Tangents, Normals and Rate of Change
The derivative at a point is the slope of the tangent there, the normal is perpendicular to it, and derivatives with respect to time give related rates.
What is Tangents, Normals and Rate of Change?
The derivative at a point is the slope of the tangent there, the normal is perpendicular to it, and derivatives with respect to time give related rates.
Key formula / rule: Tangent slope is f'(a); normal slope is -1/f'(a) when f'(a) is non-zero.
Key points
- Derive the equations of the tangent and normal at a given point.
- Apply related rates to a geometric configuration changing in time.
Common exam trap
Using the point of tangency without checking it lies on the curve.
Definitions
- Term
Tangents, Normals and Rate of Change
- Meaning
The derivative at a point is the slope of the tangent there, the normal is perpendicular to it, and derivatives with respect to time give related rates.
- Term
Tangents, Normals and Rate of Change — explanation
- Meaning
Interpreting the derivative geometrically as a slope and physically as a rate covers most application questions. Related-rates problems are built by differentiating a geometric relation with respect to time.
Learning objectives
Derive the equations of the tangent and normal at a given point.
Apply related rates to a geometric configuration changing in time.
Formulae
- Key point
Tangent slope is f'(a); normal slope is -1/f'(a) when f'(a) is non-zero.
- Key point
Related rates differentiate a relation with respect to time using the chain rule.
- Key point
A horizontal tangent means f'(a) = 0 and a vertical tangent means f'(a) does not exist.
Prerequisites
MAT-U2-LC-T2-S1-C1
Common mistakes
Using the point of tangency without checking it lies on the curve.
Substituting numerical values before differentiating in a related-rates problem.
Keywords
Tangents
Normals
Rate
Change
Practice preview
Find the point on the curve y = x^2 - 4x + 3 where the tangent is parallel to the line y = 2x + 5.…
medium
Find the angle of intersection between the curves y = x^2 and x = y^2 at the point (1, 1).…
hard
A man 2 m tall walks at a uniform speed of 5 km/h away from a lamp post 6 m high. Find the rate at which the length of his shadow is increasing.…
hard