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
The area of a circle (A) with radius (r) is given by,
A = πr2
Therefore, the rate of change of the area with respect to its radius r is
\begin{align}\frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r\end{align}
∴When r = 6 cm,
\begin{align}\frac{dA}{dr} = 2\pi \times 6 =12 \pi\; cm^2/s\end{align}
Hence, the required rate of change of the area of a circle is 12π cm2/s.
The correct answer is B.
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.
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y = Ax : xy' = y (x ≠ 0)
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\begin{align}\left(\frac{d^2y}{dx^2}\right)^2\;+\;cos\left(\frac{dy}{dx}\right)\;=\;0\end{align}
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\begin{align}\left(\frac{d^2y}{dx^2}\right)^3\;+ \left(\frac{dy}{dx}\right)^2+\;sin\left(\frac{dy}{dx}\right)\;+ 1=\;0\end{align}
is (A) 3 (B) 2 (C) 1 (D) not defined
How the units of rate of change of area with respect for cm2 /sec