The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.
The area of a circle (A) with radius (r) is given by,
A = πr2
Now, the rate of change of area (A) with respect to time (t) is given by,
\begin{align} \frac{dA}{dt}=\frac{d}{dt}(\pi r^2).\frac{dr}{dt}=2\pi r\frac{dr}{dt}\;\;\;[By\; Chain \;Rule]\end{align}
It is given that,
\begin{align} \frac{dr}{dt}= 3\; cm/s\end{align}
\begin{align} \therefore \frac{dA}{dt}= 2\pi r(3)=6 \pi r \end{align}
Thus, when r = 10 cm,
\begin{align} \frac{dA}{dt}= 6\pi(10)=60 \pi\; cm^2/s \end{align}
Hence, the rate at which the area of the circle is increasing when the radius is 10 cm is 60π cm2/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.
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
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 total revenue in Rupees received from the sale of x units of a product is given by
R (x) = 13x2 + 26x + 15
Find the marginal revenue when x = 7.
y = Ax : xy' = y (x ≠ 0)