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Gradients, Tangents and Normals
Gradients, Tangents and Normals
As mentioned in the Differentiation section, we can find a derivative to give the gradient of a graph at any given point.
From that information, we can create a tangent and a normal.
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View ProductHow to Compare the Tangent and Normal
Let’s denote the gradient of the tangent m_T and gradient of the normal m_N.
For a pair of tangent and normal lines at one point, we have one rule:
The two must be perpendicular.
This means that we must have
m_T\cdot m_N = -1
Example 1: Finding the Tangent
A tangent is a straight line which touches our graph, but doesn’t pass through it at the meeting point. By definition, a straight line graph (i.e. y = x) cannot have a tangent – only a curved graph can have a tangent.
So, as an example, here’s the graph of \textcolor{blue}{y = (x - 1)^2 - 2}. Find the equation of the tangent line at x = \dfrac{3}{2}.
[4 marks]
To find the tangent, we first find the gradient at our point of interest.
Since we have
\textcolor{blue}{y = (x - 1)^2 - 2}
= \textcolor{blue}{x^2 - 2x - 1}
we have a gradient of
\dfrac{dy}{dx} = 2x - 2
at any point, and \dfrac{dy}{dx} = 1 when x = \dfrac{3}{2}.
Since our gradient is a straight line, it must have the general equation
y = mx + c
When x = \dfrac{3}{2}, y = -\dfrac{7}{4} and \dfrac{dy}{dx} = m_T = 1.
Then we can plug in these values to find c:
-\dfrac{7}{4} = \dfrac{3}{2} + c
c = -\dfrac{7}{4} - \dfrac{3}{2} = -\dfrac{13}{4}
Therefore, our tangent to y = (x - 1)^2 - 2 at x = \dfrac{3}{2} is given by the equation \textcolor{red}{y = x - \dfrac{13}{4}}.
Example 2: Finding the Normal
The normal is a straight line which is exactly perpendicular to the tangent.
Say we have the same graph from Example 1, but now we wish to find the normal at the point x = \dfrac{3}{2}.
[3 marks]
Since we already know that the gradient of the tangent, m_T = 1, we can conclude that the gradient of the normal, m_N = -1.
Given, also, that the point of interest is at \left( \dfrac{3}{2}, -\dfrac{7}{4} \right), we can form a straight line equation for the normal:
-\dfrac{7}{4} = \left(-1 \times \dfrac{3}{2} \right) + c
c = -\dfrac{7}{4} + \dfrac{3}{2} = -\dfrac{1}{4}
The equation of the normal is given by \textcolor{limegreen}{y = -x - \dfrac{1}{4}}.
Gradients, Tangents and Normals Example Questions
Question 1: What is the gradient of the tangent and the normal at x = 2 for the equation f(x) = -x^2?
[2 marks]
f(x) = -x^2 gives f'(x) = -2x
When x = 2,
m_T = -4 and m_N = \dfrac{-1}{-4} = \dfrac{1}{4}
Question 2: Find the tangent to the equation y = x^3 at x = 1. Verify, also, that this tangent runs parallel to the tangent at x = -1.
[5 marks]
y = x^3 gives a gradient of
\dfrac{dy}{dx} = 3x^2
When x = 1,
\dfrac{dy}{dx} = 3 and y = 1
Then
1 = (3 \times 1) + c
c = 1 - 3 = -2
The equation of the tangent at x = 1 is y = 3x - 2.
At x = -1, \dfrac{dy}{dx} = 3 \times (-1)^2 = 3.
Therefore, we can see that the gradients of the tangents at x = 1 and x = -1 are the same, so the two must run parallel.
Question 3: Given that the derivative of y = \sqrt{x + 4} is given by \dfrac{1}{2\sqrt{x + 4}}, show that the normal at x = -\dfrac{1}{2} passes through the origin.
[5 marks]
When x = \dfrac{-1}{2}, y = \sqrt{\dfrac{7}{2}}.
We have m_T = \dfrac{1}{2\sqrt{\dfrac{7}{2}}} = \dfrac{1}{\sqrt{14}}. Then m_N = -2\sqrt{\dfrac{7}{2}} = -\sqrt{14}.
y = mx + c gives
\sqrt{\dfrac{7}{2}} = \left( -\sqrt{14} \times \dfrac{-1}{2} \right) + c
\sqrt{\dfrac{7}{2}} = \sqrt{\dfrac{14}{4}} + c
\sqrt{\dfrac{7}{2}} = \sqrt{\dfrac{7}{2}} + c
c = 0
So, the normal intercepts the y-axis at y = 0, or, more appropriately, the origin.
Specification Points Covered
G1 – Understand and use the derivative of f(x) as the gradient of the tangent to the graph of y=f(x) at a general point (x,y); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second derivatives; differentiation from first principles for small positive integer powers of x
G3 – Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, points of inflection
