Ocean engineers model external pressure (in kPa) on a submersible as a differentiable

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Ocean engineers model external pressure

p

(in kPa) on a submersible as a differentiable, one-to-one function of depth

D

(in m), written

p=H(D)

. At a depth of

180\text{ m}

the measurements are

H(180)=1.8\text{ kPa}

and

H'(180)=0.012\text{ kPa/m}

. Water temperature

T

(in

^{\circ}

C) also varies with depth and satisfies

\displaystyle \frac{dT}{dD}=-0.020\;^{\circ}\text{C/m}

at the same location. (a) Compute the value of

\displaystyle \frac{dD}{dp}

at the point where

D=180\text{ m}

. Show all work and include units. (b) Using your result from part (a), determine the instantaneous rate of change of temperature with respect to pressure,

\displaystyle \frac{dT}{dp}

, at that same point. Interpret the sign of your answer in context.

Related theory

Rates of Change
Derivatives of Inverse Functions
Advanced Differentiation
Differentiation Rules
Derivatives of standard functions via chain rule
Derivatives of composite functions
Continuity and Differentiability

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