Question

f′(x)=5f(x)+3x. If the output value of the function at x=1 is 6 estimate the value of the function at 1.006.

Answer 1

I don't know how to do it

Answer 2

you know that for small ∆x,
∆y/∆x ≈ dy/dx
so,
∆y ≈ y' ∆x
Here you have y' = 5y+3x, so at (1,6), y' = 33
then, taking ∆x = 0.006
∆y ≈ 33 * 0.006 = 0.196
That means that
y(1.006) = y(1) + ∆y ≈ 6.196