Question

Question in attachment.

Answer 1

The general solution of equations of the form $g'(x)=kg(x)$ is $g(x)=C*e^k^x$ for some constant C. 

This can be found using separation of variables. 

$\dfrac{dg}{dx} = kg$

$\dfrac{dg}{g}=kdx$ 

$\int\limits \dfrac{dg}{g} = \int\limits kdx$ 

$ln(IgI)=kx+c$

$e^l^n^(^I^g^I^)=e^{kx+c}$

$g=c\cdot e^k^x$   $Let \ C=e^c \geq 0$

In our case $k=8$ so $g(x)=C\cdot e^{8x}$ 

Let's use the fact that $g(2)=7$ to find C. 

$g(x)=c\cdot e^8^x$ 

$g(2)=c\cdot e^8^.^2$  $Plug \ x=2$

$7=c\cdot e^8^.^2$      $g(2)=7$ 

$7e^-^1^6 = c$ 

In conclusion, $g(x)=7e^{8x-16}$.