The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
The circumference of a circle (C) with radius (r) is given by
C = 2πr.
Therefore, the rate of change of circumference (C) with respect to time (t) is given by,
\begin{align} \frac{dC}{dt}=\frac{dC}{dr}.\frac{dr}{dt}\;\;\; [By\; Chain\; Rule]\end{align}
\begin{align} =\frac{d}{dr}(2\pi r).\frac{dr}{dt}\end{align}
\begin{align} =2\pi.\frac{dr}{dt}\end{align}
It is given that
\begin{align} \frac{dr}{dt}=0.7\; cm/s\end{align}
Hence, the rate of increase of the circumference 2π(0.7)=1.4π cm/s
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.
Let f : N → N be defined by
State whether the function f is bijective. Justify your answer.
Show that the Signum Function f : R → R, given by
is neither one-one nor onto.
The vertices of ΔABC are A (3, 5, −4), B (−1, 1, 2), and C (−5, −5, −2).
A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
Determine order and degree(if defined) of differential equation ym + 2yn + y' =0
Determine order and degree(if defined) of differential equation yn + 2y' + siny = 0