Math 203 Lecture 13 - Tangent planes, Linear approximation, Differentials and another series corner

In this lecture, we begin by going over some homework problems regarding limits and continuity, and then we moved on to the next topic--tangent planes.

Tangent planes to surfaces are the multidimensional equivalent to tangent lines to curves in 2D--and they are useful in a similar fashion. We can use tangent lines to approximate surfaces locally. We derive the formula to obtain a tangent plane to a surface (partial derivatives come in handy here), and use it to find a formula for a linear approximation (called a linearization) to he surface. This is analogous to linear approximation in single variable calculus.

We also talk about differentials, an idea very similar to linear approximation and we use it to solve problems involving errors in measurements.

We end the lecture with another series corner. We discuss the divergence test for series. We say what it can and cannot tell us, and do a few examples.

Next time, we will go over the multivariable chain rule. This is important in itself, but it is also important for streamlining implicit differentiation, which we shall also do.

[ravibajaj]

Thanks professor, these videos are amazing!!

[manuelsanchez]

Question, in the note from your class you used
 |f(x, y) - L| =
so I used
|x^2y^3/2x^2+y^2 - 1|

<|( x^2y^3)/(2x^2) - 1|

=|y/3 - 1|

=|0 -1|

|1| = 1


or was I just suppose to plug in the limit of the origin to the equation?


this is all @12:17

[manuelsanchez]

Q: Can I use Fx(X-Xo)+Fy(Y-Yo)+Fz(Z-Zo)=0 to fine linear approximation and to find the tangent plane?

[UnforsakenXII]

That's odd. I thought series was a calc II subject. Well, I guess it can't hurt to review.