1.23: The equation of a curve is When is increased to , increases to where and are small.

4PM1/1/June/2023 — Question 7 · 10 marks

The equation of a curve is y=e4x2x−3\displaystyle y = \sqrt{\frac{\mathrm{e}^{4x}}{2x-3}}
When x\displaystyle x is increased to (x+δx)\displaystyle (x + \delta x), y\displaystyle y increases to (y+δy)\displaystyle (y + \delta y) where δx\displaystyle \delta x and δy\displaystyle \delta y are small.
(a) Show that δy≈e2x(4x−7)(2x−3)32δx\displaystyle \delta y \approx \frac{\mathrm{e}^{2x}(4x-7)}{(2x-3)^{\frac{3}{2}}} \delta x
(7)
Given that x=2.5\displaystyle x = 2.5
(b) find an estimate, to 2 significant figures, of the value of δy\displaystyle \delta y when the value of x\displaystyle x increases by 0.2%
(3)