ПОМОГИТЕ ПОЖАЛУЙСТА!
нужно решить номера 104, 105, 110 и 111!
спасибо заранее!
More Questions From This User See All
ПОМОГИТЕ ПОЖАЛУЙСТА!
нужно решить номера 104, 105, 110 и 111!
спасибо заранее!
Copyright © 2024 SCHOLAR.TIPS - All rights reserved.
Answers & Comments
Verified answer
104) 1) y = 2x^4 - x^3 + 3x + 4, y' = 8x^3 - 3x^2 + 32) y = -x^5 + 2x^3 - 3x^2 - 1, y' = -5x^4 + 6x^2 - 6x
3) y = 6*кор3(x) + 1/x^2 = 6*x^(1/3) + x^(-2), y' = 6/3*x^(-2/3) - 2x^(-3) = 2/кор3(x^2) - 2/x^3
4) y = 2/x^3 - 8*кор4(x) = 2*x(-3) - 8*x^(1/4), y' = -6*x^(-4) - 8/4*x^(-3/4) = -6/x^4 - 2/(кор4(x^3))
5) y = (2x + 3)^8, y' = 8(2x + 3)^7 * 2 = 16(2x + 3)^7
6) y = (4 - 3x)^7, y' = 7(4 - 3x)^6 * (-3) = -21(4 - 3x)^6
7) y = кор3(3x - 2) = (3x - 2)^(1/3), y' = 1/3*(3x - 2)^(-2/3) * 3 = 1/кор3(3x - 2)^2
8) y = 1/V(1 - 4x) = (1 - 4x)^(-1/2), y' = (-1/2)*(1 - 4x)^(-3/2) * (-4) = 2/V(1 - 4x)^3
9) y = sin (0,5x), y' = cos (0,5x) * 0,5 = 1/2*cos (0,5x)
10) y = cos (-3x) = cos(3x), y' = -sin (3x)*3 = -3sin (3x)
105) 1) y = e^x - sin x, y' = e^x - cos x
2) y = cos x - tg x, y' = -sin x - 1/cos^2 x
3) y = ctg x - кор3(x), y' = -1/sin^2 x - 1/(3*кор3(x^2))
4) y = 6x^4 - 9e^x, y' = 24x^3 - 9e^x
5) y = 5/x + 4e^x, y' = -5/x^2 + 4e^x
6) y = 1/(3x^2) + 1/2*ln x = -2/(3x^3) + 1/(2x)
110) 1) f(x) = sin x*cos x, x0 = pi/6
f'(x) = cos x*cos x + sin x*(-sin x) = cos^2 x - sin^2 x = cos (2x)
f'(x0) = cos (2*pi/6) = cos (pi/3) = 1/2
2) f(x) = e^x*ln x, x0 = 1
f'(x) = e^x*ln x + e^x*1/x = e^x*(ln x + 1/x)
f'(x0) = e^1*(ln 1 + 1/1) = e*(0 + 1) = e
3) f(x) = 2cos x/sin x = 2ctg x, x0 = pi/4
f'(x) = -2/sin^2 x
f'(x0) = -2/sin^2 (pi/4) = -2/(1/2) = -4
4) f(x) = x/(1 + e^x), x0 = 0
f'(x) = (1*(1 + e^x) - x*e^x) / (1 + e^x)^2
f'(x0) = (1 + e^0 - 0*e^0) / (1 + e^0)^2 = (1 + 1 - 0)/(1 + 1)^2 = 2/4 = 1/2
111) Уравнение касательной в точке x0 выглядит так: f(x) = y(x0) + y'(x0)*(x - x0)
1) y = x^2 - 2x, x0 = 3, y(x0) = 9 - 2*3 = 3
y' = 2x - 2, y'(x0) = 2*3 - 2 = 4
f(x) = 3 + 4(x - 3) = 4x - 9
2) y = x^3 + 3x, x0 = 3, y(x0) = 27 + 9 = 36
y' = 3x^2 + 3, y'(x0) = 3*9 + 3 = 30
f(x) = 36 + 30(x - 3) = 30x - 54
3) y = sin x, x0 = pi/6, y(x0) = sin pi/6 = 1/2
y' = cos x, y'(x0) = cos pi/6 = V(3)/2
f(x) = 1/2 + V(3)/2*(x - 1/2) = V(3)/2*x + (2 - V(3))/4
4) y = cos x, x0 = pi/3, y(x0) = cos pi/3 = 1/2
y' = -sin x, y'(x0) = -sin pi/3 = -V(3)/2
f(x) = 1/2 - V(3)/2*(x - 1/2) = -V(3)/2*x + (2 + V(3))/4