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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

If a resistor of R ohms is connected across a battery of E volts with internal resistance r ohms, then the power (in watts) in the external resistor is

P = E 2 R ( R + r ) 2

If E and r are fixed but R varies, what is the maximum value of the power?

To determine

To find:

What is the maximum value of power if E and r are fixed but R varies.

Explanation

1) Concept:

i) A critical number of a function f   is a number c in the domain of f  such that either  f'c=0 or f'c does not exist.

ii) First derivative test for absolute extreme values- suppose that c is a critical number of a continuous function f defined on an interval.

If f'x>0 for all x<c and f'x<0 for all x>c, then fc is the absolute maximum value of f.

If f'x<0 for all x<c and f'x>0 for all >c, then fc is the absolute Minimum value of f.

2) Given:

P=E2RR+r2

3) Calculation:

As the power of external resistor is P=E2RR+r2

For finding the maximum value of the power, first find the critical points.

Therefore, differentiating P with respect to R by using rules of differentiation,

By using rule of constant multiplication.

P=E2RR+r2

P'R=ddRE2RR+r2=E2ddRRR+r2

By using quotient rule,

=E2R+r2ddRR-RddRR+r2R

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