Consider a function f which satisfies the following properties:
i) f(x+y) = f(x)f(y)
ii) f(0) not = 0
iii) f'(0)= 1
a) Let g be another function that satisfies properties (i)-(iii) and let k(x) = f(x)/g(x). Show that k is defined for all x (Is g(x) = 0 ever?) and find k'(x). Use this to discover the relationship between f and g.
b) For values smaller and smaller of h, evaluate ((e^h)-1)/h. Guess the limit as h---> 0.
c) Use the definition of the derivative and the above problem to guess f'(0) for f(x) = e^x
d) Can you think of a function which satisfies (i)-(iii)? Can be more than one function.
MY LOGIC:
a) How do I find what f(x) and g(x) are?
b) Getting the limit approaching 1 from using values such as(.01,.001, and .0001)
c) Using the definition of a derivative, I got from using factoring and algebra that f(0)=1. Since (b) has a limit approaching one, I can assume that f'(0) will be 1 also.
d) Anything that is not a quadratic centered on (0,0)?
i) f(x+y) = f(x)f(y)
ii) f(0) not = 0
iii) f'(0)= 1
a) Let g be another function that satisfies properties (i)-(iii) and let k(x) = f(x)/g(x). Show that k is defined for all x (Is g(x) = 0 ever?) and find k'(x). Use this to discover the relationship between f and g.
b) For values smaller and smaller of h, evaluate ((e^h)-1)/h. Guess the limit as h---> 0.
c) Use the definition of the derivative and the above problem to guess f'(0) for f(x) = e^x
d) Can you think of a function which satisfies (i)-(iii)? Can be more than one function.
MY LOGIC:
a) How do I find what f(x) and g(x) are?
b) Getting the limit approaching 1 from using values such as(.01,.001, and .0001)
c) Using the definition of a derivative, I got from using factoring and algebra that f(0)=1. Since (b) has a limit approaching one, I can assume that f'(0) will be 1 also.
d) Anything that is not a quadratic centered on (0,0)?