А) a) f'(x)=((4x-7)'(x^2+4) -(x^2+4)^2(4x-7)) /(x^2+4)^2)=
=(4x^2+16 -2x*4x +2x*7) / (x^2+4)^2)=(-4x^2+14x+16) /(x^2+4)^2);
F '(0)=16/4^2)=1
б) f '(x)=(3xtgx)'=(3x)' *tgx +(tgx)'*3x=3tgx+3x/cos^2 x;
f '(0)=3*tg0+3*0/cos^2 0=3*0+0=0;
b)f '(x)=(2e^x+3x)'=2e^x+3;
f '(ln2)=2e^ln2 +3=2*2+3=7;
c) f '(x)=(3x^2-lnx)'=3*2x-1/x=6x-1/x=(6x^2-1)/x;
f '(1)=(6*1-1)/1=5;
d)f '(x)=(√2(3x^2-1)^1/2)'=√2 *(1/2)*(3x^2-1)^(-1/2) *(3x^2-1)'=
=6x√2 /√(3x^2-1)
f '(1)=6*1*√2 /√(3*1-1)=6√2 /√2=6;
e) f '(x)=(6x/pi -5)* ctgx+( ctg x)' *(6x/pi -5)=
=6/pi *ctgx-(6x/pi-5)/sin^2 x;
f '(pi/2)=(6/pi *ctgpi/2) -((6pi/2)/pi -5) /sin^2 pi/2=0-(3-5)/1=2
2)
а)f(x)=3x-x^3; f '(x)=3-3x^2; k=f '(-2); k=3-3*(-2)^2=-9; k=9
б) f(x)=5x^2-3x+2; f '(x)=10x-3; f '(2)=20-3=17; k=17;
b)f '(x)=(3/x^2)'=(3x^(-2))'=3*(-2x^-3)=-6/x^3
f '(1)=-6/(-1)^3=6; k=6
g) (ln(2x+1)'=1/(2x+1) *(2x+1)'=2/(2x+1);
f '(0)=2/(2*0+1)=2; k=2
3)
a)f(x)=x^3-2x; x0=2
y=f(x0) +f '(x0)*(x-x0)-уравнение касательной!!!
f '(x)=3x^2-2
f '(2)=3*4-2=10; f(2)=2^3-2*2=8-4=4
y=4+10(x-2); y=10x-16;
б) f(x)=sin2x
f '(x)=2cos2x; f '(-pi/6)=2*cos(2*(-pi/6))=2*cospi/3=2*2* 1/2=1;
f(-pi/6)=sin(2*(-pi/6))=-sinpi/3=-√3/2
y=-√3/2 +(x+pi/6); y=x-√3/2+pi/6
g) f '(x)=(e^3x)'=3e^3x
f '(0)=3*e^(3*0)=3*1=3; f(0)=e^(3*0)=1
y=1+3*(x-0); y=3x+1 в) не видно!