Question

At what angle does y(1+x) :x cut x-axis

Answer 1

Answer:

Find the angle between tang... maths ... (1) cuts the axis of x at (−1,0) and (3.0). These points are obtained by putting y=0 in the equation (1). Now the ... This will help us to improve better.

Answer 2

Concept:

A curve could be a shape or a line which is smoothly drawn in an exceedingly plane having a bent or turns in it. for instance, a circle is an example of curved-shape.

Given:

we are providing the curve $y(1+x)=x$ cuts the $x-$ axis.

Find:

We have to search out the angle of the curve.

Solution:

The given curve is $y(1+x)=x$.

Firstly, we are going to separate the variables, we get

$y=\frac{x}{1+x}$

Now, we'll differentiate either side with respect to $x$, we get

$\frac{dy}{dx}=\frac{d\left(\frac{x}{1+x}\right)}{dx}$

Further, we'll use the quotient rule of differentiation that's $\frac{d\left(\frac{u}{v}\right)}{dx}=\frac{v \frac{du}{dx}-u\frac{dv}{dx}}{v^2}$, we get

$\begin{aligned}\frac{dy}{dx}&=\frac{(1+x)\frac{dx}{dx}-x\frac{1+x}{dx}}{(1+x)^2}\\ &=\frac{1+x-x}{(1+x)^2}\\ &=\frac{1}{(1+x)^2}\end$                  .......(1)

Furthermore, it's only if the curve cut the $x-$ axis.

So, we substitute $x=0$ in equation (1), we get

$\begin{aligned}\frac{dy}{dx}&=\frac{1}{1+0^2}\\ &=1\end$

So, slope is $m=\frac{dy}{dx}=1$.

As we all know that, $m=\tan \theta=1$

Now, we are going to find the angle by using that $\tan 45^{\circ}=1$, we get

$\begin{aligned}\tan \theta&=\tan 45^{\circ}\\ \theta&=45^{\circ}\\ \theta&=\frac{\pi}{4}\end$

Hence, the angle is $45^{\circ}$ when the curve $y(1+x)=x$ cuts the $x-$axis.

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