2a:null 2b:["$","title",null,{"children":"The exponential function | Mattias Villani | Observable"}] 31:T646,propertiestext = html`

${tex`e^x`} notation	${tex`\exp(x)`} notation
${tex`\lim_{x \rightarrow - \infty}e^x = 0 `}	${tex`\lim_{x \rightarrow - \infty}\exp(x) = 0 `}
${tex`\lim_{x \rightarrow \infty}e^x = \infty`}	${tex`\lim_{x \rightarrow \infty}\exp(x) = \infty`}
${tex`e^x = \sum_{n=0}^\infty\frac{x^n}{n!}`}	${tex`\exp(x) = \sum_{n=0}^\infty\frac{x^n}{n!}`}
${tex`e^x e^y = e^{x+y}`}	${tex`\exp(x)\exp(y)= \exp(x+y)`}
${tex`(e^x)^y = e^{xy}`}	${tex`(\exp(x))^y= \exp(xy)`}
${tex`\frac{e^{x}}{e^{y}} = e^{x-y}`}	${tex`\frac{\exp(x)}{\exp(y)} = \exp(x-y)`}
${tex`e^{-x} = \frac{1}{e^{x}}`}	${tex`\exp(-x)= \frac{1}{\exp(x)}`}
Inverse: ${tex`\ln(x)`} so that ${tex`\ln(e^x)=x`}	Inverse: ${tex`\ln(x)`} so that ${tex`\ln(\exp(x))=x`}
Derivative: ${tex`\frac{\mathrm{d}}{\mathrm{d}x}e^x = e^x`}	Derivative: ${tex`\frac{\mathrm{d}}{\mathrm{d}x}\exp(x) = \exp(x)`}
Integral: ${tex`\int e^x = e^x + C`}	Integral: ${tex`\int \exp(x) = \exp(x) + C`}

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**Example**. An initial amount ${tex`c=100`} in some currency is placed in a bank account with 5% annual interest, i.e. ${tex`b=1.05`}.
\nThe amount in the bank account \n - after ${tex`x=1`} year is ${tex`100 \\cdot 1.05 = 105`}\n - after ${tex`x=2`} years is ${tex`100 \\cdot 1.05 \\cdot 1.05 = 100 \\cdot 1.05^2 = 110.25`}\n - after ${tex`x=5`} years is ${tex`100 \\cdot 1.05^5 \\approx 127.6282`}\n - after ${tex`x`} years is ${tex`100 \\cdot 1.05^x`}","pinned":false,"hidden":false,"mode":"md"},{"id":133,"value":"\n\nThe (natural) [exponential function](https://en.wikipedia.org/wiki/Exponential_function) has Euler's number ${tex`e \\approx 2.71828`} as base ^{[[1](#footnote1)]}\n${tex.block`f(x)=e^x`}\nAn alternative notation for the (natural) exponential function is ${tex`f(x)=\\exp(x)`}.\n\nThe more general exponential function has an additional parameters *k* that determines the growth rate of the exponential function\n${tex.block`f(x)=e^{kx}`}","pinned":false,"hidden":false,"mode":"md"},{"id":237,"value":"The graph below lets you compare:\n- exponential growth ${tex`e^{kx}`}\n- power growth ${tex`1 + x^p`} \n- linear growth ${tex`1 + x`}\n\nNote that all three functions have the same initial value at ${tex`x=0`}. \n\nThe default setting in the graph compares the growth of the exponential function ${tex`\\exp(x)`} to that of the cubic function ${tex`1 + x^3`} and the linear growth. Note how the cubic grows faster than the exponential over the interval ${tex`x \\in (0,3)`}. But, increase the upper limit of the *x*-axis and see how the exponential function eventually overturns the cubic function. This is true for any ${tex`n`} in the power function: for any ${tex`x`} large enough, the exponential function will eventually growth faster than the power function.","pinned":false,"hidden":false,"mode":"md"},{"id":145,"value":"#### Exponential function ${tex`e^{kx}`}","pinned":false,"hidden":false,"mode":"md"},{"id":6,"value":"viewof k = Inputs.range([-3, 3], {value: 1, step: 0.1, label: \"k\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":148,"value":"#### Power function ${tex`1 + x^p`}","pinned":false,"hidden":false,"mode":"md"},{"id":218,"value":"viewof n = Inputs.range([-1, 10], {value: 3, step: 0.1, label: \"p\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":186,"value":"viewof inputs2 = Inputs.form({\n upperlim : Inputs.range([1, 100], {value: 3, label: \"upper limit of x-axis\", step: 1}),\n plotpower : Inputs.toggle({label: \"plot power function\", value: true}),\n plotlinear : Inputs.toggle({label: \"plot linear function\", value: true}),\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"Plot.plot({\n style: \"font-size: 12px;\",\n color: {\n legend: true,\n domain: plotlabels.labels,\n range: plotlabels.colors\n },\n x: {\n label: \"x\",\n domain: xdomain \n },\n y: {\n label: \"f(x)\",\n domain: ydomain,\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(funcdata, {x: \"x\", y: \"expofunc\", stroke: colors[0], strokeWidth: 3}),\n Plot.line(funcdata, {filter: inputs2.plotpower, x: \"x\", y: \"powerfunc\", stroke: colors[1], strokeWidth: 3}),\n Plot.line(funcdata, {filter: inputs2.plotlinear, x: \"x\", y: \"linfunc\", stroke: colors[2], strokeWidth: 3}),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":306,"value":"Here are some properties of the exponential function in the two different notations.","pinned":false,"hidden":false,"mode":"md"},{"id":255,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":390,"value":"Note in particular that the exponential function has the remarkable property of being its own derivative and indefinite integral (anti-derivative).🤯","pinned":false,"hidden":false,"mode":"md"},{"id":382,"value":"footnote1 = md`1. [^](#paragraph1) There is an interesting connection between the strange number ${tex`e`} and the interest rate example. In that example we assumed that the interest was paid out and re-invested at the end of each year. But what if the interest was paid out and reinvested every half year? every month? every day? or even continuously (which is of course not practically possible, but imagine it)? The number ${tex`e`} appears in the formula for the amount in the bank after *x* years when the interest is *continously compounded*, see this [wikipedia article on compound interest](https://en.wikipedia.org/wiki/Compound_interest#Continuous_compounding).`","pinned":false,"hidden":false,"mode":"ojs"},{"id":14,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is some code and data details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":16,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":18,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":20,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs"},{"id":23,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":25,"value":"function f(x, a, k){\n return a*Math.exp(k*x)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":142,"value":"function powerfunction(x, c, n){\n return 1 + c*Math.exp(n*Math.log(x))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":40,"value":"xdomain = [0,inputs2.upperlim]","pinned":false,"hidden":false,"mode":"ojs"},{"id":36,"value":"funcdata = {\n return d3.range(xdomain[0], xdomain[1], (xdomain[1] - xdomain[0])/1000).map(d => ({\n x: d, \n expofunc: f(d, 1, k), \n powerfunc: powerfunction(d, 1, n),\n linfunc: 1 + d\n }))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":179,"value":"ydomain = [0,d3.max(funcdata.map(d => d.expofunc).concat(funcdata.map(d => d.powerfunc)))]","pinned":false,"hidden":false,"mode":"ojs"},{"id":190,"value":"plotlabels = {\n let labels = [\"exponential function\"];\n let labelColors = [colors[0]]\n if (inputs2.plotpower){\n labels.push(\"power function\")\n labelColors.push(colors[1])\n }\n if (inputs2.plotlinear){\n labels.push(\"linear function\")\n labelColors.push(colors[2])\n }\n return {labels: labels, colors: labelColors}\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":259,"value":"","pinned":false,"hidden":false,"mode":"html"}],"files":"$W32","stdlib":"1","titling":"auto"}} 2f:"$2e:body"