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Peter Barendse

Understanding Tangents and Normals with Wolfram|Alpha

January 12, 2012 —
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Finding the tangents and normals of a mathematical function or relation is one of the most common exercises in any calculus course. In this post, I’ll show you the newest functionality in Wolfram|Alpha for discovering and investigating them.

The simplest example of a tangent is the “tangent line” to a one-dimensional curve in the plane. Graphically, the tangent line is a line that “just touches” the curve at some point, so that if it were moved just slightly, this one point of contact would become two.

If you ask Wolfram|Alpha for the tangent line of a specific function and point, it gives it in both graphical and algebraic/numerical form:

show me the tangent line to y = x^2 at x = pi

Notice that near the point of tangency, the line and the curve are nearly identical; the line does almost exactly what the curve does near that point. This is exactly why tangents are important: it’s often much easier to answer questions about linear functions, and tangents provide a way of approximating a complicated relationship with a linear relationship (a line or a plane). It’s also why the tangent is called the “linearization” or “linear approximation” of the relation.

For example, suppose you need to know the square root of 3.9. A very rough estimate would be 2, since 3.9 is roughly 4, and 2 = sqrt(4). Even better would be to linearize the function y = sqrt(x) at the point x = 4 and use this to find an estimate:

linearize y = sqrt[x] at x = 4

This says that the linear approximation to sqrt(x) at x = 4 is x/4 + 1. Plugging x = 3.9 into this approximation yields 3.9/4 + 1 = 1.975, which is pretty close to the actual value of sqrt(3.9), an irrational number whose first six digits are 1.97484.

You can also ask Wolfram|Alpha for the slope of a tangent line to a function at a point, which is another common calculus question (it’s actually equal to the derivative of the function at that point).

slope of the tangent line to x^2 sin(x) at -1

What about higher-dimensional functions? A function or relation with two degrees of freedom is visualized as a surface in space, the tangent to which is a plane that just touches the surface at a single point. For example, here’s the tangent plane to z = sin[xy] at x = 1, y = .9, as displayed by Wolfram|Alpha:

tangent plane to z = sin[xy] at x = 1, y = .9

The “normal” to a curve or surface is a kind of the complement of the tangent. The “normal line” to a one-dimensional curve is perpendicular to the tangent line and goes through the same point on the curve:

normal line to x^2+y^2 = 4 at (x,y) = (sqrt[2], sqrt[2])

The normal to a surface in space is also a line. It is the unique line that is perpendicular to the tangent plane at that point:

normal line to cos[x] + cos[y] - 1 at x = -0.6, y = -2.4

Tangents and normals to higher-dimensional surfaces exist as well. Of course, no plot is possible, but Wolfram|Alpha will give you algebraic and numerical representations of the tangent and normal to any multidimensional surface and any point:

tangent to z = x^2 + sin[x + y + w] at x = 1, y = 2, w = 0

normal to z = x^2 + sin[x + y] - w at x = 1, y = 2, w = 0

I hope that this new functionality will be used by both students to check their understanding and by professionals to solve practical problems. Enjoy!


It might be more useful when showing normals and tangent planes to show the 3D plots with the same scale for x, y, and z. With the squashed plot shown above, it is difficult to see that the normal line is really perpendicular to the surface.

Posted by George Woodrow III January 12, 2012 at 3:56 pm

    Thanks for the feedback George! We’ll be sure to pass along the message to our developers.

    Posted by The Wolfram|Alpha Team January 13, 2012 at 9:24 am

Very interesting but disturbing also. Why? The idea of a tangent is intuitively a stretch already from its derivation in the physical world . Extending it further, as given in this example, is mathematically rigorous,(and very pretty), but leading in a direction which is worrisome to me personally. Very few people on the Earth understand what’s going on with some of the more simple mathematical expressions/operations but everybody thinks they can use these edifices anyway. As a result, absolute foolishness is circulating in the physical sciences using these poorly understood conventions as justification. Remedy: Be careful of what you use to prove physical theories. Rely on the experts that specialize in them or experts that invented them,(like S. Wolfram for example).

Posted by Charles Fewell January 12, 2012 at 11:37 pm