\begin{align} y= \sqrt{1+x^2} : y^{'}=\frac{xy}{1+x^2}\end{align}
\begin{align} y= \sqrt{1+x^2}\end{align}
Differentiating both sides of the equation with respect to x, we get:
\begin{align} y^{'}=\frac{d}{dx}\left(\sqrt{1+x^2} \right)\end{align}
\begin{align} y^{'}=\frac{1}{2\sqrt{1+x^2}}\frac{d}{dx}\left(1+x^2\right)\end{align}
\begin{align} y^{'}=\frac{2x}{2\sqrt{1+x^2}}\end{align}
\begin{align} y^{'}=\frac{x}{\sqrt{1+x^2}}\end{align}
\begin{align}\Rightarrow y^{'}=\frac{x}{\sqrt{1+x^2}}\frac{\sqrt{1+x^2}}{\sqrt{1+x^2}}\end{align}
\begin{align}\Rightarrow y^{'}=\frac{x}{1+x^2}{\sqrt{1+x^2}}\end{align}
\begin{align}\Rightarrow y^{'}=\frac{x}{1+x^2}{y}\end{align}
\begin{align}\Rightarrow y^{'}=\frac{xy}{1+x^2}\end{align}
∴ L.H.S. = R.H.S.
Hence, the given function is the solution of the corresponding differential equation.
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 Signum Function f : R → R, given by
is neither one-one nor onto.
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.
Find the direction cosines of a line which makes equal angles with the coordinate axes.