In a study prepared in 2000, the percentage of households using online banking was projected to be
f(t) = 1.7e0.71t , (0 t 4)
where t is measured in years, with t = 0 corresponding to the beginning of 2000. (Round your answers to three decimal places.)
(a) What was the projected percentage of households using online banking at the beginning of 2002?
%
(b) How fast was the projected percentage of households using online banking changing at the beginning of 2002?
%/yr
(c) How fast was the rate of the projected percentage of households using online banking changing at the beginning of 2002? Hint: We want f ''(2). Why?
%/yr/yr
(a) just plug in t=2
(b) f' = 1.207 e^.71t
So, express f'(2) as a %
(c) f' is the rate of change. To find how fast the rate is changing, use f".
f" = 0.857 e^.71t
(a) To find the projected percentage of households using online banking at the beginning of 2002, we need to evaluate f(t) at t = 2.
f(t) = 1.7e^(0.71t)
f(2) = 1.7e^(0.71*2)
= 1.7e^(1.42)
≈ 4.630
Therefore, the projected percentage of households using online banking at the beginning of 2002 was approximately 4.630%.
(b) To find how fast the projected percentage of households using online banking was changing at the beginning of 2002, we need to find the derivative of f(t) with respect to t and evaluate it at t = 2.
f(t) = 1.7e^(0.71t)
f'(t) = 1.2e^(0.71t)
f'(2) = 1.2e^(0.71*2)
= 1.2e^(1.42)
≈ 3.156
Therefore, the projected percentage of households using online banking was changing at a rate of approximately 3.156%/yr at the beginning of 2002.
(c) To find how fast the rate of the projected percentage of households using online banking was changing at the beginning of 2002, we need to find the second derivative of f(t) and evaluate it at t = 2 (f''(2)).
f(t) = 1.7e^(0.71t)
f''(t) = 0.852e^(0.71t)
f''(2) = 0.852e^(0.71*2)
= 0.852e^(1.42)
≈ 2.309
Therefore, the rate of the projected percentage of households using online banking was changing at a rate of approximately 2.309%/yr/yr at the beginning of 2002.
To find the projected percentage of households using online banking at the beginning of 2002, we need to substitute t = 2 into the given function f(t) = 1.7e^(0.71t).
(a) Substituting t = 2 into the function:
f(2) = 1.7e^(0.71(2))
= 1.7e^(1.42)
≈ 5.831
Therefore, the projected percentage of households using online banking at the beginning of 2002 is approximately 5.831%.
To find how fast the projected percentage of households using online banking was changing at the beginning of 2002, we need to find the derivative of the function f(t) = 1.7e^(0.71t) and evaluate it at t = 2.
(b) Taking the derivative of f(t):
f'(t) = (1.7)(0.71)e^(0.71t)
Substituting t = 2 into the derivative:
f'(2) = (1.7)(0.71)e^(0.71(2))
≈ 1.210
Therefore, the projected percentage of households using online banking was changing at a rate of approximately 1.210% per year at the beginning of 2002.
To find how fast the rate of the projected percentage of households using online banking was changing at the beginning of 2002, we need to find the second derivative of the function f(t) = 1.7e^(0.71t) and evaluate it at t = 2.
(c) Taking the second derivative of f(t):
f''(t) = (1.7)(0.71)^2e^(0.71t)
Substituting t = 2 into the second derivative:
f''(2) = (1.7)(0.71)^2e^(0.71(2))
≈ 0.444
Therefore, the rate of the projected percentage of households using online banking was changing at a rate of approximately 0.444% per year per year at the beginning of 2002.