Найдите уравнение общей касательной к графикам функций F(x)=x^2+3x+5 и g(x)=x^2+4x-3. Created by keizdub. algebra-ru

1)F(x)=x²+3x+5F '(x)=2x+3Пусть a - абсцисса точки касанияF(a)=a²+3a+5F ' (a)=2a+3y₁=a²+3a+5+(2a+3)(x-a)=a²+3a+5+2ax+3x-2a²-3a= =x(2a+3)+(-a²+5) - уравнение касательной.2) g(x)=x²+4x-3 g ' (x)=2x+4Пусть c - абсцисса точки касания.g(c)=c²+4c-3g ' (c)=2c+4y₂= c²+4c-3+(2c+4)(x-c)=c²+4c-3+2cx+4x-2c²-4c= = x(2c+4)+(-c² -3) - уравнение касательной.3) Так как касательная общая, то{2a+3=2c+4 {2a-2c=4-3 {2(a-c)=1 {a-c=1/2{-a²+5= -c²-3 {c²-a²= -3-5 {a² - c² =8 {(a-c)(a+c)=8{a-c=1/2 {a-c=1/2{(1/2)*(a+c)=8 {a+c=16Складываем уравнения системы:2a=16+ (1/2)2a=33/2a=33/433/4 -c=1/2c=33/4 - 1/2c=31/4y=(2 * (³³/₄) + 3)x + (5 - (³³/₄)²) = (³³/₂ + 3)x +(5 - ¹⁰⁸⁹/₁₆)= =³⁹/₂ x - ¹⁰⁰⁹/₁₆=19.5x-63.0625y=19.5x - 63.0625 - общая касательная Ответ: у=19,5х - 63,0625 Новые вопросы в Алгебра52-0= (-61) - (- 12) = 66-76= 85 - (-16) = 95-45= 10-5= (-66) - (-38) = 55 - (-18) = (-31) - 52= (-24)-(-41) = (-84) + 75 =90+ (-77) = (-46) - 69 = (-25) - (-70) = (-6) - (-85) = 18+ (-68) = (-86) + 71= (-82) - (-58) = (-81) + 70 = (-16) + (-38) = 49- (-31) = 89-60= 85-26=…0 + (-3) = 8- (-98) = 14+ (-19) = 82 - 18 = 94 + 12 = 53 + 12 = (-81) + 99 = (-59) + 8 = 77- (-23) = 53 + (-73) =РЕШЕНИЕ СРОЧНО!!!!!!!!РЕШЕНИЕ ПОЖАЛУЙСТА!!!!!!!!Допоможіть будь ласка, це срочно. Даю 35 балівДоказать что : фото снизуУпростить выражениеа+в+а^2-в^2 розкласти на множникивычислить 4 корень 0,4^4 умножить5^8Помогите решить это вычеслениеПредыдущийСледующийВместе мы знаем большеЭтот сайт использует cookies Политика Cookies . 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Найдите уравнение общей касательной к графикам функций F(x)=x^2+3x+5 и g(x)=x^2+...

keizdub @keizdub

August 2022 1 7 Report

Найдите уравнение общей касательной к графикам функций F(x)=x^2+3x+5 и g(x)=x^2+4x-3

Answers & Comments

m11m

Verified answer

1)
F(x)=x²+3x+5
F '(x)=2x+3
Пусть a - абсцисса точки касания

F(a)=a²+3a+5
F ' (a)=2a+3

y₁=a²+3a+5+(2a+3)(x-a)=a²+3a+5+2ax+3x-2a²-3a=
=x(2a+3)+(-a²+5) - уравнение касательной.

2) g(x)=x²+4x-3
g ' (x)=2x+4

Пусть c - абсцисса точки касания.
g(c)=c²+4c-3
g ' (c)=2c+4

y₂= c²+4c-3+(2c+4)(x-c)=c²+4c-3+2cx+4x-2c²-4c=
= x(2c+4)+(-c² -3) - уравнение касательной.

3) Так как касательная общая, то
{2a+3=2c+4 {2a-2c=4-3 {2(a-c)=1 {a-c=1/2
{-a²+5= -c²-3 {c²-a²= -3-5 {a² - c² =8 {(a-c)(a+c)=8

{a-c=1/2 {a-c=1/2
{(1/2)*(a+c)=8 {a+c=16

Складываем уравнения системы:
2a=16+ (1/2)
2a=33/2
a=33/4

33/4 -c=1/2
c=33/4 - 1/2
c=31/4

y=(2 * (³³/₄) + 3)x + (5 - (³³/₄)²) = (³³/₂ + 3)x +(5 - ¹⁰⁸⁹/₁₆)=
=³⁹/₂ x - ¹⁰⁰⁹/₁₆=19.5x-63.0625
y=19.5x - 63.0625 - общая касательная

Ответ: у=19,5х - 63,0625