Find the rate of change of the distance between the origin and a moving point on the graph of y=sinx if dx/dt=2 centimeters per second.
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Coordinates of moving point are (x, sin(x))
We find distance (D) between origin and point using distance formula
D = √((x-0)²+(sinx-0)²)
D = √(x²+sin²x)
Differentiate both sides with respect to t:
dD/dt = d/dt (√(x²+sin²x))
dD/dt = d/dx (√(x²+sin²x)) * dx/dt
dD/dt = 1/2 (x²+sin²x)^(-1/2) * (2x + 2 sinx cosx) * 2 cm/sec
dD/dt = 2 (x + sinx cosx) * 2 / (2√(x²+sin²x)) cm/sec
dD/dt = 2 (x + sinx cosx) / √(x²+sin²x) cm/sec
We find distance (D) between origin and point using distance formula
D = √((x-0)²+(sinx-0)²)
D = √(x²+sin²x)
Differentiate both sides with respect to t:
dD/dt = d/dt (√(x²+sin²x))
dD/dt = d/dx (√(x²+sin²x)) * dx/dt
dD/dt = 1/2 (x²+sin²x)^(-1/2) * (2x + 2 sinx cosx) * 2 cm/sec
dD/dt = 2 (x + sinx cosx) * 2 / (2√(x²+sin²x)) cm/sec
dD/dt = 2 (x + sinx cosx) / √(x²+sin²x) cm/sec