The slope field for the equation dP/dt=0.0333333P(30−P), for P bigger than or equal to 0, is shown below.
LINK TO THE SLOPE FIELD IMAGE: http://s7.postimage.org/8xg4q4s6z/image.jpg
On a print out of this slope field, sketch the solutions that pass through (0,0); (3,12); (12,3); (−14.5,3); (−6,36); and (−6,30).
A) For which positive values of P are the solutions increasing?
Increasing for: [I got: (-14.5,-6), (0,3)]
(Give your answer as an interval or list of intervals, e.g., if P is increasing between 1 and 5 and between 7 and infinity, enter (1,5),(7,Inf).)
B) For what positive values of P are the solutions decreasing?
Decreasing for: [I got (-6,0),(3,12)]
(Again, give your answer as an interval or list of intervals, e.g., if P is decreasing between 1 and 5 and between 7 and infinity, enter (1,5),(7,Inf).)
C)What is the equation of the solution to this differential equation that passes through (0,0)?
P= [ I got P=0]
D) If the solution passes through a value of P>0, what is the limiting value of P as t gets large?
P -> [ Im not sure how to answer this one ]
Thank you.
LINK TO THE SLOPE FIELD IMAGE: http://s7.postimage.org/8xg4q4s6z/image.jpg
On a print out of this slope field, sketch the solutions that pass through (0,0); (3,12); (12,3); (−14.5,3); (−6,36); and (−6,30).
A) For which positive values of P are the solutions increasing?
Increasing for: [I got: (-14.5,-6), (0,3)]
(Give your answer as an interval or list of intervals, e.g., if P is increasing between 1 and 5 and between 7 and infinity, enter (1,5),(7,Inf).)
B) For what positive values of P are the solutions decreasing?
Decreasing for: [I got (-6,0),(3,12)]
(Again, give your answer as an interval or list of intervals, e.g., if P is decreasing between 1 and 5 and between 7 and infinity, enter (1,5),(7,Inf).)
C)What is the equation of the solution to this differential equation that passes through (0,0)?
P= [ I got P=0]
D) If the solution passes through a value of P>0, what is the limiting value of P as t gets large?
P -> [ Im not sure how to answer this one ]
Thank you.
-
A)
P(t) is increasing when dP/dt > 0
0.0333333P(30−P) > 0
P(30−P) > 0
0 < P < 30
Solution: (0, 30)
B)
P(t) is decreasing when dP/dt < 0
P < 0 or P > 30
Since we are looking for positive values of P, then interval is (30, ∞)
C)
P = 0 seems correct
D)
Limiting value of P occurs when dP/dt = 0
0.0333333P(30−P) = 0
P = 0, or P = 30
Since we are looking at solutions through a value of P > 30, then
limiting value as t gets large is P = 30
--------------------------------
Notice how answer to D) matches answers to A) and B)
When P < 30, then P increases
When P > 30, then P decreases
So we can see that P always goes toward 30
Also, we could have solved differential equation to get
P(t) = 30 e^t / (C + e^t)
And then find limit at t approaches infinity:
lim[t→∞] 30 e^t / (C + e^t)
= lim[t→∞] 30 / (C e^(-t) + 1)
= 30 / (0 + 1)
= 30
P(t) is increasing when dP/dt > 0
0.0333333P(30−P) > 0
P(30−P) > 0
0 < P < 30
Solution: (0, 30)
B)
P(t) is decreasing when dP/dt < 0
P < 0 or P > 30
Since we are looking for positive values of P, then interval is (30, ∞)
C)
P = 0 seems correct
D)
Limiting value of P occurs when dP/dt = 0
0.0333333P(30−P) = 0
P = 0, or P = 30
Since we are looking at solutions through a value of P > 30, then
limiting value as t gets large is P = 30
--------------------------------
Notice how answer to D) matches answers to A) and B)
When P < 30, then P increases
When P > 30, then P decreases
So we can see that P always goes toward 30
Also, we could have solved differential equation to get
P(t) = 30 e^t / (C + e^t)
And then find limit at t approaches infinity:
lim[t→∞] 30 e^t / (C + e^t)
= lim[t→∞] 30 / (C e^(-t) + 1)
= 30 / (0 + 1)
= 30