a function whose derivative is a constant multiple of itself must be?

Question

a function whose derivative is a constant multiple of itself must be:
a. periodic
b. linear
c. exponential ( i was thinking it was this one cause the derivative of e^x is itself????? anyone know)
d. quadratic
e. logarithmic

Answer 1

c. exponential ( i was thinking it was this one cause the derivative of e^x is itself????? anyone know)


Good instinct.
y=e^ax
y'=ae^ax
y"=a^2e^ax etc.

Answer 2

c. If f'(x)=cf(x), then f'(x)/f(x)=c and by integration, ln f(x) = cx + d, where d is a constant.
Exponentiating on both side, you get: f(x) = exp(cx)*exp(d).

There is also the particular solution f=0 which can be seen as the limit case of taking d=-infinity.

Answer 3

c) because it is a theorem: if y'=k*y then y(t) =y(0)* e^(kt)

ALTHOUGH there is another possibility, y(t)=0 for all t. This is exponential as much as it is periodic, logarithmic, linear, quadratic...........

Answer 4

d. quadratic.

derivative of kx^2 + lx + m = 2kx +l , where k, l and m are contants.