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I am having trouble imagining a scenario where finding the line tangent to a curve at some point useful. What could this line represent?

Also, I know it is convenient to work with algebraic expressions, but I have no idea how to turn a set of data into an algebraic expressions so I can actually use the calculus I learned.

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    $\begingroup$ Predicting growth. Trajectories. Accelerating a vehicle through a curve. The list is endless. $\endgroup$
    – fleablood
    Aug 5 at 16:24
  • $\begingroup$ Data doesn't just "turn... into algebraic expressions" by itself. You formulate a mathematical model that explains the measurements, and then fitting the data to that model is a very big topic in operations research and other mathematical sciences. Too big to go into in any depth here. $\endgroup$
    – hardmath
    Aug 9 at 16:20

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How is finding the tangent line of a curve at a point useful?

Basically, all of the classic physics models that we know exist because of this idea. The way to see how is by understanding the slope of the tangent line as the rate of change. In other words, the slope of the tangent at a point tells you "How fast is this quantity increasing/decreasing" at said point. With this in mind, I'll give you an example of an application I found particularly enlightening.


Newton's second law states that $$ F= ma $$ where $F$ is some force being applied to an object and $a$ is its acceleration due to said force. But what is acceleration? Well, acceleration is just the rate of change of velocity. And what is velocity? Well, velocity is just the rate of change of distance. And since "rate of change" is just the derivative (i.e. the slope of the tangent), if $x$ is the distance then $a = \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\mathrm{d}}{\mathrm{d}t}x\right)$ is just the "rate of change" of the "rate of change" of distance (hence the $2$ derivatives $\frac{\mathrm{d}}{\mathrm{d}t}$ one after the other). In other words, Newton's second law can be re-written as $$ F= m\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\mathrm{d}}{\mathrm{d}t}x\right) $$ Now, just out of curiosity, say you're thinking of a scenario where some object is subject to some constant force $-g m$ (i.e $F=-g m$). Then, in this case, Newton's second is $$ -g = \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\mathrm{d}}{\mathrm{d}t}x\right) $$ You can now ask the question. What function $x(t)$ satisfies the above equation? Well, remembering the standard calculus power rule $\frac{\mathrm{d}}{\mathrm{d}t} t^n = nt^{n-1}$ you can see that choosing $x(t) = x_0 + v_0 t - \frac{g}{2}t^2$ works since: $$ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\mathrm{d}}{\mathrm{d}t}x_0 + v_0 t - \frac{g}{2}t^2\right) = \frac{\mathrm{d}}{\mathrm{d}t}\left(v_0 - gt\right) = -g $$ And what was the point of all this? We'll, $-mg$ is just the weight of some object of mass $m$, so if we trust that Newton's laws work, then $x(t) = x_0 + v_0 t - \frac{g}{2}t^2$ should be the equation of motion for some object moving only by the force of its own weight. And indeed it is, since you may recall that this equation for $x$ is precisely the equation of motion for an object in free fall.

So to recap, just by using the concept of "slope of a tangent line", trusting some old guy's law and applying a couple of calculus rules, we can now predict how any object moves when in free fall! And I bet you can think of a ton of scenarios where this is useful.


Now, this is just the tip of the iceberg. This process of starting with some physics law, and then trying to find the equation of motion for some scenario happens time and time again. Have you heard of Maxwell's equations? They're the backbone of the classic theory of electromagnetism, and just like Newton's second law, they involve derivatives. Schrödinger's equation? Fundamental to quantum mechanics, and is also an equation that involves derivatives just like Newton's second.

Physics theory as we know it doesn't exist without studying the tangent line at a point.


How do I turn data into curves in the first place?

This process is called curve fitting, and has been studied extensively. Some common techniques of curve fitting one might encounter are "linear regression" or "least squares", but there are several more techniques that might be better or worse depending on your data.

Hope this helps!

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It shows the average gradient. This can be used to find certain values depending on what the dependent and independent variables are. With this you can apply basic trigonometry to find even more information concerning the graph. With the gradient you can use y = mx + c which can be useful in answering the question. I hope this helps. An example for turning a dataset of an algebraic equation into a curve graph would be the use of a quadratic equation.

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One use of the tangent line is as an approximation to a function at a specific point.

Suppose, for example, that you have $f(x) = x^{4/x}$, need to find $f(1.01)$, and for some reason don't have a scientific calculator handy. But if you can work out that $f'(x) = 4x^{4/x-2}(1 - \ln(x))$, then the tangent line at $x = 1$ is $\tilde{f}(x) = f(1) + f'(1)(x - 1) = 1 + 4(x - 1) = 4x - 3$, so $f(1.01) \approx \tilde{f}(1.01) = 1.04$. (The true computer-calculated answer is $1.0401940173555868$.)

How do I turn data into curves in the first place?

As a layperson, the easiest way is to use Excel (or equivalent spreadsheet program), create a scatterplot (X-Y) chart of your data, add a trendline, and check the “Display equation on chart” checkbox.

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