1. Functions with points

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

2. $\displaystyle f(t) = Ce^{kt}$

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.

$\displaystyle Ce^{2k} = 4$

$\displaystyle C(e^k)^2 = 4$

$\displaystyle C\left(\frac{7}{4}\right)^2 = 4$

$\displaystyle C = \frac{4^3}{7^2}$

3. so it would be y = 64/49*e^(ln7/4)t as the answer