calculus 1
aknown11
MATH 2413
Applications of Limits
NAME:
DATE:
IN ORDER TO RECEIVE FULL CREDIT, SHOW ALL YOUR WORK. ENCLOSE FINAL ANSWERS IN BOXES.
1. A soda is removed from the fridge and placed in a room that is 70◦F
Using Newton’s Law of Cooling we derive the temperature as a function of time in minutes:
T (t) = 70− 35e−0.018t
What is the temperature of the soda after 10 minutes?
What is the temperature of the soda after 100 minutes?
What is lim t→∞
T (t)?
Why does this make sense?
2. The zombie apocalypse has finally occurred and the government estimates that the spread of the disease will have an infection rate of 0.1. Assuming that the population of the United States is about 300, 000, 000 the population of healthy people as a function of time in days is Phealthy = 300, 000, 000e
−0.1t and the population of those either a zombie or dead is Pinfected = 300, 000, 000(1− e−0.1t).
What is the population of healthy people after 40 days?
What is the population of infected people after 40 days?
What is the population of healthy people after 100 days?
What is the population of infected people after 100 days?
What is lim t→∞
Phealthy ?
What is lim t→∞
Pinfected ?
How long until only one person is left?
3. The previous example is an extreme situation of modeling a population’s growth or decline. A more commonly used method is to consider logistic growth. This type of growth is used to maximize sustainable crop yields.
P (t) = KP0e
rt
K + P0(ert − 1)
where
K is the carrying capacity
P0 is the initial population
r is the growth rate
t is time
Find lim t→∞
P (t). Show your work.
Note: Allowing time to tend towards infinity in this case is a little silly but can be interpreted as waiting until the very end of the season to harvest the crop before it starts to die due to the changing of the seasons.
4. The concentration in milligrams per cubic centimeters of a particular drug in a patients system is given by:
C(t) = 0.16t
t2 + 4t+ 4
What is the lim t→∞
C(t)? Show your work.
What is the physical interpretation of this limit?
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