# The rate of change of the brightness after t days.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.4, Problem 71E

(a)

To determine

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

Expert Solution

The rate of change of the brightness after t days is B(t)=7π54cos(2πt5.4)_.

### Explanation of Solution

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=2πt5.4 in the above equation,

f(g(t))=cos(2πt5.4)

Thus, the derivative is f(g(t))=cos(2πt5.4).

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

g(t)=ddt(2πt5.4)=2π5.4ddt(t)=2π5.4(1)=2π5.4

Thus, the derivative is g(t)=2π5.4.

Substitute cos(2πt5.4) for f(g(t)) and 2π5.4 for g(t) in equation (3),

ddt(sin(2πt5.4))=cos(2πt5.4)(2π5.4)=2π5.4cos(2πt5.4)

Substitute 2π5.4cos(2πt5.4) for ddt(sin(2πt5.4)) in equation (2),

B(t)=0.35(2π5.4cos(2πt5.4))=0.7π5.4cos(2πt5.4)0.407cos(2πt5.4)

Therefore, the rate of change of the brightness after t days is B(t)=0.41cos(2πt5.4)_.

(b)

To determine

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

Expert Solution

The rate of increase after one day is B(1)=0.16.

### Explanation of Solution

Calculation:

From part (a), the rate of change of the brightness after t days is B(t)=0.41cos(2πt5.4).

Substitute t=1 in the above equation,

B(1)=7π54cos(2π(1)5.4)=7π54cos(2π5.4)(0.407)(0.396)=0.161172

Therefore, the rate of increase after one day is B(1)=0.16.

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