In a microfluidic device

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

In a microfluidic device, the volumetric flow rate QQ (in μ\muL/s) is an invertible, differentiable function of the pressure difference ΔP\Delta P (in Pa), expressed as Q=F(ΔP)Q=F(\Delta P). For ΔP=120 Pa\Delta P=120\text{ Pa}, laboratory measurements give F(120)=0.45 μL/sF(120)=0.45\,\mu\text{L/s} and F′(120)=0.004 μL/(s⋅Pa)F'(120)=0.004\,\mu\text{L}/(\text{s}\cdot\text{Pa}). The fluid temperature TT (in ∘^{\circ}C) satisfies dTdQ=15 ∘C⋅s/μL\displaystyle \frac{dT}{dQ}=15\,^{\circ}\text{C}\cdot\text{s}/\mu\text{L} when Q=0.45 μL/sQ=0.45\,\mu\text{L/s}. Thermal expansion causes a channel length LL (in m) to change with temperature according to dLdT=1.2×10−4 m/∘C\displaystyle \frac{dL}{dT}=1.2\times10^{-4}\,\text{m}/^{\circ}\text{C} over the relevant range. (a) Determine (F−1)′(0.45)\displaystyle \left(F^{-1}\right)'\bigl(0.45\bigr) and state its units. (b) Compute dTd ΔP\displaystyle \frac{dT}{d\,\Delta P} at the instant when ΔP=120 Pa\Delta P=120\text{ Pa}. Include units. (c) Using your preceding answers, find dLd ΔP\displaystyle \frac{dL}{d\,\Delta P} at the same instant. (d) State whether the channel length is increasing or decreasing as the pressure difference increases at that moment, and explain how your calculation supports your conclusion.

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