Algebra vs. Calculus

Algebra

Calculus

Average rate of change= Slope between two points .

Instantaneous rate of change = Slope at single point .

Slope of a secant line, a line connecting two points,

$(x_1,y_1)$and $(x_2,y_2)$.

You can see that this is truly an average rate of change of the function, $f\mathrm{(}x\mathrm{)}$, because the actual graph increases and decreases (changes) quite a bit between those two points, but the average change is increasing, going up.

Slope of the tangent line, a line that touches your equation at a single point, $(x_1,y_1)$.

You can see that the instantaneous rate of change of the function can be quite different at different moments in time, different $(x_1,y_1)$. Here the change is going up, meaning increasing.

Finding the slope of the tangent line (a line that touches the actual function at just one point, $(x_1,y_1)$) allows you to see what change is happening to the actual function at one instant in time, $(x_1,y_1)$.

The equation for finding the slope of the secant line or the average rate of change:

$\mathit{average }\mathit{rate\; of\; change}=\mathit{slope}=m=\frac{y_2–y_1}{x_2–x_1}={\overline{)\frac{f\mathrm{(}x+h\mathrm{)}–f\mathrm{(}x\mathrm{)}}{h}}}^{\mathit{New\; Heavy\; Math},\mathit{ but\; still\; the\; same\; thing}.}$

The equation for finding the slope of the tangent line or instantaneous rate of change:

$\mathit{instantaneous }\mathit{rate\; of\; change}=\mathit{slope}=m=f^{\prime }\mathrm{(}x\mathrm{)}=\frac{\mathit{dy}}{\mathit{dx}}={\overline{)\underset{h\to 0}{\lim }\frac{f\mathrm{(}x+h\mathrm{)}–f\mathrm{(}x\mathrm{)}}{h}}}^{\mathit{Heavy\; Math},\mathit{ but\; still\; the\; same\; thing}.}$

Slope means Slope

No matter which rate of change you are finding you are always finding the slope of a line, the m , from your old friend $y=\mathit{m}x+b$.

Equation for finding the slope between two points.

$m=\frac{y_2–y_1}{x_2–x_1}$

Equation for finding the slope between two points.

$m=f\prime \left(x\right)$

The major difference is that our $f\prime \left(x\right)$, our derivative equation, will vary from $f\mathrm{(}x\mathrm{)}$ to $f\mathrm{(}x\mathrm{)}$.

So, if someone asks you to find:

–          Slope of the secant line

–          Average rate of change

You would want to use your classic slope formula to find the value.

$m=\frac{y_2–y_1}{x_2–x_1}$

If someone asks you to find:

–          Slope of the tangent line

–          Instantaneous rate of change

You would need to find the derivative, $f\prime \left(x\right)$, and then use that derivative equation to find the value.

$m=f\prime \left(x\right)$