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# Curve Fitting Of Exponential Curve online calculator

This calculator is for fitting exponential curve of the type y=ae^bx. Very detailed step by step calculation is provided.

INSTRUCTION: Use ',' or new line to separate between values

## $$Equation\;of\;created\;curve\;is\; Y = 16.120101e^{0.705131x}$$

$$\displaylines{---}$$
$$\displaylines{The\;curve\;to\;be\;fitted\;is\;Y=ae^{bx} \\ \\ Take\;log\;on\;both\;side\;to\;the\;base\;e \\ \\ \log\;_{e}\;Y\;=\;\log\;_{e}\;a\;+\;bx \\ \\ It\;is\;in\;the\;form\;y=A+Bx.\;Where\;y\;=\;\log\;_{e}\;Y,\;A\;=\;\log\;_{e}\;a,\;B\;=\;b \\ \\ Now\;we\;have\;to\;apply\;Linear\;Regression \\ \\ Before\;applying\;we\;have\;to\;find\;y\;values\;for\;corresponding\;Y\;values \\ \\ Using\;y\;=\;\log\;_{e}\;Y \\ \\ }$$
$$No$$
$$Y$$
$$y$$
$$1$$
$$520$$
$$6.253829$$
$$2$$
$$266$$
$$5.583496$$
$$3$$
$$567$$
$$6.340359$$
$$4$$
$$4149$$
$$8.330623$$
$$5$$
$$6212$$
$$8.734238$$
$$6$$
$$9973$$
$$9.207637$$
$$7$$
$$11894$$
$$9.383789$$
$$8$$
$$45000$$
$$10.714418$$
$$Total$$

$$64.548389$$
$$\displaylines{}$$
$$\displaylines{Now\;we\;have\;to\;find\;mean\;of\;x\;values \\ \\ Mean = \frac{\sum_{i=1}^{n}x_{i}}{n} \\ \\ \,=\frac{4+5+6+7+8+9+10+11}{8} \\ \\ \,=\frac{60}{8} \\ \\ \Rightarrow \bar{x}= 7.5 \\ \\ Then\;we\;have\;to\;find\;mean\;of\;y\;values \\ \\ Mean = \frac{\sum_{i=1}^{n}y_{i}}{n} \\ \\ \,=\frac{6.253829+5.583496+6.340359+8.330623+8.734238+9.207637+9.383789+10.714418}{8} \\ \\ \,=\frac{64.548389}{8} \\ \\ \Rightarrow \bar{y}= 8.068549 \\ \\ }$$
$$No$$
$$x_{i}$$
$$y_{i}$$
$$(x_{i} - \bar{x})$$
$$(y_{i} - \bar{y})$$
$$(x_{i} - \bar{x})*(y_{i} - \bar{y})$$
$$(x_{i} - \bar{x})^2$$
$$1$$
$$4$$
$$6.253829$$
$$-3.5$$
$$-1.81472$$
$$6.35152$$
$$12.25$$
$$2$$
$$5$$
$$5.583496$$
$$-2.5$$
$$-2.485053$$
$$6.212633$$
$$6.25$$
$$3$$
$$6$$
$$6.340359$$
$$-1.5$$
$$-1.72819$$
$$2.592285$$
$$2.25$$
$$4$$
$$7$$
$$8.330623$$
$$-0.5$$
$$0.262074$$
$$-0.131037$$
$$0.25$$
$$5$$
$$8$$
$$8.734238$$
$$0.5$$
$$0.665689$$
$$0.332844$$
$$0.25$$
$$6$$
$$9$$
$$9.207637$$
$$1.5$$
$$1.139088$$
$$1.708632$$
$$2.25$$
$$7$$
$$10$$
$$9.383789$$
$$2.5$$
$$1.31524$$
$$3.2881$$
$$6.25$$
$$8$$
$$11$$
$$10.714418$$
$$3.5$$
$$2.645869$$
$$9.260541$$
$$12.25$$
$$Total$$




$$29.615518$$
$$42.0$$
$$\displaylines{}$$
$$\displaylines{B= \frac{\sum_{i=1}^{n}(x_{i} - \bar{x})(y_{i} - \bar{y})}{\sum_{i=1}^{n}(x_{i} - \bar{x})^{2}} \\ \\ From\;the\;table\;total\;we\;will\;get\;numerator\;and\;denominator \\ \\ \Rightarrow \frac{29.615518}{42.0} \\ \\ \Rightarrow \mathbf{\color{Red}{0.705131}} \\ \\ A = \bar{y}-B* \bar{x} \\ \\ \Rightarrow A = 8.068549-0.705131* 7.5 \\ \\ \Rightarrow \mathbf{\color{Red}{2.780067}} \\ \\ Equation\;of\;line\;\Rightarrow\;y\;=\;A+B*x \\ \\ \Rightarrow y = 2.780067+0.705131*x \\ \\ But\;A\;=\;\log\;_{e}\;a \\ \\ \Rightarrow a\;=\;e^{A} \\ \\ \Rightarrow a\;=\;e^{2.780067} \\ \\ \Rightarrow a\;=\;16.120101 \\ \\ b\;=\;B\;=\;0.705131 \\ \\ After\;putting\;back\;these\;in\;original\;equation\;Y=ae^{bx} \\ \\ We\;get\;Y\;=\;16.120101e^{0.705131x} }$$