AASL 5.4 Tangents and normals

In this lesson, the process of finding the equations of the tangent and normal lines to a curve is explained, emphasizing the connection between tangents and derivatives. The tangent line touches the curve at a single point without crossing it, while the normal line is perpendicular to the tangent.

To find these equations, the following steps are outlined:

• Identify a point on the curve, which is determined by substituting the given x-coordinate into the function.
• Calculate the gradient using the derivative of the function, where the derivative at a point gives the slope of the tangent line.
• Use the formula $y - y_1 = m(x - x_1)$ to derive the equations for both the tangent and normal lines, noting that the gradient of the normal is the negative reciprocal of the tangent's gradient.