If tangent to the curve y² + 3x – 7 = 0 at point (h, k) is parallel to line x – y = 4, then value of k is ___________

OR

For the curve y = 5x – 2x³, if x increases at the rate of 2units/sec, then at x = 3 the slope of the curve is changing at ___________

Given: Tangent to the curve y² + 3x – 7 = 0 at point (h, k) is parallel to line x – y = 4

To Find: Value of k

Differentiating the curve with respect to x we get,

⇒

⇒

At point (h, k),

⇒

And the line is parallel to the line x – y = 4

⇒ Slope of the line is = 1

So, equating both the slopes we get,

⇒

OR

Given: y = 5x – 2x³ and x increases at the rate of 2units/sec

To Find: Rate of change of curve at x = 3.

Slope of the curve is,

Rate of change of slope is =

Rate of change of x is = units/sec

So, at x = 3, the rate of change of slope is (-12 x 3 x 2) = -72 units/sec.

That means the slope is decreasing at the rate of 72 units/sec.