Mathematics

Problem

Let

f

and

g

be differentiable functions on an interval

I

, and consider the function

h(x)=3f(x)-7g(x)

. (a) Provide a clear verbal justification, based on properties of derivatives and integrals, that an antiderivative of

h

can be constructed from antiderivatives of

f

and

g

. Your explanation should cite the relationship between differentiation and antidifferentiation without merely restating definitions. (b) Suppose

F'(x)=f(x)

and

G'(x)=g(x)

for all

x

I

. Write a general expression for an antiderivative

H

h

in terms of

F

G

, and any necessary constant(s) of integration.