Suppose f(x) =a and f′(x) =b, estimatef(x+h).

Answer 1

f(a+h) ≈ f(a) + f'*h = f(a) + bh

Draw a picture, showing the tangent line at x=a and the secant line joining f(a) and f(a+h)

The slope of the secant line is exactly ∆y/∆x = (f(a+h)-f(a))/h
The slope of the tangent line is, of course, the limit as h=0.
Go back and review the first principles of derivatives; there will be illustrations showing this.