Question

For the given curve: y = 5x – 2x3, when x increases at the rate of 2 units/sec, then how fast is the slope of curve changes when x = 3?

Answer 1

Solution:

The given curve is $\sf\color{orange}y = 5x – 2x3$

∴ $\frac{d y}{d x}$ = 5 – 6x2

$\sf\color{red}i.e$. m = 5 – 6x2,

$\sf\color{aqua} \:where\: ‘m’\: is \:the\: slope$

∴ $\frac{d m}{d t}$=$—12x\frac{d x}{d t} =-12x(2) = -24x$

∴ $\left.\frac{d m}{d t}\right]_{x=3}$= -24(3) = -72.

$\sf\color{darkviolet}Hence, \:the \:rate\: of \:the\: change\: of \:the\: slope = -72.$

$\underline{\bf{Note :-}}$

the slope of the curve is decreasing at the rate of 72 units per second when x is increasing at the rate of 2 units per second.

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Answer 2

Given that, y = 5x – 2x3

Then, the slope of the curve, dy/dx = 5-6x2

⇒d/dt [dy/dx]= -12x. dx/dt

= -12(3)(2)

= -72 units per second

Hence, the slope of the curve is decreasing at the rate of 72 units per second when x is increasing at the rate of 2 units per second.

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