Suppose that you can calculate the derivative of a function using the formula f'(x) = 4f(x) + 2x. If the output value of the function at x= 4 is 2 estimate the value of the function at 4.013

Answer 1

I would estimate it using the first two terms of a McLaurin series expansion:
f(4.013) = f(4) + (4.013-4) f'(x)
= f(4) + 0.013 f'(4)
where f(4) = 2, and
f'(4) = 4f(4) + 2*4 = 16

So, f(4.013) = 2 + 0.013*16 = 2.208