The intercepts on x-axis made by tangents to the curve, y = _0^x |t|,dt, x R, which are pa…

Question

The intercepts on $x$-axis made by tangents to the curve, $y = \displaystyle\int_0^x |t|\,dt$, $x \in R$, which are parallel to the line $y=2x$, are equal to:

Accepted Answer

Since, $$y = \int\limits_0^x {\left| t \right|} dt,x \in R$$

therefore $${{dy} \over {dx}} = \left| x \right|$$

But from $$y = 2x,{{dy} \over {dx}} = 2$$

$$ \Rightarrow \left| x \right| = 2 \Rightarrow x =  \pm 2$$

Points $$y = \int\limits_0^{ \pm 2} {\left| t \right|dt}  =  \pm 2$$

$$\therefore$$ equation of tangent is

$$y - 2 = 2\left( {x - 2} \right)$$ or $$y + 2 = 2\left( {x + 2} \right)$$

$$ \Rightarrow $$ $x$-intercept $$ =  \pm 1.$$

Solutions · 1