- A tangent is a line that just touches the curve but doesn’t go through it.
The gradient of a tangent = Gradient of a curve at that point
- A normal is a straight line perpendicular (at right angle 90°) to a curve.
We know that differentiation is the process that we use to find the gradient of a point on the curve. However, we can also find the gradient of a curve at a given point by drawing a tangent at that point (which is a straight line) and finding the gradient of that line which is equal to the gradient/derivative of the curve as well.
What is a Tangent?
As shown in Fig 1, a tangent just touches the curve but doesn’t go through it. The line intersects the curve at a point and the gradient of the line is equal to the derivative of the curve at the point.
Q. Find the equation of tangent to the curve y = (6 – x)(x + 3) at the point (3, 5)
Write the equation in the form that we can differentiate:
Now differentiate it:
The gradient of the tangent is equal to gradient of the curve at point (3, 5):
Hence m(gradient) = -3 as well.
Now to find the equation of the tangent:
Point is (3, 5) and m = -3
equation of tangent at point (3, 5)
What is a Normal?
As shown in Fig 2, A normal is a straight line perpendicular (at right angle 90°) to a curve or to the tangent of the curve at that point.
If two lines, with gradients m1 and m2 are at right angles then .
Therefore if the gradient of the tangent is m1 than the gradient of the normal is
Q. Find the equation of the normal to the curve at the point where x = 2.
Firstly find the y coordinate at x=2:
Now differentiate the equation of the curve to find the gradient of the tangent at:
Hence m1 = 8
Equation of normal is where Point is (2, 2) and .
equation of normal at point (2, 2)
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