# Wonderful Calculus Quiz problem

• Nov 3rd 2008, 08:17 AM
DJ Big T
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!
• Nov 3rd 2008, 09:10 AM
Soroban
Hello, DJ Big T!

Quote:

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

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

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

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

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

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

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The answer can be further simplified:

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

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