Question

If f is continuous for all real numbers, dy/dx=f(x) and y(2)=4 then y(x)=
a) 4+ \[\int\limits_{2}^{x}\] fprime(t) dt
b) 4+ \[\int\limits_{2}^{x}\] f(t) dt
c) \[\int\limits_{2}^{x}\] f(t) dt-4
d) 4- \[\int\limits_{2}^{x}\] f(t) dt

Answer 1

Is the first line supposed to contain a capital F somewhere?

Answer 2

what do you mean?

Answer 3

All of your options contain F
but the information says f, it doesn't make sense.

Answer 4

Oh you fixed it :P k

Answer 5

haha yeah sorry about that

Answer 6

So if \(\large\rm y'=f(x)\)
We integrate to get
\(\large\rm y=\int f(x)dx+c\)
We would then use the data the provided to solve for c,
\(\large\rm 4=\int f(2)dx+c\)
solving for c gives us,
\(\large\rm c=-\int f(2)dx+4\)
plugging this back into our general solution gives us,\[\large\rm y=\int\limits f(x)dx+c=\int f(x)dx-\int f(2)dx+4\]

Answer 7

And I guess we would want to rewrite that as a single integral,
using some dummy variable like t,\[\large\rm y(x)=\int\limits_2^x f(t)dt+4\]
Mmm sorry to do the whole problem for you,
this one was confusing me a lil bit though,
needed to write out the steps to see it.

Answer 8

Any confusion?

Answer 9

Not at all! Thank you!