Science:Math Exam Resources/Courses/MATH102/December 2017/Question 16

MATH102 December 2017

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Question 16

Suppose the line of best fit through the points $$(x_{1},\ln y_{1}),(x_{2},\ln y_{2}),\cdots ,(x_{n},\ln y_{n})$$ has slope ${\textstyle 3}$ and passes through ${\textstyle (0,-7)}$. We want a function of the form ${\textstyle y=Ce^{rx}}$ approximating the data $$(x_{1},y_{1}),(x_{2},y_{2}),\cdots ,(x_{n},y_{n}).$$ Find the values of ${\textstyle C}$ and ${\textstyle r}$ that fit the data best. (Your answers should be numbers.) ${\textstyle C=}$ ${\textstyle r=}$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Take logarithms on both side of the exponential relation.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Suppose ${\textstyle y=Ce^{rx}}$ is the curve of best fit for ${\textstyle (x_{1},y_{1}),\dots ,(x_{n},y_{n})}$. Taking the logarithm, we have that $$\ln(y)=\ln(C)+rx$$ is the curve of best fit for ${\textstyle (x_{1},\ln y_{1}),\dots ,(x_{n},\ln y_{n})}$. Since the line of best fit through the points ${\textstyle (x_{1},\ln y_{1}),\dots ,(x_{n},\ln y_{n})}$ has slope ${\textstyle 3}$ and passes through ${\textstyle (0,-7)}$, we know that $$r=3$$ and $$-7=\ln(C),$$ that is, $$C=e^{-7}.$$ Answer: ${\textstyle {\color {blue}C=e^{-7},r=3}}$.