# Linear approximations and tangent planes

0/2

##### Intros

###### Lessons

**Linear Approximations & Tangent Planes Overview:**__Tangent Planes__- A review of linear approximation
- 2D = tangent lines, 3D = tangent planes
- $z=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$
- An example

__Linear Approximation with Multi-Variable Functions__- Estimating a value of a 3D surface
- Approximation for points "near" $(x_0,y_0)$,
- An example

0/0

##### Examples

###### Free to Join!

StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun — with achievements, customizable avatars, and awards to keep you motivated.

#### Easily See Your Progress

We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.#### Make Use of Our Learning Aids

#### Earn Achievements as You Learn

Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.#### Create and Customize Your Avatar

Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.

###### Topic Notes

__Notes:__

__Tangent Planes__Recall that in Calc I, linear approximation is about finding a linear equation tangent to a curve at a point, and using it to estimate values of the curve "near" that point. The equation of the tangent line at point $a$ would be:

Calc III is similar, but now it is in 3D. Instead of 2D curves, we have 3D surfaces. Instead of tangent lines, we have tangent planes. The formula for the tangent plane at point $(x_0, y_0)$is:

$z=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$

__Linear Approximation__Just like how we can estimate values of a 2D curve, can also estimate the value of a 3D surface near a point using linear approximation. We say that if we are at the point $(x_0,y_0)$, and we want to approximate a point near it (say $(x,y)$), then

$f(x,y) \approx f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$

2

videos

remaining today

remaining today

5

practice questions

remaining today

remaining today