, while the graph of $g\left( x \right)$ is its mirror image, dropping steeply from the upper left and flattening out along the x-axis on the right.$
$f\left( x \right) = {b^x}$ uchun ba'zi asosiy xususiyatlar sifatida ro'yxatlang.

Yechim

Ushbu xususiyatlarning ko'pchiligini oldingi muammodagi eskizda ko'rish mumkin.

(a) Har bir $x$ uchun $f\left( x \right) = {b^x} > 0$. Bu $b > 0$ talabining bevosita natijasidir.

(b) Har qanday $b$ uchun bizda $f\left( 0 \right) = {b^0} = 1$ mavjud.

(c) Agar $b > 1$ (masalan, yuqoridagi $f\left( x \right) = {4^x}$) biz $f\left( x \right) = {b^x}$ ortib borayotgan funktsiya ekanligini ko'ramiz va bu,
\[f\left( x \right) \to \infty {\mbox{ as }}x \to \infty \hspace{0.5in}{\rm{and}}\hspace{0.5in}f\left( x \right) \to 0{\mbox{ as }}x \to - \infty \]
(d) Agar $0 < b < 1$ (masalan, yuqoridagi $g\left( x \right) = {\left( {\frac{1}{4}} \right)^x}$) biz $f\left( x \right) = {b^x}$ kamayuvchi funktsiya ekanligini ko'ramiz va bu,
\[f\left( x \right) \to 0{\mbox{ as }}x \to \infty \hspace{0.5in}{\rm{and}}\hspace{0.5in}f\left( x \right) \to \infty {\mbox{ as }}x \to - \infty \]
E'tibor bering, oxirgi ikkita xususiyat ko'plab Hisoblash mavzularida juda muhim xususiyatdir va shuning uchun ularni doimo eslab qolishingiz kerak!

$f\left( x \right) = {{\bf{e}}^x}$, $g\left( x \right) = {{\bf{e}}^{ - x}}$ va $h\left( x \right) = 5{{\bf{e}}^{1 - 3x}}$ ni $x = - 2, - 1,0,1,2$ da baholang.

Yechim

Shunga qaramay, bu muammoning maqsadi bu turdagi funktsiyalarni baholashingiz mumkinligiga ishonch hosil qilishdir. Eslatib o'tamiz, bu masalalarda $e$ o'zgaruvchi emas, bu raqam! Aslida, ${\bf{e}} = 2.718281828 \ldots $

$h\left( x \right)$ ni hisoblashda 5 ga ko'paytirishdan OLDIN eksponentsiyani bajarganingizga ishonch hosil qiling.

	$\displaystyle x = - 2$	$\displaystyle x = - 1$	$\displaystyle x = 0$	$\displaystyle x = 1$	$\displaystyle x = 2$
$\displaystyle f\left( x \right)$	$\displaystyle 0.135335$	$\displaystyle 0.367879$	$\displaystyle 1$	$\displaystyle 2.718282$	$\displaystyle 7.389056$
$\displaystyle g\left( x \right)$	$\displaystyle 7.389056$	$\displaystyle 2.718282$	$\displaystyle 1$	$\displaystyle 0.367879$	$\displaystyle 0.135335$
$\displaystyle h\left( x \right)$	$\displaystyle 5483.166$	$\displaystyle 272.9908$	$\displaystyle 13.59141$	$\displaystyle 0.676676$	$\displaystyle 0.033690$

$f\left( x \right) = {{\bf{e}}^x}$ va $g\left( x \right) = {{\bf{e}}^{ - x}}$ grafigini chizing.

Yechim

Boshqa "eskiz" muammosida bo'lgani kabi, bu erda ham eskizni yaratish uchun oldingi misolda topilgan raqamlardan foydalanishdan boshqa narsa yo'q. Mana,

$The graphs of $f\left( x \right) = {{\bf{e}}^x}$ and $g\left( x \right) = {{\bf{e}}^{ - x}}$ on the same axis system for $ - 2 \le x \le 2$. Both graphs are always positive and both pass through $\left( {0,1} \right)$. The graph of $f\left( x \right) = {{\bf{e}}^x}$ is nearly flat along the x-axis on the left and increases steeply on the right, while the graph of $g\left( x \right) = {{\bf{e}}^{ - x}}$ is its mirror image, decreasing steeply from the upper left and flattening out along the x-axis on the right.$

E'tibor bering, ushbu grafiklardan $f\left( x \right) = {{\bf{e}}^x}$ va $g\left( x \right) = {{\bf{e}}^{ - x}}$ haqida quyidagi muhim xususiyatlarni ko'rishimiz mumkin.
\[\begin{align*} & {{\bf{e}}^x} \to \infty {\mbox{ as }}x \to \infty & \hspace{0.5in} & {\rm{and}} & \hspace{0.5in} & {{\bf{e}}^x} \to 0{\mbox{ as }}x \to - \infty \\ & {{\bf{e}}^{ - x}} \to 0{\mbox{ as }}x \to \infty & \hspace{0.25in} & {\rm{and}} & \hspace{0.25in} & {{\bf{e}}^{ - x}} \to \infty {\mbox{ as }}x \to - \infty \end{align*}\]
Bu xususiyatlar Kalkulyator kursida ma'lum bir muntazamlik bilan namoyon bo'ladi va shuning uchun eslab qolish kerak.

	\(\displaystyle x = - 2\)	\(\displaystyle x = - 1\)	\(\displaystyle x = 0\)	\(\displaystyle x = 1\)	\(\displaystyle x = 2\)
\(\displaystyle f\left( x \right)\)	\(\displaystyle f\left( { - 2} \right) = \frac{1}{{16}}\)	\(\displaystyle f\left( { - 1} \right) = \frac{1}{4}\)	\(\displaystyle f\left( 0 \right) = 1\)	\(\displaystyle f\left( 1 \right) = 4\)	\(\displaystyle f\left( 2 \right) = 16\)
\(\displaystyle g\left( x \right)\)	\(\displaystyle g\left( { - 2} \right) = 16\)	\(\displaystyle g\left( { - 1} \right) = 4\)	\(\displaystyle g\left( 0 \right) = 1\)	\(\displaystyle g\left( 1 \right) = \frac{1}{4}\)	\(\displaystyle g\left( 2 \right) = \frac{1}{{16}}\)
\(\displaystyle h\left( x \right)\)	\(\displaystyle h\left( { - 2} \right) = 16\)	\(\displaystyle h\left( { - 1} \right) = 4\)	\(\displaystyle h\left( 0 \right) = 1\)	\(\displaystyle h\left( 1 \right) = \frac{1}{4}\)	\(\displaystyle h\left( 2 \right) = \frac{1}{{16}}\)

	\(\displaystyle x = - 2\)	\(\displaystyle x = - 1\)	\(\displaystyle x = 0\)	\(\displaystyle x = 1\)	\(\displaystyle x = 2\)
\(\displaystyle f\left( x \right)\)	\(\displaystyle 0.135335\)	\(\displaystyle 0.367879\)	\(\displaystyle 1\)	\(\displaystyle 2.718282\)	\(\displaystyle 7.389056\)
\(\displaystyle g\left( x \right)\)	\(\displaystyle 7.389056\)	\(\displaystyle 2.718282\)	\(\displaystyle 1\)	\(\displaystyle 0.367879\)	\(\displaystyle 0.135335\)
\(\displaystyle h\left( x \right)\)	\(\displaystyle 5483.166\)	\(\displaystyle 272.9908\)	\(\displaystyle 13.59141\)	\(\displaystyle 0.676676\)	\(\displaystyle 0.033690\)