![]() ![]() % Compute y = a^t and its approximate derivative You can see that the derivative, in green, has the same shape as the function, in blue. ![]() Here is the code that generates the initial plot, with $a = 2$. It plots the graph of $a^t$ and its approximate derivative. I invite you to download the function and run it. #Matlab exponential softwareThe function expgui is included with the software for the book Experiments with MATLAB. If MATLAB can compute powers and roots, we can plot the graph of $2^t$. We do not have to be concerned yet about the definition of $2^t$ for irrational $t$. ![]() All floating point numbers are ratios of two integers. Theoretically, for floating point arithmetic, this is all we need to know. When I ask for the derivative of $3^t$, some will even apply the same process to get $t 3^ = 8.8250. Finding derivatives is a purely mechanical process, like adding fractions or solving quadratic equations. When I ask them "why?", they say "take the $3$, put it out in front, and subtract $1$ from the exponent". When I ask students to tell me the derivative of $t^3$, they can usually respond $3t^2$. I think that students who have taken calculus in high school or college should understand the mathematical ideas involved in exponential growth, but I'm afraid that most of them don't. The rate can even be negative, in which case it is "exponential decay". But, more precisely, a time varying quantity grows expontially if the rate of growth is proportional to size of the quantity itself. To most people "exponential growth" simply means "very rapid growth". ![]()
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