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1 The function of has a derivative given by f'(x)=- a positive constant. x = R, x 0, x = k where k is x(k-x) (a) The expression for f'(x) can be written in the form Find a and b in terms of k. (b) Hence, find an expression for f(x). а b + where a, b = R. bЄ 3 x k-x Consider P, the population of a colony of ants, which has an initial value of 1200. dP P(k-P) dt The rate of change of the population can be modelled by the differential equation where t is the time measured in days, t≥0, and k is the upper bound for the population. (c) By solving the differential equation, show that P=- 1200k (k-1200)e 5+1200 At t = 10 the population of the colony has doubled in size from its initial value. (d) Find the value of k, giving your answer correct to four significant figures. 5k (e) Find the value of t when the rate of change of the population is at its maximum.

1 The function of has a derivative given by f'(x)=- a positive constant. x = R, x 0, x = k where k is x(k-x) (a) The expression for f'(x) can be written in the form Find a and b in terms of k. (b) Hence, find an expression for f(x). а b + where a, b = R. bЄ 3 x k-x Consider P, the population of a colony of ants, which has an initial value of 1200. dP P(k-P) dt The rate of change of the population can be modelled by the differential equation where t is the time measured in days, t≥0, and k is the upper bound for the population. (c) By solving the differential equation, show that P=- 1200k (k-1200)e 5+1200 At t = 10 the population of the colony has doubled in size from its initial value. (d) Find the value of k, giving your answer correct to four significant figures. 5k (e) Find the value of t when the rate of change of the population is at its maximum.

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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1 The function of has a derivative given by f'(x)=- a positive constant. x = R, x 0, x = k where k is x(k-x) (a) The expression for f'(x) can be written in the form Find a and b in terms of k. (b) Hence, find an expression for f(x). а b + where a, b = R. bЄ 3 x k-x
Transcribed Image Text:1 The function of has a derivative given by f'(x)=- a positive constant. x = R, x 0, x = k where k is x(k-x) (a) The expression for f'(x) can be written in the form Find a and b in terms of k. (b) Hence, find an expression for f(x). а b + where a, b = R. bЄ 3 x k-x
Consider P, the population of a colony of ants, which has an initial value of 1200. dP P(k-P) dt The rate of change of the population can be modelled by the differential equation where t is the time measured in days, t≥0, and k is the upper bound for the population. (c) By solving the differential equation, show that P=- 1200k (k-1200)e 5+1200 At t = 10 the population of the colony has doubled in size from its initial value. (d) Find the value of k, giving your answer correct to four significant figures. 5k (e) Find the value of t when the rate of change of the population is at its maximum.
Transcribed Image Text:Consider P, the population of a colony of ants, which has an initial value of 1200. dP P(k-P) dt The rate of change of the population can be modelled by the differential equation where t is the time measured in days, t≥0, and k is the upper bound for the population. (c) By solving the differential equation, show that P=- 1200k (k-1200)e 5+1200 At t = 10 the population of the colony has doubled in size from its initial value. (d) Find the value of k, giving your answer correct to four significant figures. 5k (e) Find the value of t when the rate of change of the population is at its maximum.
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