E calculator Eeraised to Power of X
Our free exponents & logarithms calculator solves ecalculator eeraised power problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
e^x = 1 + x + x^2/2! + x^3/3! + ... (Taylor series)
The exponential function e^x where e is Euler's number (approximately 2.71828). It can be computed via the Taylor series which converges for all real x. The function is its own derivative and integral, making it fundamental in calculus.
Worked Examples
Example 1: Computing e^2 with Taylor Series
Problem:Calculate e^2 and verify using the Taylor series expansion to 6 terms.
Solution:e^2 = 7.389056...\n\nTaylor series: e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5!\nFor x = 2:\nTerm 0: 1\nTerm 1: 2\nTerm 2: 4/2 = 2\nTerm 3: 8/6 = 1.3333\nTerm 4: 16/24 = 0.6667\nTerm 5: 32/120 = 0.2667\nPartial sum = 7.2667 (error = 0.1224)
Result:e^2 = 7.389056 | 6-term approximation = 7.2667 (98.3% accurate)
Example 2: Continuous Compounding
Problem:Find the value of $5,000 invested at 8% for 10 years with continuous compounding using A = P*e^(rt).
Solution:A = P * e^(rt)\nA = 5000 * e^(0.08 * 10)\nA = 5000 * e^(0.8)\ne^(0.8) = 2.2255\nA = 5000 * 2.2255 = $11,127.70\n\nInterest earned: $11,127.70 - $5,000 = $6,127.70
Result:A = $11,127.70 | Interest earned = $6,127.70
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy