(a) Explanation Given: Resistivity is the reciprocal of the conductivity and is denoted by ρ . The unit of resistivity is ohm-meter. The resistivity of a given metal is, ρ ( t ) = ρ 20 e α ( t − 20 ) . Calculation: The given equation is ρ ( t ) = ρ 20 e α ( t − 20 ) . The first derivative of ρ ( t ) with respect to t is found as, d d t ( ρ ( t ) ) = d d t ( ρ 20 e α ( t − 20 ) ) = ρ 20 d d t ( e α ( t − 20 ) ) = ρ 20 ( e α ( t − 20 ) ) d d t ( α ( t − 20 ) ) = α ρ 20 e α ( t − 20 ) That is, ρ ′ ( t ) = ρ 20 α e α ( t − 20 ) . The second derivative will be, d d t ( ρ ′ ( t ) ) = d d t ( ρ 20 α e α ( t − 20 ) ) = ρ 20 α d d t ( e α ( t − 20 ) ) = α ρ 20 ( e α ( t − 20 ) ) d d t ( α ( t − 20 ) ) = α 2 ρ 20 e α ( t − 20 ) That is, ρ ″ ( t ) = ρ 20 α 2 e α ( t − 20 ) . The Taylor series expansion is f ( x ) ≈ f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ . Substitute t = 20 in ρ ( t ) = ρ 20 e α ( t − 20 ) (b) To determine To graph: The resistivity of copper and the linear and quadratic approximations for − 250 ° C ≤ t ≤ 1000 ° C , if α = 0.0039 / ° C and ρ 20 = 1.7 × 10 − 8 Ω m . (c) To determine To find: The values of t , where the linear approximation agree with the exponential expression to within one percent.

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Author: Stewart

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Chapter 11.11, Problem 32E

(a)

To determine

To find: The expression for the linear and quadratic approximation.

(b)

To determine

To graph: The resistivity of copper and the linear and quadratic approximations for −250°C≤t≤1000°C, if α=0.0039/°C and ρ20=1.7×10−8Ωm.

(c)

To determine

To find: The values of t, where the linear approximation agree with the exponential expression to within one percent.

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Chapter 11.11, Problem 31E

Chapter 11.11, Problem 33E

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The expression for the linear and quadratic approximation.

Solution Summary: The author explains the expressions for the linear and quadratic approximation. The reciprocal of the conductivity is denoted by rho .

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To find: The expression for the linear and quadratic approximation.

To graph: The resistivity of copper and the linear and quadratic approximations for −250°C≤t≤1000°C, if α=0.0039/°C and ρ20=1.7×10−8Ωm.

To find: The values of t, where the linear approximation agree with the exponential expression to within one percent.

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Chapter 11 Solutions