Long Questions (Question 2 & 3)

Question 2:
Given the equation of a curve is:
y = x² (x – 3) + 1
(a) Find the gradient of the curve at the point where x = –1.
(b) Find the coordinates of the turning points.

Solution:
(a)
$\begin{array}{l}y=x^2\left(x-3\right)+1\\ y=x^3-3x^2+1\\ \frac{dy}{dx}=3x^2-6x\\ \text{When }x=-1\\ \frac{dy}{dx}=3{\left(-1\right)}^2-6\left(-1\right)\\ =9\\ \text{Gradient of the curve is 9}\text{.}\end{array}$

(b)
At turning points, $\frac{dy}{dx}=0$
3x² – 6x = 0
x² – 2x = 0
x (x – 2) = 0
x = 0, 2

y = x² (x – 3) + 1
When x = 0, y = 1
When x = 2,
y = 2² (2 – 3) + 1
y = 4 (–1) + 1 = –3
Therefore, coordinates of the turning points are (0, 1) and (2, –3).

Question 3:
It is given the equation of the curve is y = 2x (1 – x)⁴ and the curve pass through T(2, 4).
Find
(a) the gradient of the curve at point T.
(b) the equation of the normal to the curve at point T.

Solution:
(a)
$\begin{array}{l}y=2x{\left(1-x\right)}^4\\ \frac{dy}{dx}=2x\times 4{\left(1-x\right)}^3\left(-1\right)+{\left(1-x\right)}^4\times 2\\ =-8x{\left(1-x\right)}^3+2{\left(1-x\right)}^4\\ \\ \text{At }T\left(2,4\right),x=2.\\ \frac{dy}{dx}=-16\left(-1\right)+2\left(1\right)\\ =16+2\\ =18\end{array}$

(b)
$\begin{array}{l}\text{Equation of normal:}\\ y-y_1=-\frac{1}{\frac{dy}{dx}}\left(x-x_1\right)\\ y-4=-\frac{1}{18}\left(x-2\right)\\ 18y-72=-x+2\\ x+18y=74\end{array}$