A function f is given by the formula f(x)= Ae^{kx} for constants A and k. We also know that f(3)=...

Question:

A function f is given by the formula {eq}f(x)= Ae^{kx} {/eq} for constants A and k. We also know that f(3)= 4 and f(7)= 11. Find numerical values for the constants A and k.

Simplifying criteria:

To simplify any mathematical expression is converting from one equal form to another such that the second form is less complex to the first one.

Following one rule which we used in the given example:

{eq}{{e^a} = y\,\,\, \Rightarrow a = \ln y} {/eq}

Given that: {eq}\displaystyle f(x) = A{e^{kx}} {/eq}

{eq}\displaystyle \eqalign{ & f(x) = A{e^{kx}} \cr & {\text{Given that;}} \cr & f(3) = 4,{\text{ }}f(7) = 11 \cr & \cr & f(3) = A{e^{k\left( 3 \right)}} = 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {f(3) = 4,{\text{ }}\left( 1 \right)} \right) \cr & f(7) = A{e^{k\left( 7 \right)}} = 11\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {f(7) = 11,{\text{ }}\left( 2 \right)} \right) \cr & \cr & {\text{From }}\left( 2 \right){\text{ dividing }}\left( 1 \right); \cr & \frac{{A{e^{k\left( 7 \right)}}}}{{A{e^{k\left( 3 \right)}}}} = \frac{{11}}{4} \cr & {e^{4k}} = \frac{{11}}{4}\,\,\,\, \Rightarrow 4k = \ln \left( {\frac{{11}}{4}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{e^a} = y\,\,\, \Rightarrow a = \ln y} \right) \cr & k = \frac{1}{4}\ln \frac{{11}}{4} \cr & \cr & {\text{From }}\left( 1 \right): \cr & A{e^{\left( {\frac{1}{4}\ln \frac{{11}}{4}} \right)\left( 3 \right)}} = 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {k = \frac{1}{4}\ln \frac{{11}}{4}} \right) \cr & A{e^{\frac{3}{4}\ln \frac{{11}}{4}}} = 4 \cr & A = 4{e^{ - \left( {\frac{1}{4}\ln \frac{{11}}{4}} \right)}} \cr} {/eq}