3.49: Exponential curve calculus; AP sums

0606/23/O/N/22 — Question 9 · 14 marks

3.49 diagram
(a)
The equation of a curve is y=kxe−2xy=kx\mathrm{e}^{-2x}, where kk is a constant. Find dydx\dfrac{\mathrm{d}y}{\mathrm{d}x}.
[2]
(b)
Find the coordinates of the stationary point on the curve y=10xe−2xy=10x\mathrm{e}^{-2x}.
[3]
(c)
Use your answer to part (a) to find ∫4xe−2x dx\displaystyle\int 4x\mathrm{e}^{-2x}\,\mathrm{d}x.
[3]
(d)
Find the exact value of ∫014xe−2x dx\displaystyle\int_{0}^{1}4x\mathrm{e}^{-2x}\,\mathrm{d}x.
[2]
(a)
An AP has third term 1010 and S8=116S_{8}=116. Find aa and dd.
[5]
(b)
Find the sum of nineteen terms starting with the twelfth.
[4]