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4. Exponential

{
  "topic": "exponential",
  "input_types": [
    "float"
  ],
  "input_labels": [
    "x"
  ],
  "input_descriptions": [
    "a number"
  ],
  "input_values": [
    3
  ],
  "type": "Calculate",
  "title": "Exponential",
  "category": "Arithmetic",
  "function": "function myFunc(arr) {\n        let x = arr[0];\n        return Math.exp(x);\n      }",
  "op_label": "Exponential, e<sup>x</sup>",
  "explanation": "This calculator finds the value of e raised to the power of x.",
  "content": "<h2>Calculating the Exponential of a Number</h2>\n<p>In mathematics, the exponential function is a mathematical function denoted as \\(e^x\\), where \\(e\\) is the base of natural logarithms, approximately equal to 2.71828. The exponential function is widely used in various fields such as calculus, complex analysis, and in describing growth processes.</p>\n<p>The expression \\(e^x\\) represents the value of the constant \\(e\\) raised to the power of \\(x\\). This function grows rapidly as the value of \\(x\\) increases and is fundamental in many mathematical models.</p>\n<p>For example, the expression \\(e^3\\) represents the value of \\(e\\) raised to the power of 3:</p>\n<p class=\"tabspace\">\\(e^3\\)</p>\n<p>Using a calculator or mathematical software, this value is approximately equal to 20.0855.</p>\n\n<h2>Examples</h2>\n<p>Let’s explore some examples to understand how to calculate the exponential of a number.</p>\n\n<div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">1.</span> Calculate the value of \\(e^3\\).</h3><h4 class=\"answer\">Answer</h4>\n<p>First, we identify the given value:</p>\n<p><b>Given:</b></p><ul><li>Exponent (x): 3</li></ul>\n<p>Next, we calculate the value of \\(e\\) raised to the power of 3:</p>\n<p><b>Steps:</b></p><ul><li>\\(e^3\\)</li>\n<li>Using a calculator, the value is approximately 20.0855.</li></ul>\n<p><b>Result:</b></p><p class=\"tabspace answer\">∴ The value of \\(e^3\\) is approximately 20.0855.</p></div>\n\n<div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">2.</span> Calculate the value of \\(e^2\\).</h3><h4 class=\"answer\">Answer</h4>\n<p>We start by identifying the given value:</p>\n<p><b>Given:</b></p><ul><li>Exponent (x): 2</li></ul>\n<p>Next, we calculate the value of \\(e\\) raised to the power of 2:</p>\n<p><b>Steps:</b></p><ul><li>\\(e^2\\)</li>\n<li>Using a calculator, the value is approximately 7.3891.</li></ul>\n<p><b>Result:</b></p><p class=\"tabspace answer\">∴ The value of \\(e^2\\) is approximately 7.3891.</p></div>\n\n<div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">3.</span> Determine the value of \\(e^0\\).</h3><h4 class=\"answer\">Answer</h4>\n<p>First, we identify the given value:</p>\n<p><b>Given:</b></p><ul><li>Exponent (x): 0</li></ul>\n<p>Next, we calculate the value of \\(e\\) raised to the power of 0:</p>\n<p><b>Steps:</b></p><ul><li>\\(e^0\\)</li>\n<li>By definition, any number raised to the power of 0 is 1.</li></ul>\n<p><b>Result:</b></p><p class=\"tabspace answer\">∴ The value of \\(e^0\\) is 1.</p></div>\n\n<div class=\"example\"><h3 class=\"question\"><span=\"example_n\">4.</span> Calculate the value of \\(e^{-1}\\).</h3><h4 class=\"answer\">Answer</h4>\n<p>We start by identifying the given value:</p>\n<p><b>Given:</b></p><ul><li>Exponent (x): -1</li></ul>\n<p>Next, we calculate the value of \\(e\\) raised to the power of -1:</p>\n<p><b>Steps:</b></p><ul><li>\\(e^{-1}\\)</li>\n<li>Using a calculator, the value is approximately 0.3679.</li></ul>\n<p><b>Result:</b></p><p class=\"tabspace answer\">∴ The value of \\(e^{-1}\\) is approximately 0.3679.</p></div>"
}