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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 99
Capitolo 3, Problema 99

Resistors connected in parallel If two resistors of R₁ and R₂ ohms are connected in parallel in an electric circuit to make an R-ohm resistor, the value of R can be found from the equation


1/R = 1/R₁ + 1/R₂


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If R₁ is decreasing at the rate of 1ohm/sec and R₂ is increasing at the rate of 0.5 ohm/sec, at what rate is R changing when R₁ = 75 ohms and R₂ = 50 ohms?

Guida verificata passo dopo passo
1
Start by understanding the given equation for resistors in parallel: \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \). This equation relates the total resistance \( R \) to the individual resistances \( R_1 \) and \( R_2 \).
Differentiate both sides of the equation with respect to time \( t \) to find the rate of change of \( R \). Use implicit differentiation: \( \frac{d}{dt} \left( \frac{1}{R} \right) = \frac{d}{dt} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \).
Apply the chain rule to differentiate: \( -\frac{1}{R^2} \frac{dR}{dt} = -\frac{1}{R_1^2} \frac{dR_1}{dt} - \frac{1}{R_2^2} \frac{dR_2}{dt} \).
Substitute the given rates of change: \( \frac{dR_1}{dt} = -1 \) ohm/sec (since \( R_1 \) is decreasing) and \( \frac{dR_2}{dt} = 0.5 \) ohm/sec (since \( R_2 \) is increasing).
Substitute the given values \( R_1 = 75 \) ohms and \( R_2 = 50 \) ohms into the differentiated equation to solve for \( \frac{dR}{dt} \), the rate at which \( R \) is changing.

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Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how the equivalent resistance R changes over time as the individual resistances R₁ and R₂ change. This requires applying the chain rule from calculus to relate the rates of change of R₁, R₂, and R.
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Intro To Related Rates

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations that define one variable in terms of another without explicitly solving for one variable. In this case, we will differentiate the equation 1/R = 1/R₁ + 1/R₂ with respect to time to find the rate of change of R, using the known rates of change of R₁ and R₂.
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Finding The Implicit Derivative

Reciprocal Functions

Reciprocal functions are functions of the form f(x) = 1/x, which have unique properties, especially in calculus. The equation for resistors in parallel involves reciprocal relationships, and understanding how to differentiate these functions is crucial for solving the problem. The behavior of these functions, particularly their rates of change, will play a key role in determining how R changes as R₁ and R₂ vary.
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Graphs of Secant and Cosecant Functions
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