xplain why your expression for the slope of the tangent line in (ii) does not work when x = ±3,and write down equations of those tangent lines.

To explain why the expression for the slope of the tangent line does not work at $x = \pm 3$ and to write down the equations of those tangent lines, we follow these steps:

1. Break Down the Problem

• Identify the function for which the slope of the tangent line is being considered.
• Determine why the expression for the slope is not valid at $x = \pm 3$.
• Calculate the equations of the tangent lines at these points, if applicable.

2. Relevant Concepts

• The slope of the tangent line at a point on a function $f(x)$ is given by the derivative $f'(a)$, where $a$ is the x-value of interest.
• If the function is not differentiable at a point, the expression for the slope will not yield a valid result.

3. Analysis and Detail

• The expression for the slope of the tangent line may not work at $x = \pm 3$ typically because these points could be vertical tangents, discontinuities, or cusps.
• If the derivative does not exist at these points, we cannot calculate a slope using the standard derivative expression.

Assuming $f(x)$ is a defined function where the slope fails at these points:

• Check $f'(3)$ and $f'(-3)$ to confirm they are undefined.

4. Verify and Summarize

• For $x = 3$ and $x = -3$, if these points lead to undefined slopes, then you're looking for vertical tangents or points where the curve does not have a well-defined direction.
• Thus, we find the equations of the tangent lines, which will usually take the form $x = 3$ and $x = -3$ respectively for vertical lines.

Final Answer

The slopes of the tangent lines do not work at $x = \pm 3$ due to undefined derivatives, typically indicating vertical tangents or cusps. The equations of the tangent lines at these points are:

• For $x = 3$: $x = 3$
• For $x = -3$: $x = -3$