Aká je rovnica dotyčnice f (x) = e ^ (x) / (x ^ 2-x pri x = 3?

odpoveď:

# Y = e ^ 3 / 36x + e ^ 3/12 #

# F (x) = e ^ x / (x ^ 2-x) #

# D_f = {AAX ## V ## RR ## !: X ^ 2-x = 0} = (- oo, 0) uu (0,1) uu (1, + oo) = RR-{0,1} #

# F '(x) = (e ^ x / (x ^ 2-x))' = ((e ^ x) '(x ^ 2-x) -e ^ x (x ^ 2-x)') / (x ^ 2-x) ^ 2 = #

# (E ^ x (x ^ 2x) -e ^ x (2x-1)) / (x ^ 2 x) ^ 2 = (x ^ 2e ^ x-xe ^ x-2xe ^ x + e ^ x) / (x ^ 2-x) ^ 2 = #

# (X ^ 2e ^ x-3XE ^ x + e ^ x) / (x ^ 2-x) ^ 2 #

Pre rovnicu dotyčnice na #A (3, f (3)), # požadujeme hodnoty

# F (3) = e ^ 3/6 #

# F '(3) = (9e ^ 3-9e ^ 3 + e ^ 3) / 36 = e ^ 3/36 #

Rovnica bude

# Y-f (3) = f '(3), (X-3) # #<=>#

# R-e ^ 3/6 = e ^ 3/36 (X-3) # #<=>#

# R-e ^ 3/6 = e ^ 3 / 36x-zrušiť (3) e ^ 3 / zrušenie (36) # #<=>#

# Y = e ^ 3 / 36x-e ^ 3/12 + e ^ 3/6 # #<=>#

# Y = e ^ 3 / 36x + e ^ 3/12 #

a graf