Tangents and Normals to Curves (AS)

Understand the concept of a tangent and a normal to a curve: The tangent of a curve at a particular point is the line that just grazes the curve at that point, while the normal is the line perpendicular to the tangent at the point of contact.
Familiarise yourself with the equation of a tangent and a normal: The equation of the tangent can be found by using the derivative to find the slope at a particular point. The equation can then be written as y - y1 = m(x - x1), where (x1,y1) is the point of tangency and m is the slope found. The normal’s equation can be found in a similar way but the slope is the negative reciprocal of the tangent’s slope.

Know how to calculate the angle between tangents and normals: Use the formula tan(θ) =

(m1 - m2)/(1 + m1*m2)

where m1 and m2 are the slopes of the two lines.

Practice finding the equation of the tangent and normal at a given point using differentiation: This will involve finding the derivative of the curve’s equation, substituting in the x value of the point of interest, and then using this as the slope in the equation of a line.
Work consistently with differentiating implicit functions: You may need to use implicit differentiation if you have a curve defined by an equation in x and y, rather than y in terms of x.
Remember to check the direction of the tangent or normal: For instance, a positive slope indicates an upward direction while a negative slope indicates a downward direction.
Be aware of horizontal and vertical tangents/normals: A horizontal line will have a gradient of 0, whilst a vertical line will have an undefined gradient.
Recall the product and quotient rules for differentiation: These rules are necessary for finding the derivative of more complex functions, allowing you to find the slope of the tangent and consequently the normal.
Master the basics of trigonometric functions, polynomial functions, and exponential functions: Differentiation of these functions is often involved in finding the equations of tangents and normals.
Understand the concept of a point of inflexion: A point of inflexion is a point on the curve where the curve changes concavity. At a point of inflexion, the tangent crosses the curve.