# Tangent and normal equation

Tangent a straight line that touches a curve at one single point. At that point, the gradient of the curve is equal to the gradient of the tangent.

Normal a straight line that is perpendicular to the tangent line:

For any questions with tangent and/or normal lines, use the steps described in the following example.

Finding the linear function of the tangent.

Let f (x) = x3. Find the equation of the tangent at x = 2
1. Find the derivative and fill in value of x to determine slope of tangent

f′(x) = 3x2

f′(2) = 3 · 22 = 12

2. Determine the y value

f(x) = 23 = 8

3. Plug the slope m and the y value in

y = mx + c

8 = 12x + c

4. Fill in the value for x to find c

8 = 12(2) + c ⇒ c =−16

eq. of tangent: y = 12x − 16

Note: Steps 1, 2 and 4 are identical for the equation of the tangent and normal

Finding the linear function of the normal.

Let f(x) = x3. Find the equation of the normal at x = 2

1. Find the derivative and fill in value of x to determine slope of tangent

f ′(2) = 12

2. Determine the y value

f(x) = 8

3. Determine the slope of the normal

and plug it and the y-value into y = mx + c

4. Fill in the value for x to find c

Note: Steps 1, 2 and 4 are identical for the equation of the tangent and normal

To find the gradient of a function for any value of x.

f(x) = 5x3 − 2x2 + x. Find the gradient of f(x) at x = 3.

In this case, f′(3) = 124