3.1. Step-by-Step Process

1. Find the point on the curve: Calculate 𝑦 = 𝑓(𝑥𝟏). So the point is 𝑃(𝑥𝟏,𝑦𝟏).

To start finding the equations of the tangent and normal, the first step is to identify the point on the curve where you want to find these lines. You do this by substituting the x-value 𝑥𝟏 into the function 𝑓(x) to get the corresponding y-value 𝑦 = 𝑓(𝑥𝟏). This gives you a specific point P on the curve, represented as P(𝑥𝟏, 𝑦𝟏).

Imagine you're hiking a curved path in the mountains. To describe your position on that path, you'd need to note your current coordinates (x, y) at a specific point, just like you find the point on the curve here.

2. Find the derivative: Calculate $\frac{dy}{dx}$ for the function $y = f(x)$.

Next, you'll find the derivative of the function $y = f(x)$. The derivative represents the rate at which y changes with respect to x and gives you the slope of the tangent line at any point on the curve. This calculation is crucial as it will help you determine the slope at the specific point you found in the previous step.

Think of the derivative as the speedometer in your car – it tells you how fast you are going at any moment. Just like you’d want to know your speed at a certain point on your journey, here, we compute the derivative to know how steeply the curve rises or falls at our particular point.

3. Calculate slope of tangent: Evaluate the derivative at $x = x_1$ to find $m = \left. \frac{dy}{dx} \right|_{x=x_1}$.

Once you have the derivative, you'll evaluate it at the specific x-value 𝑥𝟏 to find the slope of the tangent line at point P. This slope is denoted as m_tangent and indicates how steep the line will be as it touches the curve at that point.

Imagine you're standing on a hill and looking at the steepness right where you are. This is like calculating the slope of the tangent – it tells you how steep the climb is exactly where you’re standing.

4. Write the equation of the tangent: Use point-slope form: $y - y_1 = m_{tangent}(x - x_1)$.

Now that you have the slope of the tangent and the point P(𝑥𝟏, 𝑦𝟏), you can write the equation of the tangent line using the point-slope equation. This form allows you to create a linear equation that represents the tangent at that point.

It's like taking a pencil and sketching a straight line that just grazes a curve at a specific point. The equation you derive is the mathematical language of your sketch, describing how that line behaves.

5. Calculate slope of normal: $m_{normal} = -\frac{1}{m_{tangent}}$.

After you're done with the tangent, the next step is to find the slope of the normal line. Since the normal is perpendicular to the tangent, its slope is the negative reciprocal of the tangent line’s slope. This means if your tangent slope is m_tangent, the normal slope (m_normal) is calculated as -1 divided by m_tangent.

Think about how a ladder stands against a wall. If you want to describe the straight line going up the ladder (the tangent), you also want to know about the line going directly away from the wall (the normal). The relationship between them involves this concept of perpendicular slopes.

6. Write the equation of the normal: $y - y_1 = m_{normal}(x - x_1)$.

Finally, write the equation for the normal line using the same point-slope form. You’ll substitute the point P(𝑥𝟏, 𝑦𝟏) and the slope m_normal that you calculated earlier. This equation represents the line that runs perpendicular to your tangent line at point P.

This step is like displaying another set of instructions for climbing down the ladder that stands straight against the wall: you have your paths defined, not just for going along the curve but also for stepping away from it at right angles.

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