Chapter 9: Chapter 9

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?

Short Answer

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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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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Setup the exponential function with the data points

We have the data points $(1, y_1), (2, y_2), (3, y_3)$ and the exponential function $y=ab^x$. Let's plug the given points into the function: For point (1, $y_1$): $y_1=ab^1$ For point (2, $y_2$): $y_2=ab^2$ For point (3, $y_3$): $y_3=ab^3$

Step 2: Solve for the constants a and b

Divide the second equation by the first equation, and the third equation by the second equation: $\frac{y_2}{y_1} =\frac{ab^2}{ab^1} \Rightarrow \frac{y_2}{y_1}=b$ $\frac{y_3}{y_2}=\frac{ab^3}{ab^2} \Rightarrow \frac{y_3}{y_2}=b$ Now since both fractions equal b, we can set them equal to each other: $\frac{y_2}{y_1}=\frac{y_3}{y_2}$ Solve for $y_2$: $y_2^2 =y_1y_3$

Step 3: Verify Constants a and b

Now that we have an equation for $y_2$, we can plug back into the first equation and solve for $a$: $y_1=ab$ $a= \frac{y_1}{b}=\frac{y_1}{\frac{y_2}{y_1}} = \frac{y_1^2}{y_2}$ With $a$ known, we can solve for $b$: $b= \frac{y_2}{y_1}$

Step 4: Conclusion

Since we now have unique values for constants $a$ and $b$, we can say that the data points $(1, y_1), (2, y_2), (3, y_3)$ lie on an exponential curve of the form $y = ab^x$.

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Chapter 9: Chapter 9

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?

Short Answer

Step by step solution

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Step 1: Setup the exponential function with the data points

Step 2: Solve for the constants a and b

Step 3: Verify Constants a and b

Step 4: Conclusion

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More chapters from the book ‘Finite Mathematics and Applied Calculus’

Chapter 0

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

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