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46. The following equation is used to calculate the aver- age firing rate f of a neuron (in spikes per second) as a function of the concentration x of neurotransmit- ters perfusing its synapses. 20 e3x f(x) = 2.1 + e3x Find the horizontal asymptote; then find the values

46. The following equation is used to calculate the aver- age firing rate f of a neuron (in spikes per second) as a function of the concentration x of neurotransmit- ters perfusing its synapses. 20 e3x f(x) = 2.1 + e3x Find the horizontal asymptote; then find the values

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question 46

2.5 Sequential Limits 155 b. Determine how large x needs to be to ensure that f (x) is within 0.01 of L. 46. The following equation is used to calculate the aver- age firing rate f of a neuron (in spikes per second) as a function of the concentration x of neurotransmit- 44. The von Bertalanffy growth curve is used to describe how the size L(usually in terms of length) of an ani- mal changes with time. The curve is given by ters perfusing its synapses. 20 e3x f(x) = L() — а(1 — е -bе -о) 2.1 + e3x Find the horizontal asymptote; then find the values of x such that f(x) is within 0.5% of its asymptotic where t measures time after birth and a, b, and to are positive parameters. We will derive this curve in Chapter 6. To better understand the meaning of the parameters to and b, carry out these steps. values. 47. The following equation is used to calculate the average firing rate f of a neuron (in spikes per second) as a function of the concentration x of neurotransmitters perfusing its synapses. a. Evaluate L(to). What does this imply about the meaning of to? b. Find lim L(t). What does this limit say about the biological meaning of a? 16 e5x f(x) = 3.2 + ебх c. Graph L(t) and discuss how an organism grows according to this curve. 45. At the beginning of the twentieth century, sev- eral notable biologists, including G. F. Gause and T. Carlson, studied the population dynamics of yeast. For example, Carlson grew yeast under constant environmental conditions in a flask; he regularly monitored their population densities.* In Chapter 6, we will show that the following function describes the Find the values of x such that f(x) is within 0.5% of its asymptotic values. 48. Compare the solutions obtained to Problems 46 and 47 and decide which of these represents a tighter on-off switch of the neuron, that is, from firing at its maximum rate to being inactive. What do you con- clude in terms of which of the parameters a, b, and c in the function growth of the population: аесх 9.7417e0.53t f(x) b+ ecx N(t) 1+0.01476e.531 controls the narrowness of the range of x over which on-off switching occurs? Note that this function is called the logistic function and will be encountered in many different examples in upcoming chapters. where N(t) is the population density and t is time in hours. Find lim N(t) and discuss the meaning of this limit.
Transcribed Image Text:2.5 Sequential Limits 155 b. Determine how large x needs to be to ensure that f (x) is within 0.01 of L. 46. The following equation is used to calculate the aver- age firing rate f of a neuron (in spikes per second) as a function of the concentration x of neurotransmit- 44. The von Bertalanffy growth curve is used to describe how the size L(usually in terms of length) of an ani- mal changes with time. The curve is given by ters perfusing its synapses. 20 e3x f(x) = L() — а(1 — е -bе -о) 2.1 + e3x Find the horizontal asymptote; then find the values of x such that f(x) is within 0.5% of its asymptotic where t measures time after birth and a, b, and to are positive parameters. We will derive this curve in Chapter 6. To better understand the meaning of the parameters to and b, carry out these steps. values. 47. The following equation is used to calculate the average firing rate f of a neuron (in spikes per second) as a function of the concentration x of neurotransmitters perfusing its synapses. a. Evaluate L(to). What does this imply about the meaning of to? b. Find lim L(t). What does this limit say about the biological meaning of a? 16 e5x f(x) = 3.2 + ебх c. Graph L(t) and discuss how an organism grows according to this curve. 45. At the beginning of the twentieth century, sev- eral notable biologists, including G. F. Gause and T. Carlson, studied the population dynamics of yeast. For example, Carlson grew yeast under constant environmental conditions in a flask; he regularly monitored their population densities.* In Chapter 6, we will show that the following function describes the Find the values of x such that f(x) is within 0.5% of its asymptotic values. 48. Compare the solutions obtained to Problems 46 and 47 and decide which of these represents a tighter on-off switch of the neuron, that is, from firing at its maximum rate to being inactive. What do you con- clude in terms of which of the parameters a, b, and c in the function growth of the population: аесх 9.7417e0.53t f(x) b+ ecx N(t) 1+0.01476e.531 controls the narrowness of the range of x over which on-off switching occurs? Note that this function is called the logistic function and will be encountered in many different examples in upcoming chapters. where N(t) is the population density and t is time in hours. Find lim N(t) and discuss the meaning of this limit.
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