Practical Reason for Finding the Tangent Line Equation

[rayroshi]

I just have a very general question. I know how to find the equation for a line tangent to a curve, given a point; however, I now wonder if anyone could give me a real practical reason for wanting to do so, other than to answer questions on an exam or in a calculus text; i.e., what would be a real-life, practical example of when someone would need to do that?

So often, calculus instruction leaves out such interesting information, which makes the whole process dry and uninteresting. It reminds me of the oft-heard question that teachers must deal with: "Why do we have to know this stuff? I'll never, ever use it in my life." lol!

Since I am am old man, and not in any calculus class, I would like to know the answer strictly out of interest's sake, so my motive is, hopefully, 'purer' than that of a bored high school math student. I just want to know, because it's interesting.

 

[Steven G]

I studied pure mathematics. Pure mathematics is studied for the beauty of it with no interest in whether or not it can be applied. I am far from a mathematician but the little math I have seen can be beautiful, as much as a piece of art. Knowing calculus allows one to learn higher mathematics and really start to see the beauty in math. I studied a lot of Algebra in college and I remember being in class where our professors showed us true beauty that is rarely seen by the average person. This is one reason to study Calculus.

On the other hand Calculus was invented to further the study of Physics. To send someone to the moon requires the knowledge of the relationship between displacement, velocity and acceleration, which is derivatives. To know the force required to move something also requires calculus. The number of applications goes on and on.

Knowing where a function has a peak or a valley (where the derivative is 0) can be extremely valuable in business applications. After all you want to maximize your profit while minimizing your costs.

I hope others will come along and add much more than I can.

 

[pka]

I just have a very general question. I know how to find the equation for a line tangent to a curve, given a point; however, I now wonder if anyone could give me a real practical reason for wanting to do so, other than to answer questions on an exam or in a calculus text; i.e., what would be a real-life, practical example of when someone would need to do that?
So often, calculus instruction leaves out such interesting information, which makes the whole process dry and uninteresting. It reminds me of the oft-heard question that teachers must deal with: "Why do we have to know this stuff? I'll never, ever use it in my life." lol!
Since I am am old man, and not in any calculus class, I would like to know the answer strictly out of interest's sake, so my motive is, hopefully, 'purer' than that of a bored high school math student. I just want to know, because it's interesting.

I have a suggestion for you: Calculus the language of change by Keith Stroyan.
I began teaching calculus in high school in 1964 and retired as chair of a division of mathematical sciences in a regional university in 2005. In that time, I made it a point to attend as many conferences on Calculus Teaching as available. I also was involved with accreditation committees. On the latter visit it was made clear to me by both faculty in Economics or Pre-Med that the ideas on instance change (the derivative) is so important. Frankly this may show my own ignorance but I never knew that. So I got to know the work of mathematicians like Stoyan.

 

[Otis]

… a real practical reason for wanting to [find a tangent-line equation] …

Hi Ray. NASA engineers use tangent-line equations to ensure their $40 million probe travels in the required direction, after leaving earth orbit. Engineers who design roller coasters or water slides use tangent lines at crucial points, to determine forces (strength and direction) that risk accelerating people off the ride. Software designers use tangent-line equations to plot the trajectory of photon torpedos fired from a ship moving along a curve, in video game or movie special-effects. Automakers use tangent lines to splice together pieces of polynomial curves, to model curved surfaces for testing aerodynamic properties. Potentially, tangent line equations could arise while investigating any scenario involving change.

?

 

[rayroshi]

Thank you all for your responses. It seems that one can say the benefits of knowing tangent-line equations can range all the way from a purely intellectually aesthetic reason to the very practical applications of NASA engineers, roller coaster designers, and software programmers, as Otis stated. It was this latter category of practical application that most interested me. Thank you all again!

 

[yoscar04]

I think the first example I ever encountered was the definition of instantaneous velocity as the slope of the tangent of the position at a certain time t:
v(t)=dx(t)/dt. From Arnold: Methods of Classical Mechanics.

 

[HallsofIvy]

The key word here is "linearization". "Non-linear" problems are, in general, very difficult! If it is possible to replace the non-linear problem with a linear approximation, then it become relatively easy. Of course, a linear approximation to a non-linear function is going to be accurate only over a short interval. And the graph of the best linear approximation to f(x) in a short interval about \(\displaystyle (x_0, f(x_0))\) is the tangent line to the graph of y= f(x) at that point.