The area of a circle (A) with radius (r) is given by,
A = πr2
Now, the rate of change of the area with respect to its radius is given by,
\begin{align} \frac{dA}{dr} = \frac{d}{dr}(πr^2) = 2πr \end{align}
\begin{align} \frac{dA}{dr} = 2π (3) = 6π \end{align}
Hence, the area of the circle is changing at the rate of 6π cm2/s when its radius is 3 cm.
\begin{align} \frac{dA}{dr} = 2π (4) = 8π \end{align}
Hence, the area of the circle is changing at the rate of 8π cm2/s when its radius is 4 cm.
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
(A) 10π (B) 12π (C) 8π (D) 11π
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2