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Is the function # y = -5(1/3)^ -x# exponential growth or decay?

Answer 1

Abigail Allen

The function y = -5(1/3)^-x represents exponential decay.

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Answer 2

Isaiah Anthony

You can answer the question by calculating the first derivative. First of all, applying the rule #a^(-x)=1/(a^x)#, we have that #(1/3)^{-x}=3^x#. Then, since you can factor out constants, you have that #d/dx -5(1/3)^{-x}= -5\ d/dx (1/3)^{-x} = -5\ d/dx 3^x# As a fundamental derivative, we know that #d/dx 3^x=3^x*log(3)#.

So, the first derivative is #-5\log(3)*3^x#, which is always negative since #5\log(3)*e^x# is always positive. So, your function is always decreasing, and you have exponential decay.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.