a) f(x)=sin(3x)
f′(x)=3cos(3x)
b) g(x)=e2x⋅cosx
g′(x)=2e2x⋅cosx+e2x⋅(−sinx)=e2x(2cosx−sinx)
a) ∫2x⋅sin(x2)dx
La u=x2
⇒dudx=2x⇒du=2x dx
∫2x⋅sin(x2)dx=∫sinudu=−cosu+C=−cos(x2)+C
b) ∫1ex⋅lnxdx
La u=lnx og v′=x:
∫1ex⋅lnxdx=[lnx⋅12x2−∫1x⋅12x2]1e=[12x2⋅lnx−12∫xdx]1e=[12x2⋅lnx−12⋅12x2]1e=12[x2⋅lnx−12x2]1e=12((e2⋅lne−12⋅e2)−(12⋅ln1−12⋅12))=12((e2−12⋅e2)−(0−12))=12(e22+12)=12⋅e2+12=e2+14
f(x)=e2x−4ex , Df=\R
f′(x)=2e2x−4ex
f″(x)=4e2x−4ex
f″(x)=04e2x−4ex=04(ex)2−4ex=04ex(ex−1)=0ex−1=0ex=1x=0
Vendepunkt: (0 , f(0))=(0 , e2⋅0−4e0)=(0 , 1−3)=(0 , −3)