1.12: Tangents to exponential curves

4PM1/2/November/2020 — Question 8 · 15 marks

The curve C1\displaystyle C_1 has equation
y=5e−2x+4.y=5\mathrm{e}^{-2x}+4.
The curve C2\displaystyle C_2 has equation
y=e2x.y=\mathrm{e}^{2x}.
The curves C1\displaystyle C_1 and C2\displaystyle C_2 intersect at the point A\displaystyle A.
(a) Find the exact coordinates of A\displaystyle A.
(4)
The tangent at A\displaystyle A to C1\displaystyle C_1 intersects the x\displaystyle x-axis at the point B\displaystyle B.
(b) Show that the x\displaystyle x coordinate of B\displaystyle B is
12(5+ln 5).\frac12(5+\mathrm{ln}\,5).
(5)
The tangent at A\displaystyle A to C2\displaystyle C_2 intersects the x\displaystyle x-axis at the point D\displaystyle D.
(c) Find the area of △ABD\displaystyle \triangle ABD.
(6)