Review question

# Can we find a tangent to $y=1/(1+x^2)$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9085

## Question

Give a sketch of the curve $y=\dfrac{1}{1+x^2}$, for $x \geq 0$.

Find the equation of the line that intersects the curve at $x=0$ and is tangent to the curve at some point with $x>0$. Prove that there are no further intersections between the line and the curve. Draw the line on your sketch.

By considering the area under the curve for $0\le x\le 1$, show that $\pi>3$.

Show also, by considering the volume formed by rotating the curve about the $y$ axis, that $\ln 2>2/3$.

$\left[\text{Note:} \displaystyle\int_0^1\dfrac{1}{1+x^2} \:dx = \dfrac{\pi}{4}.\right]$