Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector . Be sure to use a unit vector for the direction vector. 26 f ( x , y ) = x / ( x − y ) ; P ( 4 , 1 ) ; 〈 − 1 , 2 〉
Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector . Be sure to use a unit vector for the direction vector. 26 f ( x , y ) = x / ( x − y ) ; P ( 4 , 1 ) ; 〈 − 1 , 2 〉
Computing directional derivatives with the gradientCompute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector.
26
f
(
x
,
y
)
=
x
/
(
x
−
y
)
;
P
(
4
,
1
)
;
〈
−
1
,
2
〉
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
The spread of an infectious disease is often modeled using the following autonomous differential equation:
dI
-
- BI(N − I) − MI,
dt
where I is the number of infected people, N is the total size of the population being modeled, ẞ is a constant determining the rate of
transmission, and μ is the rate at which people recover from infection.
Close
a) (5 points) Suppose ẞ = 0.01, N = 1000, and µ = 2. Find all equilibria.
b) (5 points) For the equilbria in part a), determine whether each is stable or unstable.
c) (3 points) Suppose ƒ(I) = d. Draw a phase plot of f against I. (You can use Wolfram Alpha or Desmos to plot the function, or draw the
dt
function by hand.) Identify the equilibria as stable or unstable in the graph.
d) (2 points) Explain the biological meaning of these equilibria being stable or unstable.
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