The exponential feature is believed to be a property in the form of \(f(t) = A0e^{kt}\). The exponential characteristic passes via two given factors in the x-y aircraft. You want to allow the values in the calculator to determine the exponent characteristic. By setting the values of (t1,y1) and (t2,y2), the exponential calculator finds the exponent characteristic.
On the relative time (three, 6), consider two functions (y1, y2) and their respective values (6, 7). Their behavior at t = 6 has to be judged. .
Time 1 (t1): 3
y1 = Function at Time1: 6
Time 2 (t2): 6
y2 = Function at Time2: 7
The time to evaluate = 6
TThe usual form of the exponential function is:
f(t) = A0e^kt
We want to remedy the following equation:
\({y_1}={A_0e^{kt_1}}\)
\({y_2}={A_0e^{kt_2}}\)
The Exponential feature can be evaluated by using the following steps:
Step 1:
In the first step, divide y1 and y2 to cancel A0.
\(\dfrac{y_1}{y_2} = \dfrac{A_0e^{kt_1}}{A_0e^{kt_2}}\)
\( \dfrac{y_1}{y_2} = \dfrac{\require{cancel}\cancel{A_0}e^{kt_1}}{\require{cancel}\cancel{A_0}e^{kt_2}}\)
\(\dfrac{y_1}{y_2} = \dfrac{e^{kt1}}{e^{kt_2}}\)
Step 2:
You need to assess the second one step to find the values of k.
\(\dfrac{y_1}{y_2} = \dfrac{e^{kt_1}}{e^{kt_2}}\)
\(\dfrac{y_1}{y_2} = e^{kt_1}.e^{kt_2}\)
\(\dfrac{y_1}{y_2} = e^{k(t_1 - t_2)}\)
\(In ({\dfrac{y_1}{y_2}}) = In(e^{k(t_1 - t_2)})\)
\(In ({\dfrac{y_1}{y_2}}) = e.k(t_1 - t_2)\)
\(k = \dfrac{1}{t_1 - t_2} In ({\dfrac{y_1}{y_2}})\)
Step 3:
\(A_0e^{kt_1}\)
Or
\(A_0 = y_1e^{-kt_1}\)
\(A_0 = y_1e^{-({\dfrac{1}{t_1 - t_2} In ({\dfrac{y_1}{y_2}})})t_1}\)
\(A_0 = \require{cancel}\cancel{y_1} × \dfrac{y_2}{\require{cancel}\cancel{y_1}e^{kt_2}}\)
\(A_0 = y_2e^{-kt_2}\)
Step 4:
\(k = \dfrac{1}{3 - 6} In ({\dfrac{6}{7}})\)
k = 0.1309
Now we have:
\(A_0 = y_2e^{-kt_2}\)
\(A_0 = 7 × e^{-0.1309 × 6}\)
Step 5:
The final exponential function is:
\(f(t) = A_0e^{kt}\)
\(f(t) = 4.0428e^{0.1309t}\)
Step 6:
Now you want to investigate the conduct of the exponential function at t = 6.
\(f(6) = 4.0428e^{0.1309×6}\)
\(f(6) = 7\)
you could use the exponential equation calculator to validate the results in a count of seconds.
The two styles of exponential functions are exponential boom and exponential decay. The terrible growth is represented with the aid of the exponential decay that also can be calculated with the aid of the exponential feature calculator.
