Equations of Tangents and Normals

1. Defining Tangents and Normals

Before we start calculating, let's be clear on what we're trying to find.

• A tangent is a straight line that just touches a curve at a single point and has the same gradient as the curve at that point.
• A normal is a straight line that passes through the same point but is perpendicular to the tangent.

To find the equation of any straight line, we need two things:

1. A point on the line ($(x_1,y_1)$).
2. The gradient of the line ($m$).

Once we have these, we can use the formula: $y-y_1=m(x-x_1)$.

2. Finding the Equation of a Tangent

Let's break down the process of finding the tangent's equation into a reliable, repeatable method.

The 3-Step Method for Tangents

Given a curve $y=f(x)$ and a point with x-coordinate $x_1$:

1. Find the Gradient (m):

   • First, differentiate the equation of the curve to find the gradient function, $\frac{dy}{dx}$.
   • Then, substitute the given $x$-coordinate ($x_1$) into $\frac{dy}{dx}$ to calculate the numerical gradient of the tangent, $m_{tangent}$.

2. Find the Point ((x₁, y₁)):

   • You have the x-coordinate, $x_1$.
   • To find the y-coordinate, $y_1$, substitute $x_1$ into the original equation of the curve, $y=f(x)$. Be careful not to use the derivative here!

3. Find the Equation:

   • Substitute the gradient $m_{tangent}$ and the point $(x_1,y_1)$ into the straight-line equation $y-y_1=m(x-x_1)$.
   • Rearrange the equation into the form required by the question (usually $y=mx+c$ or $ax+by+c=0$).

Let's watch a video that works through this exact process.

3. Finding the Equation of a Normal

The process for finding the normal is nearly identical, with one extra but crucial step. Since the normal is perpendicular to the tangent, their gradients are related.

Key Rule for Perpendicular Gradients:
If the tangent has gradient $m_{tangent}$, the normal has a gradient $m_{normal}$ which is the negative reciprocal:
$m_{normal}=-\frac{1}{m_{tangent}}$

The 4-Step Method for Normals

1. Find the Tangent Gradient: Differentiate $y=f(x)$ and substitute the $x$-coordinate to find $m_{tangent}$.
2. Find the Normal Gradient: Calculate $m_{normal}=-\frac{1}{m_{tangent}}$.
3. Find the Point ((x₁, y₁)): Substitute the $x$-coordinate into the original equation $y=f(x)$ to find the $y$-coordinate.
4. Find the Equation: Use the point $(x_1,y_1)$ and the normal's gradient, $m_{normal}$, in the equation $y-y_1=m(x-x_1)$.

The following video is an excellent worked example, as it also demonstrates a situation where you are only given the x-coordinate, forcing you to complete Step 3 yourself.

4. Reading and Practice

Now that you've seen the methods in action, let's consolidate your understanding and put it into practice. The following resource provides a clear, written summary of everything we've covered and is an excellent reference for revision.

After reviewing the methods, it's time to apply them. The best way to secure this skill is through practice.

Conclusion

Well done. You've now connected your new differentiation skills to a very practical application in coordinate geometry. This type of multi-step problem, combining different areas of maths, is exactly what you need to master for a high grade.

Key Takeaways:

• The equation of a straight line is found using a point and a gradient.
• To find the tangent's gradient, differentiate the curve's equation and substitute the $x$-coordinate.
• To find the normal's gradient, calculate the negative reciprocal of the tangent's gradient ($m_{normal}=-1/m_{tangent}$).
• To find the y-coordinate of the point, always substitute the $x$-coordinate into the original equation of the curve.
• The formula $y-y_1=m(x-x_1)$ is your most direct tool for building the final equation.

Preview of the Next Lesson:
Today we looked at the gradient at a specific point. But what about points where the gradient is special, for example, where it is exactly zero? These are called turning points (maximums or minimums). In the next lesson, we will use differentiation to locate the coordinates of turning points on a curve and determine their nature. This is another major application of calculus and a high-value skill for your exam.