I want to know about exponential curve?

Question

my equation=a+b*e^(-t/T)

Answer 1

An exponential function diverges from a base value by a factor of e (or converges to a base or asymptote value, reducing the difference by a factor of 1/e) for each time interval known as a "time constant". In the equation you provided, the time constant is T, the base value is 'a', b is a scale factor for the rate of increase, and the time variable is t. This equation represents an asymptotic or converging function since the power of e is negative. In each successive interval of T, the function's difference from 'a' decreases by a factor of 1/e. For an example of a real-world application of this equation, consider a capacitor of C farads being charged from 0 volts by a 5-volt source through a resistor of R ohms. The time constant of charging is R*C seconds. The equation for the voltage V on the capacitor is V = 5 - 5 * e^(-t/(R*C)). When T = 0, e^(-t/(R*C)) = 1 and V = 0; when T is large, e^(-t/(R*C)) approaches 0 and V approaches 5.

Answer 2

f(t) = a+b*e^(-t/T)

1. When t = 0,
f(0) = a + b*e^(0/T)
= a + b*1
= a + b
So f(t) intercept is (0. a + b).

2. As t→∞ e^(-t/T) (= 1/e^(t/T)) →1/e^∞(= 0)
So as t→∞ f(t)→a + b*0 = a
ie as t increases f(t) is asymptotic to f(t) = a.

3. As t→-∞e^(-t/T) →e^+∞→∞
So as t→-∞ f(t)→∞
ie as t decreases f(t) increases without bound if b>0 and decreases without bound if b<0