A point is moving along the graph of the given function at the rate dx/dt. Find dy/dt for the given values of x. y = 2x2 + 5; dx/dt = 3 centimeters per second when x=-1 , x=0 , and x=1

Answer 1

The point is moving with rate of dx/dt. We want to find dy/dt based on the fact that we have a function relating y to x and we are given dx/dt at specific values of x.

This is the quintessential related-rates problem...

y = 2x² + 5

dy/dx = 2*2*x = 4x (power rule)

But, we need to think about the derivative as a ratio of two very small numbers dy and dx. In this sense we can use this "concept" to multiply and/or divide by these quantities to build "new quantities"...

dy/dx = dy/dt * dt/dx --> Nothing more than "inserting dt" to be cancelled.

dy/dx = dy/dt / (dx/dt) --> Changed multiplication into division by use of the reciprocal.

We have dy/dx and we can evaluate dy/dx at x= -1, 0, 1

dy/dx | x=1 --> 4

dy/dx | x=0 --> 0

dy/dx | x=-1 --> -4

at x = 1 : 4 = dy/dt / (dx/dt) --> dy/dt = 4 * dx/dx = 12 cm/second

at x = 0 : 0 = dy/dt / (dx/dt) --> dy/dt = 0 cm/second

at x = -1 : -4 = dy/dt / (dx/dt) --> dy/dt = -12 cm/second