What is the difference between average slope and Instant slope(Instantaneous Rate of Change)

Question

I'm starting to learn calculus, and I'm getting confused about what average slope and instant slope(instantaneous rate of change)do and what they're differences are after looking at several sources on the internet. I know that average slope is $\frac{Δy}{Δx }$ and instant slope is $\frac{dy}{dx}$. Are these formulas correct? And if they are, what difference is there between Δy and dy?

Answer 1

$\Delta y$ and $\Delta x$ represent actual numbers. If you have two points on the graph of a function, then $\Delta y$ is the change in their $y$-coordinates, and $\Delta x$ is the change in their $x$-coordinates. So, when you divide change in $y$ by change in $x$, i.e. $\frac{\Delta y}{\Delta x}$, you get the slope of the line that connects them. As you move the points closer together (i.e. make $\Delta x$ smaller and smaller), the line no longer connects two points, but becomes a line tangent to the graph of the function. The slope of that line is written as $\frac{dy}{dx}$, where here we think of $\frac{dy}{dx}$ as a single symbol, and not a fraction. Take a look at this Desmos link for a way to visualize this. The $h$ slider controls $\Delta x$, so as you make $h$ smaller and smaller you can see the slope $\frac{\Delta y}{\Delta x}$ get closer to 2, which is $\frac{dy}{dx}$, the slope of the tangent line at $x = 1$.