Find dx/dt with 3 given datapoints

Question

If t = 1 and x = 1.00; If t = 2 and x = 0.45; If t = 3 and x = 0.29;

How do you find dx/dt = ... if the function is assumed exponential?

Thanks!

Answer 1

Assume that $x=f(t)=A\cdot B^t+C$.

Then, $\displaystyle \frac{dx}{dt} = f'(t)=A \cdot B^t \cdot \ln B$

First, observe that $A \cdot B^1 +C=1, \quad A\cdot B^2+C=0.45, \quad A\cdot B^3+C=0.29$

This will give,

$A\cdot B= 1-C, \quad A\cdot B^2 =0.45-C, \quad A\cdot B^3 =0.29-C$.

But we know $\displaystyle \frac{A \cdot B^2}{A\cdot B} = \frac{A \cdot B^3}{A\cdot B^2}=B$.

So, $\displaystyle \frac{0.45-C}{1-C} = \frac{0.29-C}{0.45-C}=B$

From here, you can find the value of $C$.

Then you can find $B$ and $A$. Hence you can calculate $f'(t)$.