3. Finding Tangents and Normals to Curves

Calculate 𝑦 = 𝑓(𝑥1). So the point is 𝑃(𝑥1, 𝑦1).

In the first step of finding the tangent and normal lines to a curve, you need to determine the coordinates of the point on the curve where you want to find these lines. Here, you substitute your chosen value of 𝑥 into the function 𝑓(𝑥) to calculate the corresponding 𝑦-coordinate. This gives you the point 𝑃(𝑥1, 𝑦1) which is essential as it will be used to write the equations of the tangent and normal.

Imagine you are on a journey and you want to plot a point on the map at a specific location. This step is like determining your exact coordinates on the map (latitude and longitude) based on your starting point (𝑥) and your route (𝑓(𝑥)).

Calculate 𝑑𝑦/𝑑𝑥 for the function 𝑦 = 𝑓(𝑥).

The second step involves calculating the derivative of the function, denoted as 𝑑𝑦/𝑑𝑥. The derivative represents the slope of the curve at any point and is critical in determining the slope of the tangent line. By knowing the function, you can differentiate it to find the rate at which 𝑦 changes with respect to 𝑥.

Think of the derivative as the speedometer in a car. Just as the speedometer tells you how fast you are driving at a certain moment, the derivative tells you how steep the curve is at a specific point.

Evaluate the derivative at 𝑥 = 𝑥1 to find 𝑚_tangent = |𝑑𝑦/𝑑𝑥| at 𝑥 = 𝑥1.

In this step, you substitute our specific 𝑥-coordinate (𝑥1) into the derivative you calculated earlier to evaluate the slope at that point. This slope, denoted as 𝑚_tangent, tells you how steep the tangent line will be at point 𝑃.

Imagine standing on a hill; the slope of the hill at your feet is similar to what you're finding here. Knowing how steep the hill is at your position (the slope) will help you understand how to walk flat or how to build something there.

Use point-slope form: 𝑦 − 𝑦1 = 𝑚_tangent (𝑥 − 𝑥1).

The tangent line's equation is generated using the point-slope format, which takes into account the slope calculated in the previous step and the coordinates of point 𝑃. Plugging 𝑚_tangent, 𝑦1, and 𝑥1 into the formula provides you with the equation of the tangent line at that specified point.

It’s like composing a song where you have a melody (the slope) and notes (the point) together to create harmony. You use both to express a particular sound (the equation of the tangent) that represents that moment perfectly.

Since the normal is perpendicular to the tangent: 𝑚_normal = −1/𝑚_tangent.

The normal line at a point on the curve runs perpendicular to the tangent line. To find its slope (𝑚_normal), you take the negative reciprocal of the slope of the tangent (𝑚_tangent). This relationship is essential for determining how the normal line behaves in relation to the tangent.

Think of the normal line like the balance beam in gymnastics; if one side is sloped down, the other must slope up at a right angle, showing how things can balance out in opposite directions.

Use point-slope form: 𝑦 − 𝑦1 = 𝑚_normal (𝑥 − 𝑥1).

Finally, similar to finding the tangent line, you write the equation for the normal line at point 𝑃. You use the point-slope form again, this time substituting in the slope of the normal (𝑚_normal) alongside the point’s coordinates to create the equation of the normal line.

This step is akin to drawing a straight line on a piece of graph paper; just like connecting two dots with a ruler to get a straight line, here you’re utilizing mathematical points and slopes to craft another line (the normal) that has a distinctly different angle.

--