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Original
2026-01-01
Modified
2026-02-28
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<p>The concept of exponential growth says that something always grows in<a>relation</a>to its current value. It is similar to the concept of doubling. For example, If a rabbit population doubles every month, the<a>numbers</a>would go 2, then 4, then 8, 16, 32, 64, 128, 256, and so on.</p>
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<p>The concept of exponential growth says that something always grows in<a>relation</a>to its current value. It is similar to the concept of doubling. For example, If a rabbit population doubles every month, the<a>numbers</a>would go 2, then 4, then 8, 16, 32, 64, 128, 256, and so on.</p>
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<p>Let us consider a special tree that grows exponentially. Then, </p>
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<p>Let us consider a special tree that grows exponentially. Then, </p>
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<p>Its height in mm is ex. </p>
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<p>Its height in mm is ex. </p>
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<p>Where e stands for Euler’s number. </p>
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<p>Where e stands for Euler’s number. </p>
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<p>E = 2.718</p>
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<p>E = 2.718</p>
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<p>Therefore, let us now increase the value of x exponentially and see what happens. </p>
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<p>Therefore, let us now increase the value of x exponentially and see what happens. </p>
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<p>1-year-old \(e^1\) = 2.7 mm high, which is really tiny.</p>
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<p>1-year-old \(e^1\) = 2.7 mm high, which is really tiny.</p>
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<p>5-year-old \(e^5 \) = 148 mm high, which is as high as a cup. </p>
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<p>5-year-old \(e^5 \) = 148 mm high, which is as high as a cup. </p>
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<p>10-year-old \(e^{10}\) = 22 m high, which is as tall as a building.</p>
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<p>10-year-old \(e^{10}\) = 22 m high, which is as tall as a building.</p>
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<p>15-year-old \( e^{15}\) = 3.3 km high, which is as tall as 10 stacked Eiffel towers.</p>
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<p>15-year-old \( e^{15}\) = 3.3 km high, which is as tall as 10 stacked Eiffel towers.</p>
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<p>20-year-old \(e^{20}\) = 485 km high, which reaches up into the space.</p>
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<p>20-year-old \(e^{20}\) = 485 km high, which reaches up into the space.</p>
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<p>There is no tree that can grow that tall. Therefore, when someone says that it grows exponentially, we have to think about what they could be meaning. But sometimes, some things can grow exponentially, for a while. </p>
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<p>There is no tree that can grow that tall. Therefore, when someone says that it grows exponentially, we have to think about what they could be meaning. But sometimes, some things can grow exponentially, for a while. </p>
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<p>Therefore, we use a general<a>formula</a>, </p>
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<p>Therefore, we use a general<a>formula</a>, </p>
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<p>\(y(t) = a × e^{kt}\)</p>
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<p>\(y(t) = a × e^{kt}\)</p>
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<p>Where y(t) = value at time "t"</p>
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<p>Where y(t) = value at time "t"</p>
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<p>a = value at the start</p>
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<p>a = value at the start</p>
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<p>k =<a>rate</a>of growth (when >0) or decay (when <0)</p>
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<p>k =<a>rate</a>of growth (when >0) or decay (when <0)</p>
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<p>t = time</p>
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<p>t = time</p>
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<p>For example, </p>
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<p>For example, </p>
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<p>Let us say that 2 months ago, we had 3 mice, and now we have 18. Find the value of the rate of growth. </p>
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<p>Let us say that 2 months ago, we had 3 mice, and now we have 18. Find the value of the rate of growth. </p>
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<p>Let's start with the formula, </p>
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<p>Let's start with the formula, </p>
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<p>\(y(t) = a × e^{kt}\)</p>
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<p>\(y(t) = a × e^{kt}\)</p>
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<p>We know that, </p>
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<p>We know that, </p>
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<p>a = 3, t = 2 and here, y(2) = 18.</p>
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<p>a = 3, t = 2 and here, y(2) = 18.</p>
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<p>Therefore, </p>
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<p>Therefore, </p>
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<p>\(18 = 3 × e^{2k}\)</p>
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<p>\(18 = 3 × e^{2k}\)</p>
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<p>\(6 = e^{2k}\)</p>
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<p>\(6 = e^{2k}\)</p>
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<p>Taking the natural logarithm on both sides, </p>
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<p>Taking the natural logarithm on both sides, </p>
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<p>ln(6) = ln(e2k)</p>
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<p>ln(6) = ln(e2k)</p>
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<p>ln(ex)=x, so:ln(6) = 2k</p>
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<p>ln(ex)=x, so:ln(6) = 2k</p>
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<p>2k = ln(6)</p>
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<p>2k = ln(6)</p>
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<p>\(k = \frac{ln(6)}{2}\)</p>
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<p>\(k = \frac{ln(6)}{2}\)</p>
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