Question

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
  
negative 31.
​(Round to the nearest integer as​ needed.)
Part 3
For intervals beyond x​ = 8, the rate of change will
▼
but will lie between 0 and
  
enter your response here. As x​ increases, the exponential function will
▼
increase but at a faster rate.
increase but at a slower rate.
decrease but at a faster rate.
decrease but at a slower rate.
​(Round to the nearest integer as​ needed.)

Answer 1

Part 1: Average rate on 0 < x < 4 = (f(4)-f(0))/4 = (500·0.85^4 − 500)/4 ≈ -60.

Part 2: Average rate on 4 < x < 8 = (f(8)-f(4))/4 = (500·0.85^8 − 500·0.85^4)/4 ≈ -31.

Part 3: For intervals beyond x = 8 the average rate of change will still be negative but will lie between 0 and -31 (i.e., closer to 0 than -31). As x increases, the exponential function will decrease but at a slower rate (the average rates approach 0 from below).