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.
Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.
(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)
Let f, g and h be functions from R to R. Show that
(f + g)oh = foh + goh
(f . g)oh = (foh) . (goh)
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.
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.
An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?
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
If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.
The order of the differential equation
\begin{align}2x^2\frac{d^2y}{dx^2}\;- \;3\frac{dy}{dx}\;+ y=\;0\end{align}
is (A) 2 (B) 1 (C) 0 (D) not defined