The anti derivative of cos 3x is a function of x whose derivative is cos 3x.
It is known that,
\begin{align} \frac {d}{dx} (sin 3x) = 3 cos3x \end{align}
⇒ \begin{align} cos 3x =\frac {1}{3} \frac {d}{dx}(sin 3x) \end{align}
∴ \begin{align} cos 3x = \frac {d}{dx}\left(\frac {1}{3}sin 3x\right) \end{align}
Therefore, the anti derivative of cos3x is
\begin{align} sin 2x \;is \frac {1}{3}sin 3x \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.
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
y = cosx + C : y' + sinx = 0
Find gof and fog, if
(i) f(x) = | x | and g(x) = | 5x – 2 |
(ii) f(x) = 8x3 and g(x) = x1/3 .
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?