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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

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=fg 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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