The data in the table are generated by the function $f ( x ) = a e ^ { - b x }$ . Approximate the unknown constants a and b to four decimal places. x
1 2 3 4
F ( x )
0)71939 0.41687 0.24157 0.13998
A) $a = 1.2414 , a = 0.5456$
B) $a = 0.0000 , a = 0.5456$
C) $$a = 0.5456 , a = 0.5456$$
D) $a = 1.2414 , a = 0.0000$
E) $a = 0.5456 , a = 1.2414$

Question

Quizplus · Accepted Answer

The data in the table are generated by the function $f ( x ) = a e ^ { - b x }$. To approximate the unknown constants a and b, we can use the natural logarithm to linearize the function:$$\ln(f(x)) = \ln(a) - bx$$Then, we can use linear regression to find the slope and y-intercept of the line that best fits the data points. The slope of the line will be -b, and the y-intercept will be $\ln(a)$.Using a calculator or spreadsheet, we can find that the slope of the line is -0.5456 and the y-intercept is 0.2231. Therefore, we have:$$b = 0.5456$$$$\ln(a) = 0.2231$$Solving for a, we get:$$a = e^{0.2231} = 1.2414$$Therefore, the approximate values of the constants are $a = 1.2414$ and $b = 0.5456$.

The Data in the Table Are Generated by the Function f(x)=ae−bxf ( x ) = a e ^ { - b x }f(x)=ae−bx

The Data in the Table Are Generated by the Function $f ( x ) = a e ^ { - b x }$