The curve $y=5-\mathrm{e}^{x}$ has a tangent at $A$, where the $x$-coordinate of $A$ is \(\l

Question

The curve $y=5-\mathrm{e}^{x}$ has a tangent at $A$, where the $x$-coordinate of $A$ is $\ln2$. The tangent meets the $y$-axis at $B$, and the curve meets the $x$-axis at $C$. Find the equation of the tangent at $A$.[3] Find the coordinates of $B$ and $C$.[3] Hence find the exact area of the shaded region bounded by the curve, the tangent and the coordinate axes.[5]

Accepted Answer

(i) $y=-2x+3+2\ln2$ (ii) $B=(0,3+2\ln2)$, $C=(\ln5,0)$ (iii) $(\ln2)^2+5\ln5-2\ln2-3\text{ units}^2$

Exponential curve with a tangent and the coordinate axes

Exponential curve with a tangent and the coordinate axes

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