Question

If f and g are continuous functions such that then in terms of g is:

Answer 1

g'(x)=f(4x)

Answer 2

no i mean g'(x)=4f(x)

Answer 3

then \[\int\limits_{1}^{2}f(2x)dx\] in terms of g is

Answer 4

So there's no confusion...

If f and g are continuous functions such that g'(x)=4f(x), then \[\Large \int\limits\limits_{1}^{2}f(2x)dx \] in terms of g is?

Answer 5

correct

Answer 6

g'(x)=4f(x)

Since the derivative of g is 4f, then the integral of 4f(x) must be g(x) (i pulled out the constant 4)
\[\Large \int\limits g'(x) dx= 4 \int\limits f(x) dx\] or using limits
\[\Large 4 \int\limits_{a}^{b} f(x) dx = g(b) - g(a)\]
divide both sides by 4
\[\Large \int\limits\limits_{a}^{b} f(x) dx =\frac{ 1 }{4 } \left[ g(b) - g(a) \right]\]

Answer 7

Now... lets make a substitution first.

u = 2x
so
du = 2 dx or du/2 = dx

Change the limits using u = 2x. When x=2, u = 4. When x=1, u = 2.

\[\Large \frac{ 1}{ 2}\int\limits\limits\limits\limits\limits\limits\limits_{2}^{4}f(u)du = \frac{ 1 }{ 2 } \times \frac{ 1 }{4 }\left[ g(4) - g(2) \right]\]