y(x0) = 2-5+3=1
y'=2*3x^2-5*2x+3 = 6x^2-10x+3
y'(1) = 6-10+3=-4+3=-1
y=1+(-1)*(x-1)= 1-x+1=2-x
ответ:
) а) f(x) = 1/5x5 - x3 + 4.
f'(х) = 1/5 * 5 * х4 – 3х² = х4 – 3х².
б) f(x) = (3x – 1)/x3.
производная произведения: (f * g)' = f' * g + f * g'.
f'(х) = (3x – 1)' * x3 + (3x – 1) * (x3)' = 3 * x3 + (3x – 1) * 3x² = 3x3 + 9x3 – 3x² = 12x3 – 3x².
в) f(x) = 1/(2cosx).
производная дроби: (f/g)' = (f' * g - f * g')/g^2.
f'(х) = (1' * 2cosx - 1* (2cosx)')/( 2cosx)^2 = (0 - 1* (-2sinx))/2cos²x = sinx/cos²x.
2) а) f(x) = xsinx.
f'(х) = х' * sinx + х * (sinx)' = sinx + хcosx.
x = п/2; f'(п/2) = sinп/2 + п/2cosп/2 = 1 + п/2 * 0 = 1.
б) f(x) = (2x - 3)6.
f'(х) = 6(2х – 3)5 * (2х – 3)' = 6(2х – 3)5 * 2 = 12(2х – 3)5.
х = 1; f'(1) = 12(2 * 1 – 3)5 = 12 * (-1)5 = -12.
3) а) f(x) = 2sinx – x.
f'(х) = 2cosx – 1.
f'(х) = 0; 2cosx – 1 = 0.
2cosx = 1.
cosx = ½.
х =±п/3 + 2пn, n – целое число.
b) f(x) = x5 + 20x².
f'(х) = 5х4 + 20х.
f'(х) = 0; 5х4 + 20х = 0.
х(5х3 + 2) = 0.
отсюда х = 0.
или 5х3 + 2 = 0; 5х3 = -2; х3 = -2/5; х = 3√(-2/5).
объяснение:
у = f(x°) + f'(x°) × ( x - x° )
f(x°) = f(1) = 2×1³ - 5×1² + 3×1 = 2 - 5 + 3 = 0
f'(x°) = ( 2x³ - 5x² + 3x ) ' = 6x² -10x +3
f'(1) = 6×1² - 10×1 + 3 = 6 - 10 + 3 = -1
y = 0 - 1 × ( x - 1 ) = -x + 1
ответ: у = -х + 1