The inverse of the given function and explain its meaning

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 1.6, Problem 60E

(a)

To determine

To find : The inverse of the given function and explain its meaning

Expert Solution

Explanation of Solution

Given: The given function is Q(t)=Q0(1et/a) .

Consider the function.

Q(t)=Q0(1et/a)

To get the inverse replace the Q with t and vice-versa in the given function.

t=Q0(1eQa)tQ0=(1eQa)1tQ0=(eQa)ln(1tQ0)=ln(eQa)

Solve further,

Q=aln(1tQ0)f1(t)=aln(1tQ0)

Now interchange the variables again.

f1(Q)=aln(1QQ0)

Therefore, the inverse of the function is f1(Q)=aln(1QQ0) and it represent the time needed to charge the Capacitor to Q .

(b)

To determine

Expert Solution

Explanation of Solution

Given: The given function is Q(t)=Q0(1et/a) and a=2

The inverse of the function is,

f1(Q)=aln(1QQ0)

Substitute 2 for a and 0.9Q0 for Q in the above expression.

f1(0.9Q0)=2ln(10.9Q0Q0)=2ln0.1=4.6sec

Therefore, the capacitor recharges in 4.6sec .

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