Asked by rebkekah
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
or 0 less than x less than 4, the average rate of change will be
enter your response here.
(Round to the nearest integer as needed.)
All Answers 1
Answered by
GPT-5 mini
AI
f(x) = 500(0.85)^x.
Average rate on 0 < x < 4:
[f(4) − f(0)]/4 = [500(0.85^4) − 500]/4 ≈ (261.003 − 500)/4 ≈ −59.75 ≈ −60.
For 4 < x < 8 the average rate ≈ [500(0.85^8) − 500(0.85^4)]/4 ≈ −31.
As x increases further the average rate of change gets closer to 0 (becomes less negative), since the decay slows and f(x) → 0 (also f'(x)=500 ln(0.85)(0.85)^x → 0).
Average rate on 0 < x < 4:
[f(4) − f(0)]/4 = [500(0.85^4) − 500]/4 ≈ (261.003 − 500)/4 ≈ −59.75 ≈ −60.
For 4 < x < 8 the average rate ≈ [500(0.85^8) − 500(0.85^4)]/4 ≈ −31.
As x increases further the average rate of change gets closer to 0 (becomes less negative), since the decay slows and f(x) → 0 (also f'(x)=500 ln(0.85)(0.85)^x → 0).
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