PURE MATHEMATICS - Differential Calculus

 

Tangents & Normals

 

 

Tangents

Normals

Equation Tangent

Equation Normal

 

 

 

Tangents

 

The gradient of the tangent to the curve y = f(x) at the point (x1, y1) on the curve is given by:

the value of dy/dx, when x = x1 and y = y1

 

 

tangent and normal

 

 

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Normals

 

Two lines of gradients m 1, m 2 respectively are perpendicular to eachother if the product,

 

 

 

 

 

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Equation of a Tangent

 

The equation of a tangent is found using the equation for a straight line of gradient m, passing through the point (x1, y1)

 

 

 

 

To obtain the equation we substitute in the values for x1 and y1 and m (dy/dx) and rearrange to make y the subject.

 

 

 

Example

 

Find the equation of the tangent to the curve y = 2x2 at the point (1,2).

 

 

 

 

 

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Equation of a Normal

 

The equation of a normal is found in the same way as the tangent.

 

The gradient(m2) of the normal is calculated from;

 

 

 

(where m1 is the gradient of the tangent)

 

so,

 

 

 

Example

 

Find the equation of the normal to the curve:

 

  y = x2 + 4x + 3,   at the point (-1,0).

 

 

 

 

 

 

 

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