Determine the value of dy/dt at x=3 when y=2x^2-5x and dx/dt=2.
1 Answer
embizze | High School Teacher | (Level 2) Educator Emeritus
Determine `(dy)/(dt)` at x=3 if `y=2x^2-5x` and `(dx)/(dt)=2` .
Differentiate `y=2x^2-5x` with respect to `t` using implicit differentiation.
`(dy)/(dt)=4x(dx)/(dt)-5(dx)/(dt)` Here we can apply the general power rule: if `u` is a differentiable function of `t` then `d/(dt) u^n=n*u^(n-1)u'`
Substitute for the known values of x and `(dx)/(dt)` to get:
`(dy)/(dt)=4(3)(2)-5(2)=24-10=14`
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The derivative of y with respect to t of `y=2x^2-5x` at x=3 with `(dx)/(dt)=2` is 14.
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This result can be interpreted as follows: let p be a point on the parabola given by `y=2x^2-5x` . The point is moving at 2 units per unit time with respect to the x-axis. We are asked to find the rate of change with respect to the y-axis when x=3. The result shows that the change in y is 14 units per unit time when x=3.