Exponential Function Calculator

Our exponential function calculator instantly calculates the function that passes from the two points on an x-y plane. Get to know whether the function is exponentially increasing or decaying with respect to time by using this tool.

What Is an Exponential Function?

A function in the form of f(t) = A0e^kt is known to be the exponent function. The exponential function passes through two given points in the x-y plane. You need to provide the values of two points in the x-y plane to calculate the values of the exponent function. The exponential calculator computes the exponent function by inserting the values of (t1,y1) and (t2,y2).

How to Find the Exponential Function?

Consider two functions (y1, y2)  and their respective values (4, 5) at the relative time (2,5). We want to evaluate their behavior at the time t = 5. 

Given:

Time 1 (t1): 2

y1 = Function at Time1: 4

Time 2 (t2): 5

y2 = Function at Time2: 5

The time to evaluate = 5

Solution:

The generic form of the exponential function is:

f(t) = A0e^kt

We need to solve the following equation:

$$ {y_1}={A_0e^{kt_1}}$$

$$ {y_2}={A_0e^{kt_2}}$$

The Exponential function can be evaluated by 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 evaluate the second step to find the values of k.

$$ \dfrac{y_1}{y_2} = \dfrac{e^{kt_1}}{e^{kt_2}} $$

$$ \dfrac{y_1}{y_2} = e^{kt_1}.e^{kt_2} $$

$$ \dfrac{y_1}{y_2} = e^{k(t_1 – t_2)} $$

$$ In ({\dfrac{y_1}{y_2}}) = In(e^{k(t_1 – t_2)}) $$

$$ In ({\dfrac{y_1}{y_2}}) = e.k(t_1 – t_2) $$

$$ k = \dfrac{1}{t_1 – t_2} In ({\dfrac{y_1}{y_2}}) $$

The exponential calculator can compute the rate of decay and growth of the functions.

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:

$$ k = \dfrac{1}{2- 5} In ({\dfrac{4}{5}}) $$

k = 0.0744

Now we have:

$$ A_0 = y_2e^{-kt_2}$$

$$A_0 = 5×e^{-0.0744×5}$$

Write an expression in exponential form calculator to evaluate the values of the exponential function and the graphical representation of it.

Step 5:

The final exponential function is:

$$f(t) = A_0e^{kt}$$

$$ f(t) = 3.4468e^{0.0744t}$$

Step 6:

Now you need to analyze the behavior of the exponential function at “5”.

$$f(5) = 3.4468e^{0.0744×5}$$

$$ f\left(5\right) = 5 $$

You can use the exponential equation calculator to validate the results in a matter of seconds.

Final Graph:

Working of Exponential Function Calculator:

Using this exponential function formula calculator requires the following inputs to calculate results:

Input:

• Enter the time and the values of the function
• Enter the second function values and their relative time
• Enter the point to evaluate (Optional)
• Hit the calculate button

Output:

• The step-by-step calculation of the exponential function
• The graphical representation of the exponential growth and decay

FAQs:

What are the Two Types of Exponential Functions?

The two types of exponential functions are exponential growth and exponential decay. The negative growth is represented by the exponential decay that can also be calculated by the exponential function calculator.

References:

From the source of the wikipedia.org: Exponential function, Graph

From the source of the mathinsight.org: Parameters of the exponential, The exponent