Question

1)A function g is defined for all real numbers and has the following property:
g(a+b) - g(a) = 4ab+2b2 find g'(x)
A)4
B)-4
C)2x^2
D)4x
E)does not exist
not so sure where to start
2)If d/dx[f(x)] = g(x) and d/dx[g(x)] = f(3x, then d^2/dx^2 [f(x^2)] is
A)4x^2 f(3x^2) + 2g(x^2)
B)f(3x^2)
C)f(x^4)
D)2xf(3x^2) + 2g(x^2)
E)2xf(3x^2)
i was able to get (A) but not sure if its correct

3)lim as h-> 0 3(1/2 +h)^5 - 3(1/2)^5/h
A)0
B)1
C)15/16
D)limit does not exist
E)limit cannot be determined

thanks for the helps guys. ill really appreciate it

Answer 1

1)
Recall the definition of the derivative.

g'(x) = lim(h->0) (g(x+h)-g(x))/h
using the information in the definition of g, we have
= lim(x->0) (4xh + 2h^2)/h = 4x+2h = 4x
so, D

2)
d/dx f(x^2) = 2x d/dx f(x^2) = 2x g(x^2)
d/dx 2x g(x^2) = 2g(x^2) + 2x f(3x^2)
so, D

3)
this limit is just 3*d/dx(x^5) at x=1/2
= 3(5x^4) at x=1/2
= 3(5/16)
= 15/16
so, C