The area of a circle (A) with radius (r) is given by,
A = πr^{2}
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 cm^{2}/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 cm^{2}/s when its radius is 4 cm.
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.
Check the injectivity and surjectivity of the following functions:
(i) f : N → N given by f(x) = x^{2}
(ii) f : Z → Z given by f(x) = x^{2}
(iii) f : R → R given by f(x) = x^{2}
(iv) f : N → N given by f(x) = x^{3}
(v) f : Z → Z given by f(x) = x^{3 }
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.
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
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.