A submerged sensor moves along a path described implicitly by the equation a)
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A submerged sensor moves along a path described implicitly by the equation
a) (x2+y2)(1+cy)=16, where c is a constant related to water density.
At time t=0, the sensor is located at the point (2,2).
(a) Determine the value of c that ensures the initial position (2,2) satisfies the implicit relation.
(b) For the value of c found in part (a), compute dxdy at (2,2).
(c) The sensor’s horizontal velocity at t=0 is 3 m/s. Using your result from part (b), find the sensor’s vertical velocity at t=0.
(d) Discuss whether the sensor is moving toward increasing or decreasing circumference (distance from the origin) at t=0, and justify your conclusion using derivatives.