Using the Differential Equation Let P ( t ) be the population (in millions) of a certain city t years after 2015 , and suppose that P ( t ) satisfies the differential equation. a. P ' ( t ) = .03 P ( t ) , P ( 0 ) = 4 b. Use the differential equation to determine how fast the population is growing when it reaches 5 million people. c. Use the differential equation to determine the population size when it is growing at the rate of 400 , 000 people per year. Find a formula for P ( t ) .
Using the Differential Equation Let P ( t ) be the population (in millions) of a certain city t years after 2015 , and suppose that P ( t ) satisfies the differential equation. a. P ' ( t ) = .03 P ( t ) , P ( 0 ) = 4 b. Use the differential equation to determine how fast the population is growing when it reaches 5 million people. c. Use the differential equation to determine the population size when it is growing at the rate of 400 , 000 people per year. Find a formula for P ( t ) .
Solution Summary: The author analyzes how the population is growing at a rate of 0.15 million people per year, when it reaches 5 million.
Using the Differential Equation Let
P
(
t
)
be the population (in millions) of a certain city
t
years after
2015
, and suppose that
P
(
t
)
satisfies the differential equation.
a.
P
'
(
t
)
=
.03
P
(
t
)
,
P
(
0
)
=
4
b. Use the differential equation to determine how fast the population is growing when it reaches
5
million people.
c. Use the differential equation to determine the population size when it is growing at the rate of
400
,
000
people 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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