Tangents and normals : Cartesian type

Common Question Formats

Drawing the tangent and normal lines to a curve at a given point on the Cartesian plane.
Using the derivative to find the equation of a tangent or normal at a specific point.
Determining the intersection point of tangents and normals with curves or axes.
The application of the gradient-perpendicular relationship between tangents and normals.
Finding the equation of the normal or tangent with the given slope or a given point.

Strategies for Addressing Questions

To find the tangent or normal line, first differentiate the equation to find the gradient at a particular point.
The gradient of a curve at a given point is the same as the gradient of the tangent at that point. Use this alongside y = mx + c to find the equation of the tangent.
To find the equation of the normal, use the fact that the gradient of the normal is the negative reciprocal of the gradient of the tangent (or the curve at that point). This stems from the fact that normals are perpendicular to tangents.
If an equation does not simplify easily, look for visual cues from graphed curves.
Substitution will often be required, whether to find a specific point on the curve or to simplify the process of finding the equation of a tangent or normal.

Potential Pitfalls

Forgetting to substitute the x-value into the derivative to find the gradient of the tangent at a particular point. This gradient is not equivalent to the derivative itself.
Neglecting the relationship between the gradients of perpendicular lines. If m1 is the gradient of one line, the gradient of the line perpendicular to it would be -1/m1.
When finding the equation of the tangent or normal, always check if it’s written in y = mx + c form. If not, you may need to rearrange.
Not noting the distinction between normals and tangents. They are distinct line types with unique properties, but they are closely related through their perpendicularity.
Failing to check solutions by verifying if the tangent or normal does indeed intersect the curve at the correct point and with the correct orientation (in the case of tangents touching the curve and normals being perpendicular at the point of intersection).