Find the function f(t) = ce^(kt)
A) (2,4) and (3,7)
Now is there anyone out there that can take this step by step please. I want to arrive at the answer with you.THANKS
$\displaystyle f(t) = Ce^{kt}$
you know $\displaystyle f(2) = 4$ ...
$\displaystyle 4 = Ce^{k \cdot 2} = C e ^{2k}$
you also know $\displaystyle f(3) = 7$ ...
$\displaystyle 7 = Ce^{k \cdot 3} = C e^{3k}$
set up a ratio using the two equations ...
$\displaystyle \frac{C e^{3k}}{C e ^{2k}} = \frac{7}{4}$
doing the division, note that C divides out to be 1, and the division on the left side simplifies to ...
$\displaystyle e^k = \frac{7}{4}$
natural log both sides ...
$\displaystyle k = \ln\left(\frac{7}{4}\right)$
you now have k ... go back to either of the first equations and solve for C.
$\displaystyle Ce^{2k} = 4$
$\displaystyle C(e^k)^2 = 4$
$\displaystyle C\left(\frac{7}{4}\right)^2 = 4$
$\displaystyle C = \frac{4^3}{7^2}
$