§ 38: 38.2

38.2 Складіть рівняння дотичної до графіка функції f у точці з абсцисою x₀, якщо:

1) f(x₀) = f(1) = 2 • 1² – 3 • 1 = 2 – 3 = –1; f(x) = 2 • 3x² – 3 = 6x² – 3;

f(x₀) = f’(1) = 6 • 1² – 3 = 3; у = 3(х – 1) – 1; у = 3х – 3 – 1;

у = 3х – 4.

2) f(x₀) = f(π2) = cosπ2 = 0; f'(x) = (cosx)’ = –sinx;

f’(x₀) = f’(π2) = –sin(π2) = –1; y = –1(x – π2) + 0;

y = –x + π2.

3) f(x₀) = f(–1) = 4•(-1)2+3•(-1) = 4-3 = 1 = 1.

f’ = 124x2+3x • (4x² + 3x)’ = 8x+324x2+3x;

f’(x₀) = f’(–1) = 8 • -1+324 • 12+3 • (-1) = -8+324-3 = -52 = –2,5;

y = –2,5(x + 1) + 1 = –2,5x – 2,5 + 1; y = –2,5x – 1,5.

4) f(x₀) = f(3) = 32- 4 • 33-2 = 9-121 = –3;

f’(x) = x2 -4x'x-2-(x2 -4x)(x-2)'(x-2)3 = 2x-4x-2-x2 +4x(x-2)3 = 2x2-4x-4x+8-x2 +4x(x-2)3 = x2- 4x+8(x-2)3; f’(x₀) = f’(3) = 32-4 • 3+8(3-2)2 = 9 – 12 + 8 = 5; y = 5(x – 3) – 3; y = 5x – 15 – 3; y = 5x – 18.