Part A)
f(x) = 5^x
f(0) = 5^0
f(0) = 1
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f(x) = 5^x
f(1) = 5^1
f(1) = 5
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f(x) = 5^x
f(2) = 5^2
f(2) = 5*5
f(2) = 25
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f(x) = 5^x
f(3) = 5^3
f(3) = 5*5*5
f(3) = 125
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Rate of change for section A = (f(1) - f(0))/(1 - 0)
Rate of change for section A = (5 - 1)/(1 - 0)
Rate of change for section A = 4/1
Rate of change for section A = 4
Rate of change for section B = (f(3) - f(2))/(3 - 2)
Rate of change for section B = (125 - 25)/(3 - 2)
Rate of change for section B = 100/1
Rate of change for section B = 100
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Part B)
From part A) above, we found,
Rate of change for section A = 4
Rate of change for section B = 100
Which means that section B's rate of change is 25 times greater (since 100/4 = 25, or 25*4 = 100)
Answer for part B: 25
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Extra: Explain why one rate of change is greater than the other.
The rate of change for section B is larger because the exponential function is growing faster as x increases. This is shown visually by the sharper and steeper incline as the function curve goes upward. The function starts off with relatively slower growth but it accelerates in speed.
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