Download Differential Equations 1. Doubling time. Show that the doubling time

Differential Equations
1. Doubling time. Show that the doubling time for an exponential function is independent of
the starting time (by following these steps).
(a) Let P (t) = P0 ekt . Starting at time t, let ∆t be the additional time needed until the
population doubles: i.e. P (t + ∆t) = 2P (t). Calculate the doubling time ∆t .
(b) Does the doubling time depend on the time t at which you started or is it the same no
matter at which time you started?
(c) Suppose that the population is given by a quadratic function P (t) = t2 . Calculate the
doubling time if you start at t = 1 with P (1) = 1. Calculate the doubling time if you
start at t = 2 with P (2) = 4. Calculate the doubling time if you start at t = 3 with
P (3) = 9.
(d) For P (t) = t2 , is the doubling time constant no matter which value of t you start at or
does it depend on the value of tand P (t).
2. For a function f (t), solve the differential equation
a formula for the solution f (t).
df
dt
= 4f by separation of variables and get
3. Consider the differential equation given by Newton’s Law of Cooling. Let the temperature of
the object be denoted by T (t) in degrees fahrenheit and t=time is in minutes. Suppose the
temperature of the room is 75 degrees fahrenheit. Then
dT
= −k(T − 75)
dt
(a) Use the separation of variable technique (divide both side by (T-75) ) to find the formula
for T (t). You can check your answer on the web site.
(b) Use these equations to do problem 13 in Sect 10.4
(c) As t → +∞, what does T (t) approach?
(d) According to this model, does the temperature of the turkey ever reach 75 degrees?
4. Sect 10.4 #9, 11
5. Sect 8.2 #2, 8, 14, 42. Do these with the techniques you learned from your groups (given in
Sect 8.2). You can use the Integral Tables to double check your answer but not to solve the
problem.
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