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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Transcribed Image Text:(1) Determine if the series are divergent or are absolutely or conditionally convergent:
1
(n² + 1)³/2*
(a)
(b)
(c)
iM8 iM8 IM8
(b)
2√n
3n
Σ(−1)n ln(n)
n
n=1
(2) Determine the radius of convergence:
(a)
Σ(2n + 1)(2x)2n.
n=0
∞
Σn!xn.
n=0
(3) Calculate the following indefinite integrals:
(a)
Javi
[2√1-3²
(b)
√√äln(x²) da
x²
(c)
x² + 6x +8
√1 - x² dx
J
dx
![(4)
The development of the population P in specific small city is analysed. The investigation
revealed that the rate of the change of population per year can be modeled as
P'(t) =
solution f.
10000
(t+ 2)²
where t is the time in years t from today. What is the expected difference from today's population
in the long run (this is for t →∞)?
(5)
Find the area of the region bounded by the curved y = x² — 1 and y = 2x + 2.
(6)
Calculate the volume of the solid of revolution that is formed by revolving the curve
y = 2 + sin(x) over the interval [0, 2π] around the x-axis.
(7) Consider the function ƒ defined on R that fulfils the following conditions:
ƒ'(x) = D · f(x) for all x = R and ƒ(0) = fo
where D and fo are given non-zero constants. This type of equation arises in many applications
when modelling decay (D < 0) or growth (D > 0). Your task is to find f.
(a)
Calculate the coefficients c₂ of the power series representation of the
∞
n=0
(b)
Calculate the radius of convergence of the power series of part (b).
(c)
Use the Taylor series for the exponential function et to derive the Taylor series
of the function g(x) = foeª and compare the result with the power series you obtained in
part (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0b3755f-901a-432f-bc9e-85470cc9e964%2Fd7ee364e-7e01-43d9-9802-95a1b23c4580%2F2ajcyd_processed.png&w=3840&q=75)
Transcribed Image Text:(4)
The development of the population P in specific small city is analysed. The investigation
revealed that the rate of the change of population per year can be modeled as
P'(t) =
solution f.
10000
(t+ 2)²
where t is the time in years t from today. What is the expected difference from today's population
in the long run (this is for t →∞)?
(5)
Find the area of the region bounded by the curved y = x² — 1 and y = 2x + 2.
(6)
Calculate the volume of the solid of revolution that is formed by revolving the curve
y = 2 + sin(x) over the interval [0, 2π] around the x-axis.
(7) Consider the function ƒ defined on R that fulfils the following conditions:
ƒ'(x) = D · f(x) for all x = R and ƒ(0) = fo
where D and fo are given non-zero constants. This type of equation arises in many applications
when modelling decay (D < 0) or growth (D > 0). Your task is to find f.
(a)
Calculate the coefficients c₂ of the power series representation of the
∞
n=0
(b)
Calculate the radius of convergence of the power series of part (b).
(c)
Use the Taylor series for the exponential function et to derive the Taylor series
of the function g(x) = foeª and compare the result with the power series you obtained in
part (a).
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