For the curve y=5x-2x^3,if x increases at the | vedclass

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(C) Given the curve $y = 5x - 2x^3$. The slope of the curve is given by $m = \frac{dy}{dx} = 5 - 6x^2$. We need to find the rate of change of the slop

Vedclass · Accepted Answer

$-72$

Vedclass · Answer

(C) Given the curve $y = 5x - 2x^3$. The slope of the curve is given by $m = \frac{dy}{dx} = 5 - 6x^2$. We need to find the rate of change of the slope with respect to time $t$,which is $\frac{dm}{dt}$. Differentiating $m$ with respect to $t$ using the chain rule: $\frac{dm}{dt} = \frac{d}{dt}(5 - 6x^2) = -12x \cdot \frac{dx}{dt}$. Given that $\frac{dx}{dt} = 2 	ext{ units/sec}$ and we need to evaluate this at $x = 3$: $\left(\frac{dm}{dt}ight)_{x=3} = -12(3) 	imes 2 = -72$. Thus,the rate of change of the slope at $x = 3$ is $-72 	ext{ units/sec}^2$.

For the curve $y=5x-2x^3$,if $x$ increases at the rate of $2 \text{ units/sec}$,the rate of change in the slope of the curve at $x=3$ is ......../sec

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