In an optics experiment

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

Problem

In an optics experiment, the angle of incidence θ\theta (in radians) is modeled by θ(x)=arctan⁡ ⁣(2x1−x2),\theta(x)=\arctan\!\Bigl(\frac{2x}{1-x^{2}}\Bigr), where the refractive parameter xx varies with time tt (in seconds) according to x(t)=t2+1−t.x(t)=\sqrt{t^{2}+1}-t. Assume t≥0t\ge 0. (a) Express dθdt\dfrac{d\theta}{dt} entirely in terms of tt. Simplify your answer as much as algebraically feasible. (b) Evaluate dθdt\dfrac{d\theta}{dt} at t=0t=0. (c) Using the sign of dθdt\dfrac{d\theta}{dt}, determine whether θ\theta is increasing or decreasing for small positive values of tt. Provide justification based on your result from part (a).

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