Cellphone Revenues The number of cellphone subscribers in China for the period 2000–2005 was projected to follow the equation N ( t ) = 39 t + 68 millon subscribers inyear t. ( t = 0 represents2000.) The average annual revenue per cellphone user was $350 in 2000. Assuming that, because of competition, the revenue per cellphone user decreases exponentially with an annual decay constant of 10%, give a formula for the annual revenue in year t . Hence, project the annual revenue and its rate of change in 2002. Round all answers to the nearest billion dollars or billion dollars per year.
Cellphone Revenues The number of cellphone subscribers in China for the period 2000–2005 was projected to follow the equation N ( t ) = 39 t + 68 millon subscribers inyear t. ( t = 0 represents2000.) The average annual revenue per cellphone user was $350 in 2000. Assuming that, because of competition, the revenue per cellphone user decreases exponentially with an annual decay constant of 10%, give a formula for the annual revenue in year t . Hence, project the annual revenue and its rate of change in 2002. Round all answers to the nearest billion dollars or billion dollars per year.
Solution Summary: The author calculates the formula for the annual revenue in year t based on the number of cellphone subscribers in China.
Cellphone Revenues The number of cellphone subscribers in China for the period 2000–2005 was projected to follow the equation
N
(
t
)
=
39
t
+
68
millon subscribers
inyear t. (
t
=
0
represents2000.) The average annual revenue per cellphone user was $350 in 2000. Assuming that, because of competition, the revenue per cellphone user decreases exponentially with an annual decay constant of 10%, give a formula for the annual revenue in year t. Hence, project the annual revenue and its rate of change in 2002. Round all answers to the nearest billion dollars or billion dollars per year.
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
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