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Problem 105

How would you check whether data points of the form $\left(1, y_{1}\right),\left(2, y_{2}\right),\left(3, y_{3}\right)$ lie on an exponential curve?

Expert verified

To check if the data points \((1, y_1), (2, y_2), (3, y_3)\) lie on an exponential curve, plug the points into the exponential function \(y=ab^x\) and solve for constants \(a\) and \(b\). We get the equations \(y_1=ab\), \(y_2 = ab^2\), and \(y_3 = ab^3\). Solve for \(b\) by setting the ratio of the second and first equations equal to the ratio of the third and second equations, giving us \(\frac{y_2}{y_1}=\frac{y_3}{y_2}\). Then, solve for \(y_2^2 =y_1y_3\), and obtain unique values for \(a = \frac{y_1^2}{y_2}\) and \(b= \frac{y_2}{y_1}\). Since we have unique values for \(a\) and \(b\), the data points lie on an exponential curve of the form \(y = ab^x\).

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Chapter 9

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Chapter 9

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