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.
Determine order and degree(if defined) of differential equation (ym)2 + (yn)3 + (y')4 + y5 =0
The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.
Show that the Signum Function f : R → R, given by
is neither one-one nor onto.
Let f : N → N be defined by
State whether the function f is bijective. Justify your answer.
Classify the following measures as scalars and vectors.
(i) 10 kg (ii) 2 metres north-west (iii) 40°
(iv) 40 watt (v) 10–19 coulomb (vi) 20 m/s2
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?
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
Let f, g and h be functions from R to R. Show that
(f + g)oh = foh + goh
(f . g)oh = (foh) . (goh)