To determine how the slope of the curve changes when
x increases at the rate of 3 units
/ sec, we need to first find the expression for the slope of the curve. The slope of the curve given by
y=2x−x2 is found by differentiating
y with respect to
x. So, let's find the derivative of
y :
‌=‌(2x−x2)=2−2xThe slope of the curve at any point is given by
2−2x. Next, we need to determine how this slope changes with respect to time when
x is increasing at the rate of 3 units
/ sec. To do this, we need to find the derivative of the slope with respect to time. We use the chain rule for this purpose:
‌(‌)=‌(2−2x)⋅‌ We already have
‌=2−2x. Differentiating this expression with respect to
x gives:
‌(2−2x)=−2 We are given that
‌=3 units
∕ sec. Substituting this value into our expression gives:
‌(‌)=−2⋅3=−6‌ units ‌/ secThus, the slope of the curve is decreasing at a rate of 6 units
/ sec. Therefore, the correct option is:
Option C: Decreasing at 6 units
∕ sec