Question

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Part 1
An exponential function has an initial value of 500 and a decay rate of​ 15%. Compare the average rate of change for the interval 0less thanxless than4 to the average rate for the interval 4less thanxless than8. What do you think will happen to the rate of change for intervals beyond x​ = 8? Explain.
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Part 1
For 0 less than x less than 4​, the average rate of change will be
  
negative 60.
​(Round to the nearest integer as​ needed.)
Part 2
For 4 less than x less than 8​, the average rate of change will be
  
enter your response here.
​(Round to the nearest integer as​ needed.)

Answer 1

f(x) = 500(0.85)^x.

For 0 < x < 4:
f(0) = 500, f(4) = 500(0.85^4) ≈ 261.003 → average rate = (261.003 − 500)/4 ≈ −59.75 ≈ −60 (given).

For 4 < x < 8:
f(4) ≈ 261.003, f(8) = 500(0.85^8) ≈ 136.245 → average rate = (136.245 − 261.003)/4 ≈ −31.19 ≈ −31.

What happens beyond x = 8:
The average rate of change over equal-length intervals will continue to be negative but its magnitude will get smaller (values approach 0), so the average rate will approach 0 from below.