E to the X Calculator

Exponential Function:

\[ e^x = \exp(x) \]

Result (e^x):

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1. What is the Exponential Function?

The exponential function e^x (where e is Euler's number ≈ 2.71828) is a fundamental mathematical function that describes growth or decay processes. It appears in many areas of mathematics, physics, and engineering.

2. How Does the Calculator Work?

The calculator computes the exponential function:

\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

Where:

\( e \) — Euler's number (~2.71828)
\( x \) — The exponent (input value)

Explanation: The function calculates e raised to the power of x, which can represent continuous growth when x is positive or decay when x is negative.

3. Importance of Exponential Function

Details: The exponential function is crucial in modeling population growth, radioactive decay, compound interest, and many natural phenomena. It's also the inverse of the natural logarithm function.

4. Using the Calculator

Tips: Simply enter any real number as the exponent x. The calculator will compute e^x with high precision.

5. Frequently Asked Questions (FAQ)

Q1: What is the value of e?
A: e (Euler's number) is approximately 2.71828 and is the base of natural logarithms.

Q2: What are some special values of e^x?
A: e^0 = 1, e^1 ≈ 2.71828, e^-1 ≈ 0.36788, and e^(ln(2)) = 2.

Q3: How is e^x related to compound interest?
A: e^x appears in continuous compounding formulas: A = Pe^(rt), where P is principal, r is rate, and t is time.

Q4: Can e^x be negative?
A: e^x is always positive for real x, though it approaches 0 as x approaches negative infinity.

Q5: What's the derivative of e^x?
A: The remarkable property of e^x is that its derivative is itself: d/dx(e^x) = e^x.