The anti derivative of sin 2x is a function of x whose derivative is sin 2x.
It is known that,
\begin{align} \frac {d}{dx} (cos 2x) = 2 sin2x \end{align}
⇒ \begin{align} sin 2x =-\frac {1}{2} \frac {d}{dx}(cos 2x) \end{align}
∴ \begin{align} sin 2x = \frac {d}{dx}\left(-\frac {1}{2}cos 2x\right) \end{align}
Therefore, the anti derivative of sin2x is
\begin{align} sin 2x \;is -\frac {1}{2}cos 2x \end{align}
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.
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 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)
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.