# Thread: Wonderful Calculus Quiz problem

1. ## Wonderful Calculus Quiz problem

Hope someone can figure this one out because I am LOST. THANK YOU!!

Find the function f(t)=ce^(kt) that passes through the given points in the plane.

A) (0,5) and (1,1)

B) (2,4) and (3,7)

If you can give me A or B that would be more than awesome. Thanks again!

2. Hello, DJ Big T!

Find the function $\displaystyle f(t)\:=\:Ce^{kt}$ that passes through
these points: .$\displaystyle [A]\;\;(0,5)\text{ and }(1,1)$

The point (0,5) is on the graph.

. . Then: .$\displaystyle 5 \:=\:Ce^{k\cdot0} \quad\Rightarrow\quad C\:=\:5$

The point (1,1) is on the graph.

. . Then: .$\displaystyle 1 \:=\:Ce^{k\cdot 1} \quad\Rightarrow\quad 1 \:=\:5e^k \quad\Rightarrow\quad e^k \:=\:\tfrac{1}{5} \quad\Rightarrow\quad k \:=\:\ln(\tfrac{1}{5})$

The function is: .$\displaystyle \boxed{f(t) \;=\;5e^{(\ln\frac{1}{5})t}}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The answer can be further simplified:

. . $\displaystyle f(t)\;=\;5e^{(\ln\frac{1}{5})t} \;=\; 5\left(e^{\ln\frac{1}{5}}\right)^t \;=\;5\left(\frac{1}{5}\right)^t \;=\;5\cdot\frac{1}{5^t}$

. . $\displaystyle f(t) \;=\;\frac{1}{5^{t-1}}$