{"id":1514,"date":"2015-11-12T18:35:28","date_gmt":"2015-11-12T18:35:28","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1514"},"modified":"2017-04-03T14:55:53","modified_gmt":"2017-04-03T14:55:53","slug":"evaluate-exponential-functions-with-base-e","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/evaluate-exponential-functions-with-base-e\/","title":{"raw":"Evaluate exponential functions with base e","rendered":"Evaluate exponential functions with base e"},"content":{"raw":"

As we saw earlier, the amount earned on an account increases as the compounding frequency increases. The table below\u00a0shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.<\/p>\r\n

Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies.<\/p>\r\n\r\n
Frequency<\/th>\r\n[latex]A\\left(t\\right)={\\left(1+\\frac{1}{n}\\right)}^{n}[\/latex]<\/th>\r\nValue<\/th>\r\n<\/tr><\/thead>
Annually<\/td>\r\n[latex]{\\left(1+\\frac{1}{1}\\right)}^{1}[\/latex]<\/td>\r\n$2<\/td>\r\n<\/tr>
Semiannually<\/td>\r\n[latex]{\\left(1+\\frac{1}{2}\\right)}^{2}[\/latex]<\/td>\r\n$2.25<\/td>\r\n<\/tr>
Quarterly<\/td>\r\n[latex]{\\left(1+\\frac{1}{4}\\right)}^{4}[\/latex]<\/td>\r\n$2.441406<\/td>\r\n<\/tr>
Monthly<\/td>\r\n[latex]{\\left(1+\\frac{1}{12}\\right)}^{12}[\/latex]<\/td>\r\n$2.613035<\/td>\r\n<\/tr>
Daily<\/td>\r\n[latex]{\\left(1+\\frac{1}{365}\\right)}^{365}[\/latex]<\/td>\r\n$2.714567<\/td>\r\n<\/tr>
Hourly<\/td>\r\n[latex]{\\left(1+\\frac{1}{\\text{8766}}\\right)}^{\\text{8766}}[\/latex]<\/td>\r\n$2.718127<\/td>\r\n<\/tr>
Once per minute<\/td>\r\n[latex]{\\left(1+\\frac{1}{\\text{525960}}\\right)}^{\\text{525960}}[\/latex]<\/td>\r\n$2.718279<\/td>\r\n<\/tr>
Once per second<\/td>\r\n[latex]{\\left(1+\\frac{1}{31557600}\\right)}^{31557600}[\/latex]<\/td>\r\n$2.718282<\/td>\r\n<\/tr><\/tbody><\/table>

These values appear to be approaching a limit as n<\/em>\u00a0increases without bound. In fact, as n<\/em>\u00a0gets larger and larger, the expression [latex]{\\left(1+\\frac{1}{n}\\right)}^{n}[\/latex] approaches a number used so frequently in mathematics that it has its own name: the letter [latex]e[\/latex]. This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.<\/p>\r\n\r\n

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A General Note: The Number e<\/em><\/h3>\r\n

The letter e<\/em> represents the irrational number<\/p>\r\n\r\n

[latex]{\\left(1+\\frac{1}{n}\\right)}^{n},\\text{as}n\\text{increases without bound}[\/latex]<\/div>\r\n

The letter e <\/em>is used as a base for many real-world exponential models. To work with base e<\/em>, we use the approximation, [latex]e\\approx 2.718282[\/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707\u20131783) who first investigated and discovered many of its properties.<\/p>\r\n\r\n<\/div>\r\n

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Example 7: Using a Calculator to Find Powers of e<\/em><\/h3>\r\n

Calculate [latex]{e}^{3.14}[\/latex]. Round to five decimal places.<\/p>\r\n\r\n<\/div>\r\n

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Solution<\/h3>\r\n

On a calculator, press the button labeled [latex]\\left[{e}^{x}\\right][\/latex]. The window shows [e<\/em>^(]. Type 3.14 and then close parenthesis, (]). Press [ENTER]. Rounding to 5 decimal places, [latex]{e}^{3.14}\\approx 23.10387[\/latex]. Caution: Many scientific calculators have an \"Exp\" button, which is used to enter numbers in scientific notation. It is not used to find powers of e<\/em>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Try It 9<\/h3>\r\n

Use a calculator to find [latex]{e}^{-0.5}[\/latex]. Round to five decimal places.<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>\r\n<\/section>

Investigating Continuous Growth<\/h2>\r\n

So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e <\/em>is used as the base for exponential functions. Exponential models that use e<\/em>\u00a0as the base are called continuous growth or decay models<\/em>. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.<\/p>\r\n\r\n

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A General Note: The Continuous Growth\/Decay Formula<\/h3>\r\n

For all real numbers t<\/em>, and all positive numbers a<\/em>\u00a0and r<\/em>, continuous growth or decay is represented by the formula<\/p>\r\n\r\n

[latex]A\\left(t\\right)=a{e}^{rt}[\/latex]<\/div>\r\n

where<\/p>\r\n\r\n