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Math
CalculusSingle Variable Calculus: Early Transcendentals, Volume I8th Edition

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Chapter 3.4, Problem 81E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
Problem 43E
Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
Problem 49E
Problem 50E
Problem 51E
Problem 52E
Problem 53E
Problem 54E
Problem 55E
Problem 56E
Problem 57E
Problem 58E
Problem 59E
Problem 60E
Problem 61E
Problem 62E
Problem 63E
Problem 64E
Problem 65E
Problem 66E
Problem 67E
Problem 68E
Problem 69E
Problem 70E
Problem 71E
Problem 72E
Problem 73E
Problem 74E
Problem 75E
Problem 76E
Problem 77E
Problem 78E
Problem 79E
Problem 80E
Problem 81E
Problem 82E
Problem 83E
Problem 84E
Problem 85E
Problem 86E
Problem 87E
Problem 88E
Problem 89E
Problem 90E
Problem 93E
Problem 94E
Problem 95E
Problem 96E
Problem 97E
Problem 98E
Problem 99E
Problem 100E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by ±0.35. In view of these data, the brightness of Delta Cephei at time t, where t is measured in days, has been modeled by the function

B ( t ) = 4.0 + 0.35 sin ( 2 π t 5.4 )

(a) Find the rate of change of the brightness after t days.

(b) Find, correct to two decimal places, the rate of increase after one day.

(a)

To determine

To find: The rate of change of the brightness after t days.

Explanation

Given:

The function B(t)=4.0+0.35sin(2πt5.4).

Derivative rule:

(1) Constant Multiple Rule: ddx[cf(x)]=cddx[f(x)]

(2) Sum Rule: ddx(f+g)=ddx(f)+ddx(g)

(3) Quotient Rule: ddx(fg)=gddx(f)−fddx(g)(g)2

Result used: Chain Rule

If g is differentiable at x and f is differentiable at g(x), then the composite function F=f∘g defined by F(x)=f(g(x)) is differentiable at x and F′ is given by the product

F′(x)=f′(g(x))⋅g′(x) (1)

Calculation:

Obtain the derivative of B(t).

B′(t)=ddt(B(t))=ddt(4.0+0.35sin(2πt5.4))

Apply the sum rule (1) and the constant multiple rule (2),

B′(t)=ddt(4.0)+ddt(0.35sin(2πt5.4))=0+0.35ddt(sin(2πt5.4))

B′(t)=0.35ddt(sin(2πt5.4)) (2)

Obtain the derivative ddt(sin(2πt5.4)) by using the chain rule as shown in equation (1),

Let g(t)=2πt5.4 and f(u)=sinu  where u=g(t).

ddt(sin(2πt5.4))=f′(g(t))⋅g′(t) (3)

The derivative of f′(g(t)) is computed as follows,

f′(g(t))=f′(u)=ddu(f(u))=ddu(sinu)=cosu

Substitute u=2πt5

(b)

To determine

To find: The rate of increase after one day and correct to two decimal places.

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