What is meant by "increases exponentially with time"?

Question

The following qeustion states "you may assume Phoebe's speed increases exponentially with time", but only provides 2 data points from which to derive the model. Is this question lacking enough constraints (data points)? As I am sure I can create many exponential models that pass through those 2 points ($(0,4230)$ and $(10,6850)$). Or, does the statement "increases exponentially with time" costrain the model to the general form $f(x)=a*b^x$? My gut feeling was that $f(x)=a*b^x+c$ would also be valid form per the question's verbage.

Phoebe's Next Rocket Problem Phoebe Small is out Sunday driving in her rocket ship. She fills up with fuel at the Scorpion Gulch Rocket Fuel Station, and takes off. When she starts the last stage of her rocket, she is going 4230 miles per hour. Ten seconds later she is going 6850 mph. While the last stage is running, you may assume Phoebe's speed increases exponentially with time. Derive a model, $f(x)$, for the rocket's velocity while the last stage is running.

Answer 1

The function $f(x)=a\cdot b^x+c$ is not exponential. In fact, if $c$ is quite large it hardly increases at all. You are expected to assume $f(x)=a\cdot b^x$. Plug the $x,f(x)$ values into this to give two equations in two unknowns, $a,b$ and solve them simultaneously.

Answer 2

The statement "increases exponentially with time" does indeed constrain the model to the general form $ f (x) = a \cdot b^x $. A possible definition of "increases exponentially with time" would be that $ f (x) $ solves a differential equation of the form: $$ \forall x, \quad f' (x) = \beta \cdot f(x) $$ for some value of the parameter $\beta$. Your first model may be written as: $$ f(x) = a \cdot b^{x} = a \cdot \left( e^{\log b} \right)^x = a \cdot e^{\log b \cdot x} $$ which indeed solves the above differential equation for $ \beta = \log b $. On the other hand, for your second model the derivative $ f' (x) $ is not proportional to $ f ( x ) $ because the constant term drops when we take the derivative. Thus, by this definition, your second model would not be considered to increase exponentially with time.