Question

Let f(x) = 9x^2- 6x.(a) Use the limit process or derivative to find the slope of the line tangent to the graph of f at x = 3

Answer 1

Use the limit derivative to find slope:

`f(x)=9x^2-6x`

The limit definition of a derivative is given by:

`f(x)=\lim _(h\to0)(f(x+h)-f(x))/(h)`

Here, the derivative is:

`\begin{gathered} f^1(x)=\lim _(h\to0)(9(x+h)^2-6(x+h)-(9x^2-6x))/(h) \\ =\lim _(h\to0)(9(x^2+2xh+h^2)-6x-6h-9x^2+6x)/(h) \\ =\lim _(h\to0)(9x^2+18xh+9h^2-6x-6h-9x^2+6x)/(h) \\ =\lim _(h\to0)(18xh+9h^2-6h)/(h) \\ =\lim _(h\to0)(18xh)/(h)+(9h^2)/(h)-(6h)/(h) \\ =18x+\lim _(h\to0)(9)/(h)-6 \\ =18x-6 \end{gathered}`
`f^1(x)=18x-6`

We have now found the gradient of the tangent at any point. All that's left is to evaluate it at x = 3

`\begin{gathered} f^1(x)=18x-6 \\ y=18(3)-6 \\ y=54-6 \\ y=48 \end{gathered}`

Hence the slope of the equation = 48