12 Below is a faulty argument that appears to show that the gradient of the curve \(y = x ^ { 2 }\) at the point \(( 3,9 )\) is 1 .
Consider the chord joining \(( 3,9 )\) to the point \(\left( 3 + h , ( 3 + h ) ^ { 2 } \right)\)
The gradient is \(\frac { ( 3 + h ) ^ { 2 } - 9 } { h } = \frac { 6 h + h ^ { 2 } } { h }\)
When \(h = 0\) the gradient is \(\frac { 0 } { 0 }\) so the gradient of the curve is 1
- Identify a fault in the argument.
- Write a valid first principles argument leading to the correct value for the gradient at (3, 9).
- Find the equation of the normal to the curve at the point ( 3,9 ).