ordinary differential equations - Euler Method - Why is my intuition incorrect?

$$\dfrac{dA}{dt} = 0.5 \times A \times (1 - \dfrac{A}{100}) - 10$$ with $A(0) = 70$ and we want to use Euler's method to get an approximate value for $A(10)$, with a step size of 1.

So the answer sheet says you basically have to use $\text{Ans} + 0.5 \times \text{Ans} \times (1-\dfrac{\text{Ans}}{100}) - 10$ with the first $\text{Ans}$ being $70$, and then of course repeat 10 times.

But I'm wondering, doesn't this actually give you $\dfrac{dA(10)}{dt}$? How is this a correct method?

Answer

The Euler method does not give you $\frac {dA}{dt}$. You give it a formula for $\frac {dA}{dt}$, such as the one in your question. Then from any given point, like your start of $(0,70)$ it puts a straight line through the point with slope $\frac {dA}{dt}$ of that point. From your expression, $\frac {dA}{dt}|_{(0,70)}=0.5$ so we step one unit in $t$ at a slope of $0.5$, giving the $A$ value of the next point as $70+0.5\cdot 1=70.5$. Now we are at $(70.5,1)$, we calculate $\frac {dA}{dt}$ at this point and take another step along the $t$ axis, and so on until we get to $t=10$