1. In what ways are the formulas for compound interest, exponential growth, exponential decay, a geometric sequence, and the exponential function similar?
2. Is there one formula that could be used for all scenarios? Which formula, and why?
3. How does the value of 'a' in y = c(a)x relate to another variable in each of the other formulas?
2006-10-15 06:57:49 · 2 answers · asked by mochaspice16 1 in Science & Mathematics ➔ Mathematics
1. The rate of change of each quantity is proporational to the quantity itself.
2. Yes, y(x) = C*exp(kx), where C and k are constants. For compound interest and exponential growth, k is positive. For exponential decay, k is negative. For a geometric sequence, k can be positive or negativee. Exponential growth with any base (2^x, 10^x, etc) can be described using the exponential function, using a^x = exp((ln a)*x). It can be used for all cases because dy/dx = k*y; that is, the rate of change of the function is proportional to its value, a property that all of these scenarios share.
3. This question is not very clear, in my opinion. In the formula from part 2, y = Ce^kx, C is obtained from the initial value (it is the initial value, if it corresponds to x=0).
2006-10-15 07:08:18 · answer #1 · answered by James L 5 · 0⤊ 0⤋
TLFH(too long for help)
2006-10-15 07:00:41 · answer #2 · answered by openpsychy 6 · 0⤊ 0⤋