I'm going through Morris Kline's "Calculus, an Intuitive and Physical Approach".
One of the problems is about Maginal Cost, and it states (page 126, ex 11): If the cost C of producing $x$ units of an article is $50+3x+2x^2$, what is the MC when $x=50$? What is the cost of producing the $51^{st}$ unit?
Just taking the derivative of our C function, we get $C'= 4x+3$, which gives us $C'(50)=203$.
Now in order to get the price of the $51^{st}$ unit, I subtract $C(50)$ from $C(51)$, and I get $5405-5200=205$, so the price of the $51^{st}$ unit is not exactly given to us by the rate of change at $x=50$, which intuitively makes sense to me since our $C$ function is not linear and we're considering discrete intervals (units).
However, in the book solutions it is clearly stated "When $x=50$, $\frac{dC}{dx}=203$. This is the cost of producing the $51^{st}$ unit."
It seems clear to me that "This" refers to 203. There is no mention anywhere of the number 205.
So I'm confused. Is this an error in the text, and if not, where did I go wrong? Why is my intuition not correct and how is the rate of change over a discrete interval giving us the exact value of a continuous function at a point outside of that interval?