How do you find the average rate and instantaneous rate given a kinetics plot?

What is average rate? What is instantaneous rate? How do you find these? My teacher described finding the tangent to a graph. I am not sure what is meant by this.




This is more of a question of calculus than chemistry, but in a first order reaction 
$\ce{A -> B}$, the average rate is:


$$\frac{\Delta A}{\Delta t} = \frac{A\_2 - A\_1}{t\_2 - t\_1}$$


![enter image description here](https://i.stack.imgur.com/sXmHq.png)


 Graciously borrowed from [here](http://science.widener.edu/~svanbram/chem146/ch15/lm_c15_graphs.pdf)


So, the average rate is the slope of a secant line between the two points, 


$$ \frac {\Delta A}{\Delta t} = \frac{1 - 0.5}{400 - 200}$$


The instantaneous rate is the derivative of the plot:


$$\frac{dA(t)}{dt}$$ or $$\lim\_{t\rightarrow 0} \frac{\Delta A}{\Delta t}$$


For the above first order rate law, the equation is in the form:


$$ A(t) = e^{-kt + ln(A\_0)} $$ so the derivative is simply


$$ \frac{dA(t)}{dt} = -ke^{-kt + ln(A\_0)} $$


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