Question

If $m$ is the slope of a tangent to the curve $e^y=1+x^2$, then

• A. $\left|m\right|>1$
• B. $m>1$
• C. $m>-1$
• D. $\left|m\right|\le 1$

Answer 1

Answer: (D)

$\left|m\right|\le 1$

$e^y=1+x^2$

$\frac{dy}{dx}=\frac{2x}{1+x^2}$

So $|m|=|\frac{2x}{x^2+1}|$

Now, since $x^2\ge 0$

$\Rightarrow x^2+1\ge 1$

$\Rightarrow \frac{1}{x^2+1}\le 1$

$\Rightarrow \frac{2x}{x^2+1}\le 1$

$|m|\le 1$