3.3.1 Find all critical points of the following functions. ...
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3.3.2 Find all critical points of the following functions. ...
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3.3.3 Find all critical points of the following functions. ...
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3.3.4 Find all critical points of the following functions. ...
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3.3.5 Find all critical points of the following functions. ...
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3.3.6 Find all critical points of the following functions. c (θ) = cos(2πθ).
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3.3.7
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. ...
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3.3.8
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. ...
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3.3.9
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. ...
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3.3.10
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. ...
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3.3.11
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. ...
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3.3.13
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. Find
the second derivative at the critical points of the following
functions. Classify the critical points as minima and maxima. Use the
second derivative to draw an accurate graph of the function for the
given range. ... (as in Exercise 1) for 0 ≤ x ≤ 1.
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3.3.13
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. Find
the second derivative at the critical points of the following
functions. Classify the critical points as minima and maxima. Use the
second derivative to draw an accurate graph of the function for the
given range. ... (as in Exercise 1) for 0 ≤ x ≤ 1.
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3.3.14
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. Find
the second derivative at the critical points of the following
functions. Classify the critical points as minima and maxima. Use the
second derivative to draw an accurate graph of the function for the
given range. ... (as in Exercise 2) for 0 ≤ x ≤ 2.
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3.3.15
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. Find
the second derivative at the critical points of the following
functions. Classify the critical points as minima and maxima. Use the
second derivative to draw an accurate graph of the function for the
given range. ... (as in Exercise 3) for −2 ≤ w ≤ 2.
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3.3.16
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. Find
the second derivative at the critical points of the following
functions. Classify the critical points as minima and maxima. Use the
second derivative to draw an accurate graph of the function for the
given range. ... (as in Exercise 4) for 0 ≤ y ≤ 2.
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3.3.17
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. Find
the second derivative at the critical points of the following
functions. Classify the critical points as minima and maxima. Use the
second derivative to draw an accurate graph of the function for the
given range. ... (as in Exercise 5) for −1 ≤ z ≤ 1.
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3.3.18
Find the global minimum and maximum of the following functions on the
interval given. Don’t forget to check the endpoints. Find
the second derivative at the critical points of the following
functions. Classify the critical points as minima and maxima. Use the
second derivative to draw an accurate graph of the function for the
given range. c (θ) = cos(2πθ) (as in Exercise 6) for −1 ≤ θ ≤ 1.
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3.3.20 Suppose f (x ) is a positive function with a maximum at .... We can often find maxima and minima of other functions composed with f (x). For each of the functions h(x) = g( f (x)),
a. Show that h has a critical point at ...
b. Compute the second derivative at this point.
c. Check whether your function has a minimum or a maximum and explain.
d. Check your result using the function ...for x ≥ 0 which has a maximum at x = 1. Sketch a graph of f (x) and h(x) in this case. g ( f )=1 − f .
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3.3.20 Suppose f (x ) is a positive function with a maximum at .... We can often find maxima and minima of other functions composed with f (x). For each of the functions h(x) = g( f (x)),
a. Show that h has a critical point at ...
b. Compute the second derivative at this point.
c. Check whether your function has a minimum or a maximum and explain.
d. Check your result using the function ...for x ≥ 0 which has a maximum at x = 1. Sketch a graph of f (x) and h(x) in this case. g ( f )=1 − f .
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3.3.21 Suppose f (x ) is a positive function with a maximum at .... We can often find maxima and minima of other functions composed with f (x). For each of the functions h(x) = g( f (x)),
a. Show that h has a critical point at ...
b. Compute the second derivative at this point.
c. Check whether your function has a minimum or a maximum and explain.
d. Check your result using the function ...for x ≥ 0 which has a maximum at x = 1. Sketch a graph of f (x) and h(x) in this case. g ( f )=ln( f ).
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3.3.22 Suppose f (x ) is a positive function with a maximum at .... We can often find maxima and minima of other functions composed with f (x). For each of the functions h(x) = g( f (x)),
a. Show that h has a critical point at ...
b. Compute the second derivative at this point.
c. Check whether your function has a minimum or a maximum and explain.
d. Check your result using the function ...for x ≥ 0 which has a maximum at x = 1. Sketch a graph of f (x) and h(x) in this case. ...
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3.3.23 Solve the following optimization problems. Organic waste deposited in a lake at t = 0 decreases the oxygen content of the water. Suppose the oxygen content is ...for 0 ≤ t ≤ 25. Find the maximum and minimum oxygen content during this time.
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3.3.24
Solve the following optimization problems. The size of a population
of bacteria introduced to a nutrient grows according to ... Find
the maximum size of this population for t ≥ 0.
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3.3.25 No calculus book is complete without optimization problems involving fences. A
farmer owns 1000 m of fence and wants to enclose the largest possible
rectangular area. The region to be fenced has a straight canal on one
side, and a perpendicular and perfectly straight ancient stone wall on
another. The area thus needs to be fenced on only two sides. What is the
largest area she can enclose?
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3.3.26 No calculus book is complete without optimization problems involving fences. A
farmer owns 1000 m of fence and wants to enclose the largest possible
rectangular area. The region to be fenced has a straight canal on one
side, and thus needs to be fenced on only three sides. What is the
largest area she can enclose?
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3.3.27 Consider the bee faced by the problem in Subsection 3.3.2. Find the optimal strategy with the following travel times τ and illustrate the graphical method of solution. For each particular value of τ , find the equation of the tangent line at the optimal t and show that it goes through the point (−τ, 0). τ = 2.0.
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3.3.28 Consider the bee faced by the problem in Subsection 3.3.2. Find the optimal strategy with the following travel times τ and illustrate the graphical method of solution. For each particular value of τ , find the equation of the tangent line at the optimal t and show that it goes through the point (−τ, 0). τ= 0.5.
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3.3.29 Consider the bee faced by the problem in Subsection 3.3.2. Find the optimal strategy with the following travel times τ and illustrate the graphical method of solution. For each particular value of τ , find the equation of the tangent line at the optimal t and show that it goes through the point (−τ, 0). τ = 0.1.
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3.3.30 Consider the bee faced by the problem in Subsection 3.3.2. Find the optimal strategy with the following travel times τ and illustrate the graphical method of solution. For each particular value of τ , find the equation of the tangent line at the optimal t and show that it goes through the point (−τ, 0). Find the solution in general (without substituting a value for τ ). What is the limit as τ approaches 0? Does this answer make sense?
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3.3.31 Suppose that the total food collected by a bee follows ... where c is some parameter. If τ = 1.0, find the optimal departure time in the following circumstances. Sketch the plot from the graphical method. c = 2.0.
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3.3.32 Suppose that the total food collected by a bee follows ... where c is some parameter. If τ = 1.0, find the optimal departure time in the following circumstances. Sketch the plot from the graphical method. c = 1.0.
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3.3.33 Suppose that the total food collected by a bee follows ... where c is some parameter. If τ = 1.0, find the optimal departure time in the following circumstances. Sketch the plot from the graphical method. c = 0.1.
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3.3.34 Find the solution in general (without substituting a value for c). What does the parameter c mean
biologically (think about how long it takes the bee to collect half the
nectar)? Explain in words why the bee leaves sooner when c is smaller.
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3.3.35 Mathematical
models can help us estimate values that are difficult to measure.
Consider again a bee sucking nectar from a flower, with ... We also measure that the bee remains a length of time t on the flower. Estimate the travel time τ assuming that the bee understands the Marginal Value Theorem for the following values of t. t = 1.0.
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3.3.36 Mathematical
models can help us estimate values that are difficult to measure.
Consider again a bee sucking nectar from a flower, with ... We also measure that the bee remains a length of time t on the flower. Estimate the travel time τ assuming that the bee understands the Marginal Value Theorem for the following values of t. t = 0.1.
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3.3.37 Mathematical
models can help us estimate values that are difficult to measure.
Consider again a bee sucking nectar from a flower, with ... We also measure that the bee remains a length of time t on the flower. Estimate the travel time τ assuming that the bee understands the Marginal Value Theorem for the following values of t. t = 4.0.
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3.3.38 Mathematical
models can help us estimate values that are difficult to measure.
Consider again a bee sucking nectar from a flower, with ... We also measure that the bee remains a length of time t on the flower. Estimate the travel time τ assuming that the bee understands the Marginal Value Theorem for the following values of t. Find the solution in general (without substituting a value for t).
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3.3.39 We never showed that the value found in computing the optimal t with the Marginal Value Theorem is in fact a maximum. For each of the following forms for the function F(t), find the second derivative of R(t) and the point where ...and check whether the solution is a maximum. Suppose ... and travel time is τ = 1.
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3.3.40 We never showed that the value found in computing the optimal t with the Marginal Value Theorem is in fact a maximum. For each of the following forms for the function F(t), find the second derivative of R(t) and the point where ...and check whether the solution is a maximum. Suppose F(t) is any function with F ″ (t)<0 and travel time is τ =1.
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3.3.41 Animals
must survive predation in addition to maximizing their rate of food
intake. One theory assumes that they try to maximize the ratio of food
collected to predation risk. Suppose that different flowers with nectar
of quality n (the rate of food collection) attract P(n) predators. For example, flowers with higher-quality nectar (large values of n) might attract more predators (large value of P(n)). Bees must decide which flowers to select. For each of the following forms of P(n), find the function the bees are trying to maximize, and find the optimal n. Suppose that .... Find the optimal n for the bees.
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3.3.42 Suppose that P(n)= 1 + n. Find the optimal n for the bees and draw a graph like that for the Marginal Value Theorem. Does this make sense? Why is the result so different?
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3.3.43 Find the condition for the maximum for a general function P(n) by solving for P_ (n). Use this condition to find the optimal n for the cases P(n)= ...and P(n)=1 + n.
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3.3.44 Find a graphical interpretation of the condition in the previous problem and test it on P(n)= ...and P(n)=1 + n.
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3.3.45 Find the maximum harvest from a population following the discrete-time dynamical system ... for the given values of r .
a. Find the equilibrium population as a function of h. What is the largest h consistent with a positive equilibrium?
b. Find the equilibrium harvest as a function of h.
c. Find the harvesting effort that maximizes harvest.
d. Find the maximum harvest. r = 2.0.
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3.3.46 Find the maximum harvest from a population following the discrete-time dynamical system ... for the given values of r .
a. Find the equilibrium population as a function of h. What is the largest h consistent with a positive equilibrium?
b. Find the equilibrium harvest as a function of h.
c. Find the harvesting effort that maximizes harvest.
d. Find the maximum harvest. r = 1.5.
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3.3.47 Find the conditions for stability of the equilibrium of ... for the following values of r . Show that the equilibrium N ∗ is stable when h is set to the value that maximizes the long-term harvest. Graph the updating function and cobweb. r = 2.5, as in the text.
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3.3.48 Find the conditions for stability of the equilibrium of ... for the following values of r . Show that the equilibrium N ∗ is stable when h is set to the value that maximizes the long-term harvest. Graph the updating function and cobweb. r = 1.5, as in Exercise 46.
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3.3.49
Calculate the maximum long-term harvest for an alternative model of
competition obeying the discrete-time dynamical system ... Try
the following steps for the given values of the parameters r and k.
a. Find the equilibrium as a function of h.
b. What is the largest value of h consistent with a positive equilibrium?
c. Find the harvest level that produces the maximum long term harvest.
d. Sketch a graph of P(h) and compute the value at the maximum.
e. How do the results compare with those using the discrete time dynamical system in the text? With r = 2.5 and k = 1.
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3.3.50
Calculate the maximum long-term harvest for an alternative model of
competition obeying the discrete-time dynamical system ... Try
the following steps for the given values of the parameters r and k.
a. Find the equilibrium as a function of h.
b. What is the largest value of h consistent with a positive equilibrium?
c. Find the harvest level that produces the maximum long term harvest.
d. Sketch a graph of P(h) and compute the value at the maximum.
e. How do the results compare with those using the discrete time dynamical system in the text? With r = 1.5 and k = 1.
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3.3.51
The model of fish harvesting studied in the text includes nothing
about harvesting cost. Suppose that the population follows ...
as in the text, but that the payoff is ... where c is the cost per unit effort of harvesting. Find the optimal harvest for the following values of c, the associated equilibrium population N∗, and the associated payoff P(h). Do your answers all make sense? What should the fisherman do if c becomes too large? c = 0.1.
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3.3.52
The model of fish harvesting studied in the text includes nothing
about harvesting cost. Suppose that the population follows ...
as in the text, but that the payoff is ... where c is the cost per unit effort of harvesting. Find the optimal harvest for the following values of c, the associated equilibrium population N∗, and the associated payoff P(h). Do your answers all make sense? What should the fisherman do if c becomes too large? c = 0.2.
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3.3.53
The model of fish harvesting studied in the text includes nothing
about harvesting cost. Suppose that the population follows ...
as in the text, but that the payoff is ... where c is the cost per unit effort of harvesting. Find the optimal harvest for the following values of c, the associated equilibrium population N∗, and the associated payoff P(h). Do your answers all make sense? What should the fisherman do if c becomes too large? c = 0.5.
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3.3.55
Suppose a population follows the updating function ... but
can be harvested only every second year. This means that harvest
alternates between the chosen value h and 0.
a. Find the 2-yr updating function.
b. Find the optimal harvest.
c. Compare with the results in the text. Is it better to harvest less often?
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3.3.55
Suppose a population follows the updating function ... but
can be harvested only every second year. This means that harvest
alternates between the chosen value h and 0.
a. Find the 2-yr updating function.
b. Find the optimal harvest.
c. Compare with the results in the text. Is it better to harvest less often?
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