1. Local Linear Approximation
Math 200
Week 6 - Friday
Local Linear Approximation

2. Math 200
Goals
Be able to compute the local linear approximation for a function of two or more variables at a given point.
Be able to use a local linear approximation to estimate a given quantity.

3. Math 200
From Calc 1
In Calc 1 we discussed the fact that differentiable functions are locally linear
That is, near the point of tangency, a function is approximately equal to its tangent line
Another way of saying all this is that if I keep zooming in on a differentiable function at a point, it will eventually look flat.

4. Zoom in Say we want to approximate sin(π/15)
Math 200
Say we want to approximate sin(π/15)
We can find the tangent line to f(x) = sin(x) at x=0, which is close to π/15
f’(x) = cos(x)
f’(0) = 1
The tangent to f at x=0 is y=x (call this L(x) = x)
sin(π/15) is approximately equal to L(π/15) = π/15
Zoom in

5. Math 200
Summary of Calc 1 stuff
Local Linear Approximation for single variable functions says that a differentiable function can be approximated by its tangent line
For a differentiable function f(x), the local linear approximation at x = x0 is given by
L(x) = f(x0) + f’(x0)(x - x0)
Remember: Don’t think of this a formula to be memorized; this is just the tangent line to f at x0!

6. Math 200
New stuff
If a single-variable, differentiable function can be approximated by its tangent line near the point of tangency, then a multi- variable function, f(x,y), can be approximated by its tangent plane near the point of tangency

7. Revisiting Tangent planes for functions of two variables
Math 200
Revisiting Tangent planes for functions of two variables
Consider any function of two variables, f(x,y).
To find the tangent plane at (x0,y0), we should treat the surface z = f(x,y) as a level surface of some function of three variables:
z = f(x,y) can be written as 0 = f(x,y) - z
F(x,y,z) = f(x,y) - z
Notice that Fx = fx, Fy = fy, and Fz = -1
So,
And this will be the case for any function of two variables!

8. The only thing we’ll do differently now is rename z
Math 200
We can use this to write a general formula for the tangent plane to f(x,y) at (x0,y0):
Solve for z:
Since z0 = f(x0,y0),
The only thing we’ll do differently now is rename z
z = L(x,y)

9. Don’t memorize, understand
Math 200
Don’t memorize, understand
Now, we have this formula for the local linear approximation of a function f(x,y) at (x0,y0):
But, it’s most important to remember that we approximate functions of two variables with tangent planes
And we know that the normal vector for a tangent plane comes from the gradient
You should be able to derive this formula if you forget it

10. Math 200
Example 1
Consider the function f(x,y) = eysin(x). Use local linear approximation to approximate the value of f(0.1,0.1)
We can evaluate the function f at (0,0), which is close to (0.1,0.1), so we’ll pick that as the point of tangency.
Following our newfound formula, we need f(0,0), fx(0,0), and fy(0,0)
fx(x,y) = eycos(x); fx(0,0) = 1
fy(x,y) = eysin(x); fy(0,0) = 0
f(0,0) = 0

11. Putting it all together… fx(x,y) = eycos(x); fx(0,0) = 1
Math 200
Putting it all together…
fx(x,y) = eycos(x); fx(0,0) = 1
fy(x,y) = eysin(x); fy(0,0) = 0
f(0,0) = 0
So basically 0.1
So…
Wolfram vs. Us

12. So, the z-value at (0.1,0.1) for the surface, is really close to the z-value at (0.1,0.1) on the plane

13. Math 200
Example 2
Find the local linear approximation, L(x,y), for

14. Math 200