3.5.1 Find the following limits. ...
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3.5.2 Find the following limits. ...
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3.5.3 Find the following limits. ...
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3.5.4 Find the following limits. ...
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3.5.5 Find the following limits. ...
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3.5.6 Find the following limits. ...
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3.5.7 Find the following limits. ...
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3.5.8 Find the following limits. ...
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3.5.9 For each pair of functions, say which approaches infinity faster as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.10 For each pair of functions, say which approaches infinity faster as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.11 For each pair of functions, say which approaches infinity faster as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.12 For each pair of functions, say which approaches infinity faster as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.13 For each pair of functions, say which approaches infinity faster as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.14 For each pair of functions, say which approaches infinity faster as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.15 For each pair of functions, say which approaches 0 more quickly as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.17 For each pair of functions, say which approaches 0 more quickly as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.17 For each pair of functions, say which approaches 0 more quickly as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.18 For each pair of functions, say which approaches 0 more quickly as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.19 For each pair of functions, say which approaches 0 more quickly as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.20 For each pair of functions, say which approaches 0 more quickly as x approaches infinity. Explain which rule you used to compare each pair. Compute the value of each function at x = 1, x = 10, and x = 100. How do these compare with the order of the functions in the limit? If they are different, how large would x have to be for the values to match the order in the limit? ...
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3.5.21 The following are possible absorption functions. What happens to each as c approaches infinity? Assume that all parameters take on positive values. ...
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3.5.22 The following are possible absorption functions. What happens to each as c approaches infinity? Assume that all parameters take on positive values. ...
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3.5.23 The following are possible absorption functions. What happens to each as c approaches infinity? Assume that all parameters take on positive values. ...
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3.5.24 The following are possible absorption functions. What happens to each as c approaches infinity? Assume that all parameters take on positive values. ...
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3.5.25 Find
the derivatives of the following absorption functions (from Table 3.1
with particular values of the parameters). Compute the value at c = 0 and the limit of the derivative as c approaches infinity. Are your results consistent with the figures? α( c )=5c. Reference Table 3.1 ...
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3.5.26 Find
the derivatives of the following absorption functions (from Table 3.1
with particular values of the parameters). Compute the value at c = 0 and the limit of the derivative as c approaches infinity. Are your results consistent with the figures? ... Reference Table 3.1 ...
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3.5.27 Find
the derivatives of the following absorption functions (from Table 3.1
with particular values of the parameters). Compute the value at c = 0 and the limit of the derivative as c approaches infinity. Are your results consistent with the figures? ... Reference Table 3.1 ...
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3.5.28 Find
the derivatives of the following absorption functions (from Table 3.1
with particular values of the parameters). Compute the value at c = 0 and the limit of the derivative as c approaches infinity. Are your results consistent with the figures? ... Reference Table 3.1 ...
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3.5.29 Find
the derivatives of the following absorption functions (from Table 3.1
with particular values of the parameters). Compute the value at c = 0 and the limit of the derivative as c approaches infinity. Are your results consistent with the figures? ... Reference Table 3.1 ...
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3.5.30 Find
the derivatives of the following absorption functions (from Table 3.1
with particular values of the parameters). Compute the value at c = 0 and the limit of the derivative as c approaches infinity. Are your results consistent with the figures? α( c ) = 5c(1 + c). Reference Table 3.1 ...
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3.5.31 A bacterial population that obeys the discrete-time dynamical system ...with the initial condition ... has the solution .... For the following values of r and ...,
state which populations increase to infinity and which decrease to 0.
For those increasing to infinity, find the time when the population will
reach .... For those decreasing to 0, find the time when the population will reach ... ...
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3.5.32 A bacterial population that obeys the discrete-time dynamical system ...with the initial condition ... has the solution .... For the following values of r and ...,
state which populations increase to infinity and which decrease to 0.
For those increasing to infinity, find the time when the population will
reach .... For those decreasing to 0, find the time when the population will reach ... ...
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3.5.33 A bacterial population that obeys the discrete-time dynamical system ...with the initial condition ... has the solution .... For the following values of r and ...,
state which populations increase to infinity and which decrease to 0.
For those increasing to infinity, find the time when the population will
reach .... For those decreasing to 0, find the time when the population will reach ... ...
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3.5.34 A bacterial population that obeys the discrete-time dynamical system ...with the initial condition ... has the solution .... For the following values of r and ...,
state which populations increase to infinity and which decrease to 0.
For those increasing to infinity, find the time when the population will
reach .... For those decreasing to 0, find the time when the population will reach ... ...
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3.5.35 In
the polymerase chain reaction used to amplify DNA, some sequences of
DNA produced are too long and others are the right length. Denote the
number of overly long pieces after t generations of the process by ... and the number of piece of the right length by ....
The dynamics follow approximately ... because two new overly
long pieces are produced at each step while the number of good pieces
doubles. Suppose that ... Find expressions for ...and compute the fraction of pieces that are too long after 1, 5, 10, and 20 generations of the process.
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3.5.36 In
the polymerase chain reaction used to amplify DNA, some sequences of
DNA produced are too long and others are the right length. Denote the
number of overly long pieces after t generations of the process by ... and the number of piece of the right length by ....
The dynamics follow approximately ... because two new overly
long pieces are produced at each step while the number of good pieces
doubles. Suppose that ... Find
the ratio of the number of pieces that are too long to the total number
of pieces as a function of time. What is the limit? How long would you
have to wait to make sure that less than one in a million pieces are too
long? (This can’t be solved exactly, just plug in some numbers.)
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3.5.37 Consider the discrete-time dynamical system for medication given by ... Find the equilibrium.
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3.5.38 Consider the discrete-time dynamical system for medication given by ... The solution is ... Find the limit as t →∞.
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3.5.39
Consider the discrete-time dynamical system for medication given by
... How long will the solution take to be within 1% of the equilibrium?
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3.5.40
Consider the discrete-time dynamical system for medication given by
... What are two ways to show that this equilibrium is stable?
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3.5.41
The amount of food a predator eats as a function of prey density is
called the functional response. Functional response is often broken into
three categories. • Type I: Linear. • Type II: Increasing, concave
down, finite limit. • Type III: Increasing with finite limit, concave
up for small prey densities, concave down for large prey densities.
Sketch
a picture of the type II response. Which of the absorption functions
does it resemble? What is the optimal prey density for a predator?
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3.5.42
The amount of food a predator eats as a function of prey density is
called the functional response. Functional response is often broken into
three categories. • Type I: Linear. • Type II: Increasing, concave
down, finite limit. • Type III: Increasing with finite limit, concave
up for small prey densities, concave down for large prey densities.
Sketch
a picture of the type III response. Which of the absorption functions
does it resemble? What is the optimal prey density for a predator?
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3.5.43 Now
suppose that the number of prey that escape increases linearly with the
number of prey (the prey join together and fight back). Let p be the number of prey and F(p) be the functional response. The number of prey captured is then F(p) − cp. The constant c represents
how effectively the prey can fight. Write the equation for the optimal
prey density (the value giving the maximum rate of prey capture) in
terms of F ′ (p). Find the optimal prey density if F( p) = p. Make sure to separately consider cases with c < 1 and c > 1
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3.5.44 Now
suppose that the number of prey that escape increases linearly with the
number of prey (the prey join together and fight back). Let p be the number of prey and F(p) be the functional response. The number of prey captured is then F(p) − cp. The constant c represents
how effectively the prey can fight. Draw a picture illustrating the
optimal prey density in a case with a type II functional response.
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3.5.45 Now
suppose that the number of prey that escape increases linearly with the
number of prey (the prey join together and fight back). Let p be the number of prey and F(p) be the functional response. The number of prey captured is then F(p) − cp. The constant c represents
how effectively the prey can fight. Draw a picture illustrating the
optimal prey density in a case with a type III functional response.
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3.5.46 Now
suppose that the number of prey that escape increases linearly with the
number of prey (the prey join together and fight back). Let p be the number of prey and F(p) be the functional response. The number of prey captured is then F(p) − cp. The constant c represents how effectively the prey can fight. Find the optimal prey density if .... Make sure to separately consider cases with c < 1 and c > 1.
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3.6.1 Find the leading behavior of the following functions at 0 and ∞. f ( x )=1 + x.
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