The anti derivative of sin 2x – 4e3x is the function of x whose derivative is sin 2x – 4e3x.
It is known that,
\begin{align} \frac {d}{dx} \left(-\frac{1}{2}cos 2x – \frac {4}{3} e^{3x}\right) = sin2x – 4e^{3x} \end{align}
Therefore, the anti derivative of (sin 2x – 4e3x) is \begin{align} \left(-\frac{1}{2}cos 2x – \frac {4}{3} e^{3x}\right) \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.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
Determine order and degree(if defined) of differential equation \begin{align}\frac{d^2y}{dx^2}=\cos3x + sin3x\end{align}
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